LECTURES ON CHEVALLEY GROUPS
Robert S t e i n b e r g
Yale U n i v e r s i t y
Notes prepared by
John Faulkner and Robert Wilson
i
DEPARTMENT OF MATHEMATICS
LECTURES ON CHEVALLEY GROUPS
Robert S t e i n b e r g
Yale U n i v e r s i t y
1967
Notes p r e p a r e d by
John F a u l k n e r and Robert Wilson
T h i s work w a s p a r t i a l l y s u p p o r t e d by C o n t r a c t ARO-D-3368230-31-43033.
!\!p
/'
[
$
,
Acknowledgement
1
.
-7
These n o t e s a r e d e d i c a t e d t o my w i f e , Maria.
They
might a l s o have been d e d i c a t e d t o t h e Yale Mathematics D e p a r t ment t y p i n g s t a f f wliose u n i f o r m l y h i g h s t a n d a r d s w i l l become
a p p a r e n t t o anyone r e a d i n g t h e n o t e s .
Needless t o s a y , I
am g r e a t l y i n d e b t e d t o John F a u l k n e r and Robert Wilson f o r
w r i t i n g u p a major p a r t of t h e n o t e s .
F i n a l l y , it i s a
p l e a s u r e t o acknowledge t h e g r e a t s t i m u l u s d e r i v e d f r o m my
c l a s s and from many c o l l e a g u e s , t o o many t o mention by name,
d u r i n g my s t a y a t Yale.
Robert S t e i n b e r g
J u n e , 1968
Guide to t h e Reader
T h e s e n o t e s p r e s u p p o s e t h e t h e o r y of complex
semisimple L i e a l g e b r a s through t h e c l a s s i f i c a t i o n , a s may be f o u n d , f o r example, i n t h e b o o k s
o f E. B, Dynkin, M, J a c o b s o n , o r J.-P.
i n t h e n o t e s of
kinai ire
Sophus L i e .
Serre, or
An appen-
d i x d e a l i n g w i t h t h e most f r e q u e n t l y needed
r e s u l t s about f i n i t e r e f l e c t i o n groups and r o o t
systems h a s been included.
The r e a d e r i s
advised t o read t h i s p a r t first.
can d o r a t h e r q u i c k l y .
This he
a
P"?
TABLE OF CONTENTS
%
5
5
5
Page No.
1
l6 A b a s i s f o r
?k
2.
Abasisfor
3.
The Chevalley g r o u p s
4.
S i m p l i c i t y of
%
5.
Chevalley g r o u p s and a l g e b r a i c g r o u p s
8
3
3
6.
G e n e r a t o r s and r e l a t i o n s
7.
Central extensions
8.
V a r i a n t s of t h e Bruhat lemma
G
@
3 9,
3
5
The o r d e r s of t h e f i n i t e Chevalley g r o u p s
10
Isomorphisms and automorphisms
11.
Some t w i s t e d g r o u p s
9
S 1 2
Representations
%
13.
Representations continued
%
14.
R e p r e s e n t a t i o n s concluded
Appendix on f i n i t e r e f l e c t i o n groups
L e c t u r e s on Chevalley Groups
51.
A b a s i s f o r %We s t a r t w i t h some b a s i c p r o p e r t i e s of semisimple L i e
$?,
algebras over
throughout.
and e s t a b l i s h some n o t a t i o n t o be used
The a s s e r t i o n s n o t proved h e r e a r e proved i n t h e
s t a n d a r d books on L i e a l g e b r a s , e . g . ,
t h o s e of Dynkin, Jacpbson
o r Sophus L i e ( ~ G m i n a i r e ) .
Let
.
a C a r t a n s u b a l g e b r a of
and
=
= % f o x
afo
[X e x 1 [H,X]
The
=
0
if
x,
called roots.
Write
by t h e r o o t s .
%" a s a v e c t o r space o v e r
v f o r % Q" ' t h e v e c t o r s p a c e
Then
dim
Q
V
= 4
for a l l
y(H)
y, 6 e V
.
H e
-
Then
.
Let
y e V
over
.
%.
Define
say.
The
generated
Q
Since t h e
such t h a t
H~ e
Y
f
?
( y , F ) = ( H ,Hz)
Y
for
T h i s i s a symmetric, n o n d e g e n e r a t e , p o s i t f v e
d e f i n i t e b i l i n e a r form on
V
.
Denote t h e c o l l e c t i o n of a l l r o o t s by
s u b s e t of t h e nonzero e l e m e n t s of
C
,
L
K i l l i n g form i s nondegenerate t h e r e e x i s t s an
(0)
0
We adopt
i s not a root.
y
%=
Note t h a t
he r a n k of
roots generate
all
%
Y
c Xu, Z@I
c
=
and
n e c e s s a r i l y Abelian
.
H e%
a s s a r e l i n e a r f u n c t i o n s on
,
and
a e
for a l l
x,,
%i s
Then
where
a
= a(H)X
t h e convention t h a t
(H,H;)
Q
be a semisimple L i e a l g e b r a o v e r
generates
V
V
C
.
Then
C
satisfying:
a s a v e c t o r s p a c e over
Q
.
is a
-a
(1) a e x = +
+
#-
an i n t e g e r
(2)
e C
7
<
=
a , @>
ka 4 C
for
k
.
1
2 ( a , p ) , / ( p , ~ )e
(VJrite
and
.
a , f3 e C
for a l l
2(a,f3)/(p,p)
.
These a r e c a l l e d
Cartan i n t e g e r s ) .
( 3 ) C i s i n v a r i a n t under a l l r e f l e c t i o n s
wa(o
wa
C) (where
E.
is the reflection in the
wav = v
Thus
2(v,a)/(a,a) a)
.
i s a r o o t system i n t h e s e n s e of Appendix I.
C
Conversely, if
then
-
, i.e. ,
a
hyperplane orthogonal t o
i s any r o o t system s a t i s f y i n g c o n d i t i o n ( 2 1 ,
C
i s t h e r o o t system of some L i e a l g e b r a .
C
g e n e r a t e d by a l l
wa
i s a f i n i t e group
(Appendix 1.6) c a l l e d t h e Weyl group.
If
lal,
The group
W
a s i m p l e system of r o o t s (Appendix I . 8 ) , t h e n
by t h e
wa
= 1
i
i
..
, n)
Lemma 1:. F c r e a c h r o o t
v
= o(H)
H 1. = H
for all
(i= 1
"i
by
w.H
1
0
j
= H
v
j
Hi
wi f o r
Write
-
c.
,
H e
..., 4,) .
combination of t h e
Proof:
W
is
i s generated
and e v e r y r o o t i s
W t o a s i m p l e r o o t (Appendix 1 . 1 5 ) .
congruent under
(H,H,)
(Appendix 1.16)
...., a & ]
let
0
Ha e
%
3 . .Define
Then each
Ha
be such t h a t
0
Ha = 2 / ( o , a ) Ha
and
i s an i n t e g r a l l i n e a r
.
e W
w,
<a.,ai>
J
i0
Hi
.
.
Define a n a c t i o n of
W
on
Then
Then s i n c e t h e
generate
wi
Hi
combination of t h e
Ha = Hwa
of t h e , Hi
.
a
j
j
+ p # 0
= O
N a , ~
If
if
a
define
c
+'p
and
i s t h e sequence
wH
w
E
j
W
is an i n t e g r a l l i n e a r
.
The
Xa
a
is an
.
?
a
j
choose
Xae
a , X a # O D
N
a,B
IXa ,Xpl = Na,@
i s not a r o o t .
are roots the
p - ra,
... ,
p,
Set
a - s t r i n g of r o o t s through
... , (3
+
a r e not roots.
Lemma 2 :
Now i f
f o r some w e W and some j
j
= WH = a n i n t e g r a l l i n e a r combination
= wHa
For e v e r y r o o t
If
for all
,
a = wa
arbitrary root then
Then
W
can be chosen so t h a t :
q ~ where
p
+
ia
p
(a)
[Xc7X-al
(b)
If
c;
-
P
=
Ho
and
p
.
are roots,
... , p , ... P
ra,
p
r o o t s through
+
qa
fl {
-+ a ,
is the
and ; :
.
.
a - s t r i n g of
.
N~
- q(r+l)lc+~1~/1~1~
a,?
then
-
m o o f : , See t h e f i r s t p a r t of t h e proof of Theorem 1 0 , p. 147 i n
,,
. . .,.. , .
.
a , 0 and
If
Lernma.3:
Proof:
-
P
-
wa
ra
are roots, then. q(r+l)/a+8(2/~b
We u s e two f a c t s :
maps
-
2(p
-
r
(::)
(For
+p
a
-
(::::%)
p
-
>
q = < j3,a
ra
to
+
j3
ra,a)/(a,a)a =
.
qa
so
@
+
qa = w , ( p
-
ra) =
.
p - < @ , a> c + r a )
p
I n t h e a - s t r i n g of r o o t s through
at
most two r o o t l e n g t h s occur.
p
and
in the
vq
P
C
=C
nV ,
then C
i s a rocit 'system and e v e r y r o o t
a - s t r i n g of r o o t s t h r o u g h
3!
X
belongs t o
P
.
i s two d i m e n s i o n a l ; so a system of simple r o o t s f o r
under t h e Weyl group of
the
P
C
P
t o a simple r o o t ,
a - s t r i n g of r o o t s t h r o u g h
p
C
?
Now
C
0
and hence
h a s a t most two r o o t l e n g t h s )
/
We must show t h a t
q l a + p 1 2 / / S 1 2= r
=<
..
>+
P,C
>
= (< p,a
Set
=<
A
We must show
If
(a[
- lpl
>
I<
21 ( f 3 , a ) l / l g 1 2=
<
@ , a>
if
a , @>
.
a = kp
#
B = 0
I<
then
.
a , $ 21
Since
a
Now by ( * ) :
-
1- q(ulL/[plL
q < a , >
~
..
+ l ) ( l- q l a 1 2 / ~ p 1 2 )
..
.
P,a
>(
<
=21i(B,a)[/(a12
By Schwarzvs i n e q u a l i t y
2
( a , ~ /) l a 1
=4
.
l$l <_
p
and
4 w i t h e q u a l i t y i f and o n l y
a r e r o o t s and
G
#+
we
$
> < 4 , Then s i n c e
I< @ , a > I < I < a , P >I we have < @ , a > = - 1, 0 , o r 1 . If
< p,a > = - 1 t h e n A = 0 , If < @ , a > 2 0 t h e n
I@+ 2al > I p + a1 > I @ 1 . S i n c e t h e r e a r e o n l y two r o o t l e n g t h s
+ 2a i s n o t a r o o t and hence q = 1 . S i n c e If! + a ( > la1
and I @ + a 1 > I D ( and a t most two r o o t l e n g t h s o c c u r la1 = 1 0 1
have
;
<
or
A =0
1
B = l- q[a12/lel2
1 and
$,a>+
+
Hence
a
k @ so
$,a > < a , @
lai
< 1@I ,
then
r o o t l e n g t h s would o c c u r ) .
Then
I@ -
a1
>
A s above
<
a , @>
< a , @>
q =- <
-
1, 0 ,
=
,
.
B =0
If
<
@ , a>
=<
>
<
+ p ( <_ 1 p l ( s i n c e o t h e r w i s e t h r e e
Hence ( a , @ )< 0 s o < a,g > < 0
(a
so
la1
-
P
@,a> <
4 and
.
Hence
or
1
p,a
>/<
a,P
>
.
=
a
i s n o t a r o o t and hence
I<
a,p
<
a,P
2
>( < (< P,a >I
> = - 1 . Then
lpl //at2
.
Hence
so
by ('::)
B =0 ,
r = O .
6
We c o l l e c t t h e s e r e s u l t s i n :
:
The
Hi
(i=l,2 ,
chosen a s i n Lemma 1 t o g e t h
A*.,
i n Lemma 2 form a b a s i s f o r
Z? r e l a t i v e
t
e q u a t i o n s of s t r u c t u r e a r e a s f o l l o w s ( a n d , i n p a r t i c u l
(a)
[ H i , ~"i l
=
0
(b)
IHi,Xal
=
<
(c)
[X,,X-,I
(a
[X,,XBl= -I-
a , a i >Xa
= H =
a
('+1IX
if
Proof:
(a)
CH~,X,]
=
the
Xa
and t h e
Lemma 2 ( b )
a+p
holds since
a(HB)X,
=
Hi
<
a n i n t e g r a l l i n e a r combination
a+P
a + 13
if
#
is a root.
a+$
0 and
is abelian.
a+p
(b)
i s not a root.
holds since
a , @ >X a
( c ) f o l l o w s from t h e c h o i c e of
and from Lemma 1. ( d ) f o l l o w s from
and Lemma 3.'
.
(e)
[xa,yB]
0
holds since
=
if
i s not a root.
s i s i s c a l l e d a Chevalley b a s i s .
unique up t o s i g n changes and automorphisms of
and
isomorphic t o
s t 2 (2x2
Ha
x.
It i s
span a 3 - d i m e n s i l n a l s u b a l g e b r a
m a t r i c e s of t r a c e
0).
1
For
i , 1
,
...,
diag(al,
( i ,j ) :
the roots.
Let
p s i t i o n and
Exercise:
A
A s a n example l e t
( c)
,Z+1
,
i
...,
Ei
j
,
->
s&,,,
defdne
ai
-
a
j
.
a = a ( i , j ) by
.
Then t h e
be t h e m a t r i x u n i t w i t h
,j
elsewhere.
0
#
=
II
(Q,cp)
(1)
(i,j)
If o n l y one r o o t l e n g t h o c c u r s t h e n a l l c o e f f i c i e n t s
t i v e algebra over
I
1 in the
are
Then
L e t & be a L i e a l g e b r a over a f i e l d
couple
a ( i ,j )
k
.
We s a y t h a t
cg :
k
and
->
i s a homomorphism
such t h a t :
% i s an a s s o c i a t i v e
algebra with
Yua n a s s o c i a -
1.
§
A basis for
Let
a
be a L i e a l g e b r a over a f i e l d
t i v e algebra over
k
.
We s a y t h a t
cp :
->
and
k
%,
.a
an associa-
i s a homomorphism
if:
(1) 9 i s l i n e a r .
(2) VCX,YI= cp(~)w(Y) - cp(Y)cp(X) f o r a l l
X, Y
A u n i v e r s a l e n v e l o p i n g a l g e b r a of a L i e a l g e b r a
couple
(%,cp)
(1)
(2)
such t h a t :
% i s an a s s o c i a t i v e
algebra with
i s a homomorphism of
into%
1.
.
E:
2
;e
is a
I
(a()
( 3 ) If
i
i s a n y o t h e r such c o u p l e t h e n t h e r e e x i s t s
a u n i q u e homomorphism
and
:a->
a
6
such t h a t
63 o
rn =
@1=1.
( 3 ,9 ) s e e ,
For t h e e x i s t e n c e and u n i q u e n e s s of
e.g.,
Jacobson,
Lie A l g e b r a s .
-c.
~ i r k h o f f - W i t t Theorem:
and
be a Lie algebra over a f i e l d
i t s u n i v e r s a l enveloping algebra.
(?A! ,rp)
(a) q
(b)
Let
k
Then :
is injective.
If
i s i d e n t i f i e d w i t h i t s image i n
X1,X2,
..., Xr
monomials
(where t h e
ki
,
is a linear basis f o r
kl k2
kr
X1
X2 . , . X r
and if
form a b a s i s f o r
the
%
a r e nonnegative i n t e g e r s ) .
The proof h e r e t o o can b e found i n Jacobson.
Theorem 2 :
Assme t h e b a s i s elements
iHi,Xa]
i n Theorem 1 and a r e a r r a n g e d i n some o r d e r ,
numbers
in
?!,
,
n i , ma
of a l l
2+
(::)
E.
(i= 1 , 2 ,
m
and
.,
xZ/ma?
; a e )
of
F o r e a c h c h o i c e of
form t h e p r o d u c t ,
according t o t h e given order,
The r e s u l t i n g c o l i e c t i o n i s a b a s i s f o r t h e z - a l g e b r a
g e n e r a t e d by a l l
Remark:
Theorem.
~:/m!
(m e
z+; a
are as
e E)
The c o l l e c t i o n i s a c - b a s i s f o r
.
2z
by t h e Birkhoff-Witt
The proof of Theorem 2 w i l l depend on a sequence of lemmas.
Lemma 4:
@
Every polynomial o v e r
in
.t
variables
... H*
H1,
which t a k e s on i n t e g r a l v a l u e s a t a l l i n t e g r a l v a l u e s of t h e
v a r i a b l e s i s a n i n t e g r a l combination of t h e polynomials
fi
i =1
in
(")
Hi
Proof:
-
ni <
- d e g r e e of t h e polynomial
( a n d c o n v e r s e l y , of c o u r s e ) .
Let
be such a polynomial.
f
r
f = Z f
j=o j
(y')
We r e p l a c e
HL
r
Ziand
ni e
where
,
by
t i m e s we g e t
fr
each
We may w r i t e
b e i n g a polynomial i n
fj
...,
H1,
HL
+
.
Assuming t h e lemma t r u e f o r polynomials i n
1 and t a k e t h e d i f f e r e n c e .
.
If we do t h i s
8-1 v a r i a b l e s ( i t c l e a r l y h o l d s f o r polynomials i n no v a r i a b l e s ) ,
hence f o r
fr
,
we may s u b t r a c t t h e t e r m
complete t h e proof by i n d u c t i o n on
Lemma 5:
If
Proof:
n
m =n =1
yield
,
X, Y, H
Each(Ia
)
XY = YX
+
H
x ( y n / n ? ) = (yn/n!)x
T h i s e q u a t i o n and i n d u c t i o n on
Corollary:
)
from
f
and
for
then
The c a s e
i n d u c t i o n on
.
i s a r o o t and we w r i t e
a
X - a ~a'
r
fr
is i n
m
,
+
t o g e t h e r with
y i e l d t h e lemma,
2z*
.
( Yn-1 / ( n - l ) ? ) ( ~ - n + l )
proof:
Set
n-1
+ r
on
i n Lemma 5 , w r i t e t h e r i g h t s i d e a s
H- 2n+2 j
)(Xn-J/(nj ) ?) , t h e n u s e i n d u c t i o n
( y n - j / ( n - j ) ?)
rn=n
(
a n d Lemma 4.
n
~ernrna6 :
~ e xt z b e
t h e -&span
of t h e b a s i s
{Hi,Xa]
Then u n d e r t h e a d j o i n t r e p r e s e n t a t i o n , e x t e n d e d t o
~:/rn?
preserves
2Z'
x.
of
Q,
every
a n d t h e same h o l d s f o r
any number o f f a c t o r s .
Proof:
-
Making
(xrn/m!
a
Xg =
~:/m?
2
a c t on t h e b a s i s of
and
0
(r+m-l)/m!Xp+m,
r =r(a,p)
( s e e t h e d e f i n i t i o n of
Xa * X-a = Ha
.
( r + l )( r + 2 )
, (x:/~)*x-a
=
-
,
Xa
i n Theorem
Xa O H . =
- <
B # - a
if
11,
a , a i >XG
,
2 is preserved.
i n a l l o t h e r c a s e s , which p r o v e s
The
second p a r t f o l l o w s by i n d u c t i o n on t h e number of f a c t o r s a n d :
Lemma 7:
Let
and
U
subgroups t h e r e o f .
then so i s
Proof:
AQB
Since
V
If
A
( i n U@V)
X
x
be
actson
-modules and
and
B
A and
additive
B
a r e p r e s e r v e d by e v e r y
~:/m!
.
U@V
xm/m?
t h e binomial expansion t h a t
as
X @ 1 + 1 @ X
itfollowsfrom
I: x J / j ? @ X m- j / ( m - j ) ?
a c t s as
,
whence t h e lemma.
Lemma 8 :
Let
and
a, p c S
(b)
S
be a s e t of r o o t s s u c h t h a t
,
a
+
E
X
a
+ 6
p o s i t i v e r o o t s ) , a r r a n g e d i n some o r d e r .
E
(a)
a
E
S
=> -
a
4
S
S ( e , g . t h e s e t of
Then
Xuma/mG1 lma
{
GES
> 01
a
is a b a s i s f o r t h e z - a l g e b r a
proof:
which
g e n e r a t e d by a l l
By t h e B i r k h o f f - W i t t Theorem a p p l i e d t o t h e L i e a l g e b r a f o r
(xa(u e S ]
i s a b a s i s we s e e t h a t e v e r y
complex combination of t h e g i v e n e l e m e n t s .
efficients a r e integers.
Write
most t h e same t o t a l d e g r e e .
(x ma
in
~:/m?
copies)
%@ %'@
A
=
c
We make
A
a
is a
We must show a l l coXama/ma!
+ t e r m s of a t
a c t on
and l o o k f o r t h e component of
XQ~G
A.@
X -a @
@X-a
-a
ma c o p i e s
.
*
A e
Any t e r m of
A
other than t h e first leads t o
a z e r o component s i n c e t h e r e a r e e i t h e r n o t enough f a c t o r s ( a t l e a s t
one i s needed f o r each
X -a )
o r b a r e l y enough b u t w i t h t h e wrong
%
distribution (since
p = a)
the
,
X *X
i s a n o n z e r o element o f
o n l y if
P -a
w h i l e t h e f i r s t l e a d s t o a non-zero component o n l y if
tations.
? s a r e matched u p , i n a l l p o s s i b l e permu-a
It f o l l o w s t h a t t h e component sought i s cHa 8 - @JHa
Now e a c h
Ha
Xaqs
and t h e
X
i s a p r i m i t i v e element o f
yz ( t o s e e t h i s imbed
i n a s i m p l e s y s t e m of r o o t s and t h e n u s e Lemma 1). S i n c e
preserves
yz@zz@
by Lemma 6 it f o l l o w s t h a t
A
c e
2
,
-
')
whence Lemma 8.
Any f o r m a l p r o d u c t of e l e m e n t s of
or
~:/m?
(m, n e
z+;
t o t a l degree i n t h e
XVs
'
k e
1)w i l l
i t s degree.
& of
t h e form
a
("i
n
be c a l l e d a monomial and t h e
.
~emrna9:
6,
If
m, n e
and
y c .X
z+,
( ~ ; / n ? ) (xm/m? )
an i n t e g r a l combination of
Y
(xm/m?
) (x;/n? )
Y
then
is
and monomials of l o w e r
degree.
proof:
This holds if
Assume
B # 2
y
i y + jfl
form
.
P
= y
=
By Lemma 8 a p p l i e d t o t h e s e t
i , j e
,
-
S
by Lemma 5.
y
of r o o t s of t h e
5,
arranged i n t h e order
8 + y,
y,
.. .
(xY/m!)
( x n / n ? ) i s a n i n t e g r a l combination of t e r m s o f
B
d
a (a E S)
The map X-,>
(x;/b? ) (x;/c? ) ( ~ ~ + ~) / d ?
we s e e t h a t
t h e form
leads t o a grading of t h e algebra
group g e n e r a t e d by
has degree
whence
p
o b v i o u s l y and i f
np
+
S
.
my
b, c,
b
=nP
+
with values i n t h e a d d i t i v e
The l e f t s i d e of t h e p r e c e d i n g e q u a t i o n
Hence s o d o e s each t e r m on t h e r i g h t ,
a r e r e s t r i c t e d by t h e c o n d i t i o n
+ d(B+y) +
Clearly
.
a
.
c
+
d
can be a s l a r g e a s
+
n
+ my , hence a l s o by
..
+ m
b
+
c
bp
+
2d
+
+
cy
9
= n
t h e o r d i n a r y d e g r e e of t h e above t e r m ,
only if
b
+
by t h e l a s t c o n d i t i o n , and t h e n b = n
c =n
and
+ m
c =m
and
d =
=
by t h e f i r s t ,
which p r o v e s Lemma $.
Lemma 1 0 :
Proof:
of
H
B
If
a
and
a r e r o o t s and
f i s any p o l y n o m i a l , t h e n
By l i n e a r i t y t h i s need o n l y be proved when
f
i s a power
and t h e n i t e a s i l y f o l l o w s by i n d u c t i o n on t h e two e x p o n e n t s
s t a r t i n g with t h e equation
Observe t h a t each
+
X H = (Hg - a ( H D ) ) X a
a 5
a(HB) is an integer.
Now we can prove Theorem 2.
.
By t h e c o r o l l a r y t o Lemma 5
0
rn
.
(ti)
z'hence
is in
s o i s each of t h e proposed b a s i s
az
We must show t h a t each element o f
elements.
i s an i n t e g r a l
combination of t h e l a t t e r e l e m e n t s , and f o r t h i s i t s u f f i c e s t o
show t h a t each monomial i s .
Any monomial may, by i n d u c t i o n on t h e
d e g r e e , Lemma 9 , and Lemma 1 0 , be e x p r e s s e d a s an i n t e g r a l combinat i o n of monomials such t h a t f o r each
a
X,
the
t e r m s a l l come
t o g e t h e r and i n t h e o r d e r of t h e r o o t s p r e s c r i b e d by Theorem 2 ,
a
t h e n a l s o such t h a t each
m+n
i s r e p r e s e n t e d a t most o n c e , because
.
m+n)!
brought t o t h e f r o n t ( s e e Lemma
lo),
H
The
t e r m s may now be
t h e r e s u l t i n g polynomial
e x p r e s s e d a s a n i n t e g r a l combination of
by Lemma 4 ,
(::)'s
t e r m s h i f t e d t o t h e p o s i t i o n p r e s c r i b e d by Theorem 2 ,
Hi
and Lemma 4 used f o r each
s e p a r a t e l y , t o y i e l d f i n a l l y an
Hi
i n t e g r a l combination of b a s i s e l e m e n t s , a s r e q u i r e d .
each
Let
.
Let
vector
P
an
be a semisimple L i e a l g e b r a h a v i n g C a r t a n s u b a l g e b r a
V
v e V
% such
be a r e p r e s e n t a t i o n space f o r
that
If
a
then f o r
Hv
=
x(H)v
Proof:
If
v
for a l l
H
h
H
E%.
If such a
v
h
#
+
ex,
a
,
then
if
0
a weight of t h e r e p r e s e n t a t i o n .
h
i s a weight v e c t o r b e l o n g i n g t o t h e weight
a r o o t we have
t o t h e weight
\lie c a l l a
a weight v e c t o r i f t h e r e i s a l i n e a r f u n c t i o n
e x i s t s , we c a l l t h e c o r r e s p o n d i n g
Lemma 1 1 :
.
,
h
X v
i s a weight v e c t o r b e l o n g i n g
a.
X,v # 0
.
HXav = X,(H
+
o ( H ) ) v = ( h + a)(H)X,v
.
Theorem 3 :
subalgebra
(a)
If
i s a semisimple L i e a l g e b r a h a v i n g C a r t a n
Dk( ,
then
Every f i n i t e d i m e n s i o n a l i r r e d u c i b l e
c o n t a i n s a nonzero v e c t o r
v
+
2 -module
such t h a t
weight v e c t o r b e l o n g i n g t o some weight
xn,v+ = o
(b)
(a
v
and
h
i s t h e s u b s p a c e of
Vh
c o n s i s t i n g of weight v e c t o r s b e l o n g i n g t o
h
(c)
-
Ca
,
is a
> O) .
It t h e n f o l l o w s t h a t if
dim Vh = 1
+
V
.
Moreover, e v e r y w e i g h t
where t h e
V
,
h
then
,u h a s t h e form
a q s a r e positive roots.
V = CV,rl (
a weight ) .
The weight
h
and t h e l i n e c o n t a i n i n g
v
+
Also,
are
uniquely determined.
.
(d)
X ( H ~ )&'f o r
(e)
Given a n y l i n e a r f u n c t i o n
a
>o
h
s a t i s f y i n g ( d ) , then t h e r e
i s a unique f i n i t e dimensional f - m o d u l e
(a)
i n which
is realized a s in ( a ) .
A.
Proof:
V
There e x i s t s a t l e a s t one w e i g h t on
a s an Abelian s e t of
on t h e w e i g h t s by
,u
endornorphlsms.
<
v
if
v
-
,!;
V
since
%
acts
lie i n t r o d u c e a p a r t i a l o r d e r
= CG
(a
a positive root).
S i n c e t h e w e i g h t s a r e f i n i t e i n number, we have a maximal weight
Let
h
+
v+
a
xav+ =
be a nonzero weight v e c t o r b e l o n g i n g t o
is not a w e i g h t , f o r
o
(a
>
O)
.
ct
>
0
,
h
.
Since
we have by Lemma 11 t h a t
h
.
(b)
and ( c )
w=Qv++
Mowlet
be t h e L i e s u b a l g e b r a o f
Let
of
a- , x0, %+
x- , %, x+
,
<
x- (x+)
~ e t
X_ w i t h
g e n e r a t e d by
o
\j.
and
and
v .
Z
rL
<
0 (a
>
0)
.
be t h e u n i v e r s a l e n v e l o p i n g a l g e b r a s
respectively.
By t h e B i r k h o f f - W i t t
2-h a s a b a s i s [ a < O Xam ( a ) ] ?u
has a basis
[ fi H r i ] , 2
' h a s a b a s i s [ > Xap ( c ) ] and 2 , t h e
i=l
theorem,
0
a.
,
u n i v e r s a l enveloping a l g e b r a of
3
has a basis
% =%-v 2 f Now W i s i n v a r i a n t u n d e r /2C- . A l s o ,
V = X v + =a-bk"
v+ =%s i n c e V i s i r r e d u c i b l e , %+v+ = 0
and Q0v+
= @
. Hence V = W and ( b ) and ( c ) f o l l o w .
Hence,
V+
V+
(d)
Hn
X-a
.
Ha; X a ,
subalgebra,
(e)
Corollary:
Hence, by t h e t h e o r y o f r e p r e s e n t a t i o n s o f t h i s
h(H
) a z +
C"
(See Jacobson,
Lie
A l g e b r a s , pp. 83-65. )
See SBminaire "Sophus L I E , " Expos6 no 17.
If
,:
i s a weight and
a
a root, then
:((Ha)
=
Remark:
<
R,a
h, v
>
+
E Zf o r
C,
B
F
E
E.
.
a r e c a l l e d t h e h i g h e s t w e i g h t , a h i g h e s t weight
vector, respectively.
By Theorem 2 , we know t h a t t h e
by
a
T h i s f o l l o w s from ( b ) and ( d ) of Theorem 3 and
Proof:
B(H,)
i s i n t h e 3-dimensional s u b a l g e b r a g e n e r a t e d by
~:/m!
( a E ,Y, m F
z+)
-algebra
has a H - b a s i s
Uz
9
We s e e t h a t
p o i n t s of
f k ( P ) = 1 and
takes t h e value zero a t a l l other
fk
& ' w i t h i n a box w i t h edges
k s u f f i c i e n t l y l a r g e , t h i s box c o n t a i n s
If
I
i s a v e c t o r space over
V
c-basis for
V
,
we s a y
For
z-basis
is a l a t t i c e i n
M
.
is a finitely
M
which h a s a
V
P
.
S
and
generated ( f r e e A b e l i a n ) subgroup of
which i s a
and c e n t e r
2k
.
V
We can now s t a t e t h e followj..ng c o r o l l a r i e s t o Theorem 2.
C o r o l l a r y 1:
( a ) Every f i n i t e dimensional
lattice
.
.. ,
.
'{b)
M
M
f-module
i n v a r i a n t under a l l
contains a
V
~:/m?
( a a X, m a
z+)
;
/Z1z-
i s i n v a r i a n t under
Every such l a t t i c e i s t h e d i r e c t sum of i t s weight
components; i n f a c t , e v e r y such a d d i t i v e group is.
Proof:
(al; By t h e theorem o f complete r e d u c i b i l i t y of r e p r e s e n t a -
t i o n s of semisimple L i e a l g e b r a s
I
over a f i e l d of c h a r a c t e r i s t i c
(See Jacobson, L i e A l g e b r a s , p . 7 4 ) , we may assume t h a t
v+
V
is
2-v+
.
z
M i s f i n i t e l y g e n e r a t e d o v e r 2 s i n c e o n l y f i n i t e l y many monomials
i n a- f a i l t o a n n i h i l a t e v + . S i n c e ?A!-V+
v and s i n c e 2z
z
irreducible.
Using Theorem 3 , we f i n d
and s e t
M =
=
spans
o v e r (f
,
M
we s e e t h a t
spans
V
over
.
Before
completing t h e proof of ( a ) , we w i l l f i r s t show t h a t i f
Tcivi
ni a
= 0
z,
with
nl
#
EQ
,
such t h a t
1
0
,
c.
vi
E
M
and
Zcini
vl
= 0
be such t h a t t h e component of
#
.
uvl
0
,
then there exist
To s e e t h i s , l e t
in
V+
i s nonzero.
18
Then
of
Eciuvi
uvi-
.
0 implies
in
choice of
y
=
Le
v+
.
u
.
Zcini
where
= 0
ni o
We have
n 1. v
i s t h e component
by Lemma 12 and
F i n a l l y , suppose a b a s i s f o r
be minimal such t h a t t h e r e e x i s t
4
4-
M
cini=O.
We s e e t h a t
0 = nl
vl,
...,
M
is a l a t t i c e i n
(b)
S = { pXlh
V
M
Let
vL
vl, v 2 ,
=
V
...,
h
to
-
.
n.v ) ,
11
Hence,
.
#
p]
.
l i k e t h e p r o j e c t i o n of
v
vL
be any subgroup of t h e a d d i t i v e group of
a weight,
p r o j e c t i o n of
M
I: ci(nlvi
i=2
Let
f
be as i n Lemma 13
and
on
F
4
L: civi
i=l
we have a c o n t r a d i c t i o n t o t h e choice of
0 by
i s not a basis f o r
G
that
#
nl
V
fi
is i n
V
onto
V
Y
.
M , and M
Thus, i f
u
V
with
acts
v e M
,
the
i s t h e d i r e c t sum of
i t s weight components.
be f a i t h f u l l y r e p r e s e n t e d on a f i n i t e dimensional
C o r o l l a r y 2 : Let
V
.
f o r a l l weights
L
v e c t o r space
Let
M
be a l a t t i c e i n
V
i n v a r i a n t under
of t h e g i v e n r e p r e s e n t a t i o n ] .
In particular,
4z i s
PI -
independent of
(But, of course,
z i s n o t inde-
Giw
$endent o f t h e : * a p r e s e n t a t i o n . )
,.:
:.:
... ...
proof:
We re~allthz-5 a s s o c i a t e ? w i t h t h e r e p r e s e n t ? - t i o n on
-.
:,
...*
...,.
,
V
.I*
t h e y e i s a rep*-r:;sntation on t h e d u a l space
v".
the
< x,.&y-.> = - <
...,..
<+.' :
of
called
V
,
&:
[&
.I.
.
.c o r ~ t r a g r e d i e n tr e p r e s e n t a t i o n g i v e n by
---<.
@:?!
;
..
0' I
..
,
x e V
:here
y .c V
I
'.a.
,
2 and where
4, e
=
<
.1.1:
,;
:
?*,:>:
v a l u e of t h e l i n e a r f u n c t i o n
lattice in
.
V '
.
J,
y
; i.e.:
0.1 1
&
*,:
at
x
,
< M,I<''
"/.
c-
.'-
%
....
rp r e s e r v e s PC i f and o n l y i f
:.:.
.,,-;.,
.,
F,
T
J
0 v*
i s isc:::or-phic
denotes t h e
x,ya i
,
.8I
.
'
fi 8
Jr
&x,yq' >
If
is t h e d u a l
I(
c -i C ~ . . -
,-A>..!
4
&
4
p r e s e ~ ~ r e s5
We know t h a t
EI?<('\T? ;fiG ;ithat, t;he tensor. product
;
with
"'.
of t h e t x o r ; - : p ~ e . ; z n - t . c . t l s ~co:-re
s
sponds t o t h e i - e p ~ e s e n t a t i o n
$-:
4,: A
--+-[ & > A :
Y
i C L 2 cu
-.Lie Al~ebx.2.::
----y
"I C ,
;
- z n d ; ~ > j of
,
:
-,
., ..
3
(See Jacobeoa,
.I,
E M ) -.- ?
LLOC:
.
i n End(Vj
1.''
2
j-s
l ~ t t i c ei n
s i n c e t h e t e i i z o r x - c u ~ c toi' i:-~:ol a t t i c e s i s a l a t t i c e , Also,
2 since 4
'
4
is a l a t t i c e In
C End(M) 2nd dim A > dim
---4
because z.:tl..
7 .
.
r,
- 2 .
.'.,..
and
11.
"
14.
-r, !
,(
,!
A >
--(,,.G2
. .e "c:
k
- ~ r a c a - r x r n -
spans
x
t ;iL
2:
.latti ~e
2 Xu /n3 s
and
n = I
rn
.
Hence,
.d
2
'
11
-- X -a /n
.
?,<.
<I,
.3
Ox' o v e r y
a d ( x-2G. / 2 ! ) ( x a / n )
-
.,-
- - # , :*7L )
<--
The,
dz . S i n c e
./,> N::: Sy Lenlna
::;,s i n
i:
. 7 - > :
i:-
e
w
.- -- -
//
for
n
>T--
a
2
LIZ .
4
--- ,,he
Hence
2/n3 c (l,),
C ra nL:=Z'x
.
a
L
xz
zpzc
reserves
z
7 , and hence 2.L
z
A-,
.
adj o i n ,
h , n 21 ,
rnd(V)
yn-nncnnt .rt4
r n
then
- ( a d xa/n12 ( X -a / n )
xhich implies
2/n2 e
H
Example:
g'
i
1
X, Y,
and
Let
V
=
and
H
.
are
+
-
be t h e 3 dimensional L i e a l g e b r a g e n e r a t e d by
Let
and l e t
M
be t h e l a t t i c e i n
and
[H,Y]
=
-
.
2Y
2 spanned by
X, Y,
Then s i n c e t h e o n l y weights of t h e a d j o i n t r e p r e s e n t a t i o n
a , 0 with
Here
H
with
B(H) = 1 .
Hence
.
cz?
= s t 2 on a 2 dimensional v e c t o r space
VD
xq r,
corresponds t o
,g7=zx+ZY+ Z ( H / 2 )
Now
a ( ~ =) 2
is isomorphic w i t h
l!
[x,Y] = H , [ H , X ] = 2X,
with
H
1
1-
n
1
1
I
and t h e w e i g h t s a r e
T
D,O
x qz Z X +
Z Y + & - I and
=
We a r e now i n a p o s i t i o n t o t r a n s f e r o u r a t t e n t i o n t o a n
k
arbitrary field
Mp
t
fE ,
k
.
We have a l r e a d y d e f i n e d t h e l a t t i c e s
M p = V/,
l a t t i c e s as-z-modules
fl M
,
and
Xa
and c o n s i d e r i n g
we can form t h e t e n s o r p r o d u c t s , vk = M
y z ~ z k , V kI t = %
O z k ,
and
Xk=
.
Considering t h e s e
k
as a z - m o d u l e ,
azk , x k = x z @ z k ,
kxi=Zxa @
zk .
t h e n have :
j
Corollary 3:
1
(a)
vk
k
= iV,< ( d i r e c t sum) and
k
I
(b)
=
X,k
Proof
#
k
dimkViL = d i m Q
~ k ~ +%
_ k
( d i r e c t sum), each
0, d i r n k x k = dimc%,
an
T h i s f o l l o w s from C o r o l l a r i e s 1 and
v ,t:
W
e
.
IF
F
,
y
21
6 7.
The C h e v a l l e y g r o u p s .
of t h e form
vk
We wish t o s t u d y automorphisms of
exptXa ( t a k , c e C)
,
where
8 , ~ h er i g h t s i d e of t h e above e x ~ r e s s i o ni s i n t e r ~ r e t e da s f o l l o w s .
x ? / ~ . Pa
Since
C
-u7,we
n n
h X,/nj
'
iwe g e t a n a c t i o n of
zero f o r
ns
@-
&
4
M Q zZ [ A ]
on
M
and hence on
Oh
=M O H k
g i v e n by A
->
,
t
s i n c e X:
acts
C hn X n /n:? a c t s
.
[h] B z k
Following t h i s
zZ[ A ] O Z k
we g e t a n a c t i o n of
I-
We w i l l w r i t e
~ , ( t )f o r
1
[ ~ , ( t )t a k ] ( c l e a r l y
exptX,
G
into
n n
C t xa/n?
n=O
C3
'Faf o r
and
~ , ( t ! is additive i n
o b j e c t of s t u d y i s t h e group
Thus,
r-X3
-
M @
l a s t a c t i o n by t h e homomorphism of
.&
I , vk
.
.
M
~ n / n ? on
n s s u f f i c i e n t l v l a r g e . we s e e t h a t
Bp7[ h ]
M
on
have a n a c t i o n of
t).
g e n e r a t e d by a l l
t h e group
Our main
a, ( a a
X)
.
We w i l l c a l l i t a C h e v a l l e y group.
Exercise :
Interpret
Lemma 14:
Let
LA B = AB, r A B = BA
exp A
Q
(H a
A, , t
L
be a n a s s o c i a t i v e a l g e b r a ,
dA be t h e d e r i v a t i o n of
and
\
_1
- /H
C t" (
~3
,
a
B a
,
a.
E:
k, t
#-
A
1).
and l e t
dA = -LA - rA where
suppose
exp d A ,
exp CAP
exp r A ,
have meaning and t h a t t h e u s u a l r u l e s of e x p o n e n t i a t i o n
apply.
Then
exp dA = 'exp
Proof:
exp dA = exp -LA e x p ( - - r A )= 2 exp
.
A r exp(-A)
A r exp(-A)
(= con j u g a t i o n by exp A ) .
'
Lemma 15:
Let
a , f3
be r o o t s w i t h
a
of f o r m a l power s e r i e s i n two v a r i a b l e s
+ 6# 0
t, u
.
Then i n t h e r i n g
over%
we have t h e i d e n t i t y
( e x p tX,,
where
exp;:ul( ) =
B
( A , B ) = ABA-~B-'
taken over a l l r o o t s
i.a
,
n exp c
,
z
[ [ t , u ] ],
Z
tiuJ X
i a + j-0
ij
where t h e product on t h e r i g h t i s
+ jfi
( i ,j e
z+)
arranged
i n some f i x e d
? s a r e i n t e g e r s depending on a , P ,
and
ij
t h e chosen o r d e r i n g , b u t n o t on t o r u
Furthermore Cll = N
o r d e r , and where t h e
c
.
a7P
proof:
In
%[[t,u]]
(~XP
tXa,
where
in
cij
E:
.
such t h a t
We n o t e t h a t
set
f(t,u) =
exp u x p )
exp ( - c i j t i u j
x
~1
We s h a l l show t h a t we may choose t h e
f(t,u) =1
t d
.
( e x p t X a ) = t Xa exp t X a
Thus, u s i n g t h e product r u l e we g e t
+ I: ( e x p t X , ,
exp u X )
B
.
~
+
cijqs
~
~
-tXa
We b r i n g t h e t e r m s
I
using, e.g.,
and
( * ) - ~ ~ ~ k t ~ tuo %
th~
e f~r o+n t~
the relations
( e x p u X 9 ) ( - ' t ~=
, )( e x p ad u X B ) ( - t X 0- exp u X R
( s e e Lemma 1 4 ) and
We g e t a n e x p r e s s i o n of t h e form
Because
f(t,u)
grading
t
-
->
with
A f ( t, u )
i s homogeneous of d e g r e e
,
-> - 0 , x
A
f o r m u l a s such a s t h o s e above we s e e t h a t
ckL
(:::)
term
in
t
u
and
.
u
i s involved i n t h e
k ,L 2 l
cijss
f o r which
f(t,u) = 0
implies
To show t h a t t h e
ti uj
c o e f f i c i e n t of
efficient is
m u l t i p l e s of
i
+
<k +
j
X
+
f7L
so t h a t
c
c
11
=N
a,P
c
ij
%
?
s
pkL
ckL e
such t h a t
@
A =0
.
.
? s a r e i n t e g e r s , we examine t h e
i n t h e d e f i n i t i o n of
ia+Jf3
Xkc+&@ w i t h
ij
ij
+
.
C
f ( t , u ) = f (0,u) = 1
f(t,u)
.
T h i s co-
( t e r m s coming from e x p o n e n t i a l s of
k
+
%
<
i
X
. Hence,
ij ia+jp €
Z
i = j = 1 , t h e c o e f f i c i e n t is -
see t h a t
If
-c
c
k
(-Ck~
+ P ~ L ) ~X k a + ~ Rw i t h
u s i n g t h e $ e & e a g r a ~ h + - oc r d e r i n g of t h e
d
>
k .e
A = C
Thus
.
[[t ,u]]
i s a l s o , and from
Now we may i n d u c t i v e l y determine v a l u e s of
t
f
above but o t h e r w i s e o n l y i n terms o f d e g r e e
a polynomial i n
Then
F
0 relative t o the
- > Y ,
a
A
+
j)
cij
ell
.
Using i n d u c t i o n , we
F
Xa+@
,
by Theorem 2.
+ N0 , p
X
a+p
'
(a)
Examples:
a
If
+
i s not a r o o t , t h e r i g h t s i d e of t h e
@
.
formula i n Lemma 1 5 i s 1
+ JB ,
(b)
the right side is
form
ia
with
r=r(a,p)
If
e x p tXa
xa(t), e t c . ,
by
roots:
a
+
(a)
,
(c)
i s t h e only r o o t of t h e
S
.
e S
1 i n case ( a )
,
,
of r o o t s c l o s e d i f
-
a
yI
Let
e S
a,
(b)
xS.
subgroup of
Proof:
+ p
P
4S
,
Let
a +
implies
e S
E
S
implies
S
a
+ P
e C
a
{a]
,
a
a
.
e I
We s e e t h a t
.
P
-
-
(0.
4S
,
.
Let
Xs
e S
and
a 8 I
S
then
be a c l o s e d s e t of r o o t s s u c h t h a t
t h e n e v e r y element of
Xa(ta)
,
,
t, u e k
S an ideal if
be a n i d e a l i n t h e c l o s e d s e t
a
a e S
fixed order.
where
,
i n case (b).
and
,
i s a normal
T h i s f o l l o w s i m m e d i a t e l y f r o m Lemma 15.
Lemma 1 7 :
a
If
'
[r+l)
.
d e n o t e t h e g r o u p s g e n e r a t e d by a l l
respectively).
-
I
+-
The f o l l o w i n g a r e examples o f c l o s e d s e t s of
( a ) , ( b ) and ( c ) above a r e i d e a l s i n
Lemma 16:
=
e t c . i n t h e f o r m u l a i n Lemma 1 5 a r e r e p l a c e d
Pr = [ a l h t a > r , r >
-1
(c)
l3 e S
N
e x p (+
- tu)
and
We s h a l l c a l l a s u b s e t I o f a c l o s e d s e t
a e I
and
a70
a7B
If a l l t h e r o o t s have one
P = s e t of a l l p o s i t i v e r o o t s .
simple r o o t .
,
tu
then t h e r e s u l t i n g equation holds f o r a l l
We c a l l a s e t
implies
exp N
a s i n Theorem 1.
length, t h e right side i s
Corollary:
+ 0
a
If
ta e k
Ys
a e S
implies
c a n be w r i t t e n u n i q u e l y a s
and t h e p r o d u c t i s t a k e n i n any
.
I
I
Proof:
-
We s h a l l first prove t h e lemma i n t h e c a s e i n which t h e
<
ordering i s consistent with heights; i . e . , h t a
If
i s t h e f i r s t element o f
al
,
S
then
S
-
{a,]
-
-
ideal i n
S
.
t h e s i z e of
~ence
S
,
XS = X X s - b l l .
al
X g = rt X ' a
we s e e
Now suppose
y e
Ks
t h e r e i s a weight v e c t o r
Xa
that
v
#
0
1
v 6Vhh
X
t
a1
,
v e
.
Now
yv = v
,
and
z
y =
M
rt
h t fl
a
implies
<P
.
is an
Using i n d u c t i o n on
'
.
ya(tc)
xka l # O ,
Since
corresponding t o a weight
h
such
Xa v + z where v e Vh ,
"1 1
i s a sum of t e r m s from o t h e r weight
+
t
1
spaces.
e k
t
Hence
i s u n i q u e l y determined by
y
a1
s-{a,) ,
.
Since
we may complete t h e p r o o f o f t h i s
c a s e by i n d u c t i o n .
The p r o o f Lemma 1 7 f o r a n a r b i t r a r y o r d e r i n g f o l l o w s immediately
from:
Lemma 1g:
Let
be a group w i t h subgroups
X ,
x2,..., X
r
such t h a t :
(a)
(b)
If
p
X = yl g 2 . . .xr 7
.
1
i + l .. .
r
i s any p e r m u t a t i o n o f
w i t h u n i q u e n e s s of e x p r e s s i o n .
with uniqueness of expression.
i s a normal subgroup o f
1, 2 ,
... , r
then
X
for
X- = x p l X p2
P'
a d d i t i v e group of
'k
P
C o r o l l a r y 2 : Let
U =
X .
an isomorphism of t h e
.
onto
be t h e s e t of a l l p o s i t i v e r o o t s and l e t
a w i t h u n i q u e n e s s of e x p r e s s i o n , where
t h e p r o d u c t i s t a k e n o v e r a l l a e P a r r a n g e d i n any f i x e d o r d e r .
I
Then
xa(t) is
->
t
C o r o l l a r y 1: The map
Corollary 3:
U =
i s u n i p o t e n t and i s s u p e r d i a g o n a l r e l a t i v e t o an
U
a p p r o p r i a t e choice of a b a s i s f o r
vk .
Similarly,
U- =
g-
is u n i p o t e n t and i s s u b d i a g o n a l r e l a t i v e t o t h e same c h o i c e of b a s i s .
Proof:
Choose a b a s i s of weight v e c t o r s and o r d e r them i n a manner
c o n s i s t e n t w i t h t h e f o l l o w i n g p a r t i a l o r d e r i n g of t h e w e i g h t s :
p
v
precedes
Corollary 4 :
ht a
with
I
If
2
.
i
be t h e group g e n e r a t e d by a l l
Ui
i s normal. i n
(b)
(U,ui)
(c)
U
C
Ui+l
,
U
.
in particular,
(U,U)
C
U2
.
is nilpotent.
If
QUR w i t h Q and
tHen U = x g X R and
P =
R
c l o s e d s e t s such
XQnx = l .
R
if
a
a
We have t h e n :
Ui
Q n R = f l ,
that
i s a sum of p o s i t i v e r o o t s .
I
1 let
i
(a)
Corollary 5 :
-
if
i s a s i m p l e r o o t , one can t a k e
Q = { a ] and
R = P
(e.g.,
- {a)
.
Example:
If
t o pairs
( i ,j ) i
,
=
#
,
j
we have s e e n t h a t t h e r o o t s c o r r e s p o n d
the positive r o o t s t o pairs
- Eij , the usual matrix unit.
and t h a t we may t a k e
Xij
xij ( t ) = 1 + t E i j .
We s e e t h a t
is
U
SLL+l
Lemma 1 9 :
,
if
t h e group of
i ,k
L
+
1 s q u a r e m a t r i c e s of d e t e r m i n a n t 1.
( Xi ( t)
,
Xjk(u) )
t
a
a
and any
and
t
.@.
define
t e k"'
ha(t) = wa(t)wa(l)-l
.
Then :
(a) wa(t)X
w ( t ) - l= c t - < Q ' a > ~ w
13 a
a
where
;
+
c =c(a,fi) = - 1
t,k
If
=
c ( a , -?)
v e v k
kf
v
k
P
Vw
,,,
ar'
t
w ( t ) v = t - < p , a>v
a
ha(t)
.
there exists
independent of
by
i s independent of
and t h e r e p r e s e n t a t i o n chosen, and
c(a,g)
(c)
,
Thus,
are distinct.
For any r o o t
wa(t) = -
(b)
j
= { a l l unipotent, superdiagonal
Thb n o n t r i v i a l commutator r e l a t i o n s a r e :
= Xik(tu)
<
U- = { a l l x n i p o t e n t , subdiago.na1 m a t r i c e s ) , and t h a t
matrices),
G
( i ,j ) i
such t h a t
.
a c t s v v d i a g o n a l l y 3on
.
vk1C
a s multiplication
(Note t h a t
wa
i s b e i n g used t o d e n o t e both t h e d e f i n e d
automorphism and t h e r e f l e c t i o n i n t h e h y p e r p l a n e o r t h o g o n a l
a).
to
Proof:
-
We prove t h i s assuming
.
k =
The t r a n s f e r of c o e f f i -
c i e n t s t o a n a r b i t r a r y f i e l d i s almost immediate.
w ( t) H w G ( t ) - l = wcH
We show f i r s t t h a t
for all
01.
H
B
x.
By l i n e a r i t y i t s u f f i c e s t o prove t h i s f o r
a(H) = 0
If
H = H
Xa
then
commutes w i t h
f o r if
Ha ,
so t h a t both s i d e s equal
H
t h e l e f t s i d e , because of Lemma 2
a
t i o n s of X ( t ) and
C:
w ( t . ),
a
< X , Ya,
H
.
and t h e d e f i n i -
i s a n element of t h e t h r e e
> whose v a l u e depends on
a
c a l c u l a t i o n s w i t h i n t h i s a l g e b r a , n o t on t h e r e p r e s e n t a t i o n
dimensional a l g e b r a
chosen.
H
C?
Taking t h e u s u a l r e p r e s e n t a t i o n i n
sL
2
'
we g e t
so t h a t t h e desired equation fqllows.
We n e x t prove ( b ) .
and
wa ( t)
From t h e d e f i n i t i o n s of Y , ( t )
it f o l l o w s t h a t i f
vv=wa(t)v,
then
va=
t iv i
X
c
i-
5
where
V. B
1
V,l+ia
I.
( t h e sum i s a c t u a l l y f i n i t e s i n c e t h e r e a r e o n l y f i n i t e l y many
weights).
Then f o r
H e %,
w ( t ) ( w H)v = ?:{w H ) V "
a.
a
a
t o t h e weight w ,:: = ,!: C!
i n t h e sum o c c u r s f o r
Hvit = Hw ( t ) v = w ( t ) w ( t ) - ' ~ w ~ ( t =) v
0:
a
Ci
= ( w , ! r ) ( H ) v x . Hence vi? c o r r e s p o n d s
C.
<
i =
tt,ci.
>
c!
.
- < !!,a >
Thus t h e o n l y nonzero t e r m
.
29
By ( b ) a p p l i e d t o t h e a d j o i n t r e p r e s e n t a t i o n
= ct
-
<R,a>
"
X
where
c e
WCL
of
a n d of t h e r e p r e s e n t a t i o n c h o s e n .
t
yz
automorphism o f
for a l l
so
y
so
c =
+-
Hence
= wa(-t)
v
i s an
wa ( - t ) - lwQ ( - 1 ) v = t
= wa(l)HRwcL(1)-' =
<;<,a>
for
@0
v
wa(l)
,
h c L ( t ) = wa(-t)-'wa(-1)
w ( - 1 ) v = (-1)- <$,a>v 9 .
C.
p r o v i n g ( c 1.
.
Then:
h (t)o-l= h
g
(a)
C1
p r o d u c t of
( t )= a n e x p r e s s i o n a s a
wup
hvs , independent of t h e r e p r e s e n t a t i o n
space.
(b)
oC! x R ( t ) w-1c = X, O ( ~ tw )i t h
,
c
as in
a-
Lemma 13(a).
(c)
Proof:
h c ( t ) x R ( u ) h a ( t )-1 = Y, (t<R ,a> u )
To p r o v e ( a ) we a p p l y b o t h s i d e s t o
-I
Q
hl (t)iuo
u ?
.=,
a
t
xz
so t h a t
and
a
Write
wa(l)
i s a p r i m i t i v e e l e m e n t of
Y
Hw,B
w ( - t ) v = ( - t )- <,!$,a>
Lemma 2 0 :
Now
and i s independent
~ h i c h ~ p r o v e( sa ) .
wa(t)-'
Note t h a t
X
.
1
,
c(a,p) = c(c,-0)
By ( b )
and
(?
<wfrR>
c'.'
.
v e Vk
,! l
.
-1
v
( b y Lemma 19 ( c ) a p p l i e d
.
= cX
Lemma 1 9 ( a ) cu a X ? LO-'a
wop
.
Exponentiating t h i s g i v e s ( b ) .
By Lemma 1 9 ( c ) a p p l i e d t o t h e a d j o i n t r e p r e s e n t a t i o n
<p ,a>
h a ( t ) x n h a ( t )-1 = t
XR . Exponentiating t h i s g i v e s ( c ) .
Denote by ( R ) t h e f o l l o w i n g s e t o f r e l a t i o n s :
xia+jB
( c i j t i'u j 1
( xa ( t ) , x R ( u ) ) =
with the
a
LD
a
( t )=
'i j
( a + f~ # 0 )
a s i n Lemma 15.
xa(t)x
( - t - l ) r a ( t.)
-a
h ( t) w=- l
some e x p r e s s i o n a s a p r o d u c t o f
9
a
h g s ( i n d e p e n d e n t of t h e r e p r e s e n t a t i o n s p a c e ) .
(R7)
-1
bcxe(t)wa
(ct)
=
c
a s i n Lemma l F ( a ) .
wap
Since a l l t h e r e l a t i o n s i n ( R ) a r e independent of t h e
r e p r e s e n t a t i o n s p a c e c h o s e n , r e s u l t s proved u s i n g o n l y t h e
r e l a t i o n s ( R ) w i l l be i n d e p e n d e n t of t h e r e p r e s e n t a t i o n s p a c e
R e s u l t s proved u s i n g o t h e r i n f o r m a t i o n w i l l be l a b e l e d ( U )
(usually f o r uniqueness).
Lemma 21:
.
H
by
Let
be t h e group g e n e r a t e d by a l l
U
t h e g r o u p g e n e r a t e d by a l l
U
.
and
H
(a)
U
(b)
U n H = l .
Proof:
B
V
hence = 1
.
Example:
In
B
and
a n y element of
SLn
Let
B = UH
N
ha(t)
H
UnH
t h e group generated
B
.
(E
preserves
Relative t o an a p p r o p r i a t e
i s b o t h d i a g o n a l and u n i p o t e n t ,
= !diagonal m a t r i c e s ] ,
U = {unipotent
B = {superdiagonal matrices].
be t h e group g e n e r a t e d by a l l
h ( t)
t h e subgroup g e n e r a t e d by a l l
1
and
( a ) holds.
superdiagonal matrices],
Lemma 2 2 :
and
,
0)
(u
S i n c e c o n j u g a t i o n by
b a s i s of
ha(t)
>
Then:
i s normal i n
i s normal i n
I
a (a
~
a.
,
wa(t),
H
be
W t h e Weyl group.
and
Then :
(a) H
(b)
i s normal i n
.
(El
T h e r e e x i s t s a homomory,hism
su-ch t h a t
(c)
N
y(wa) = H w a ( t )
i s a n isomorphism.
cp
of
W
onto
for a l l roots
a
N/H
.
(E)
(u)
S i n c e by ( ~ 6 c) o n j u g a t i o n by w
Proof:
( R 4 ) and ( R 5 )
Hwa(t) = Hwa(t) wa(l)"
of
.
t
f
a
But
Wa
Thus
.
a
.
-1
GB
€
so
,
LO
0
UI
wa(l) e
Aence
( )
4-1
A
wa,wgwa
p : W
morphism
3a
and
d
=1
CS
.
#\
(by (R3))
=w
.
wa.fl
form a defining s e t f o r
A
A
i s independent
h A - 1
v'awowa
0:
( )
a n d by
Since
Hwa(t)
Then s i n c e
~2
1w ( l ) w ( - 1 ) e w
,
w,
( a ) holds.
w a ( l ) = Hwa(l)
= Hwa(t)
w
p - wrj(l)
Also
(*)
A
Write
A
wa(-1)
,
w a ( t ) = h , ( t ) w,
H
preserves
0:
->
N/H
By Appendix I V .
W
.
such t h a t
40 t h e
relations
Thus t h e r e e x i s t s a homoq3wa
A
= w,
= Hw, ( t)
.
is
y
c l e a r l y onto.
Suppose
w e k e r cp
get
...
wa ( 1 ) (1)
~ ~
1
2
%a
for all
Lemma 23:
Proof:
.
Hence
a
implies
If
c
If
wa ( l ) w a (1)
reflections, then
by
.
1
2
we g e t
Y f
%wa = C; a
and
We know t h a t
w =w
.. .
Xwa
w
a2
= h e H
... ,
.
a p r o d u c t of
Conjugating
and c o n j u g a t i n g by
for all r o ~ t sG
.
h
xa
we
wa = a
Since
w = 1 t h e proof i s completed by:
p
a r e d i s t i n c t roots then
is nontrivial.
If
a
t h e same s i g n , t h e r e s u l t f o l l o w s f r o m Lemma 17.
xc #&
and
9
have
I f t h e y have
o p p o s i t e s i g n s , t h e n one i s s u p e r d i a g o n a l u n i p o t e n t , t h e o t h e r
s u b d i a g o n a l ( r e l a t i v e t o a n a p p r o p r i a t e b a s i s ) , and t h e r e s u l t
again follows.
Convention:
wB (Bw)
we w i l l w r i t e
If
Lemma 24:
B u BwaB
Proof:
n e N
If
nB (Bn)
S =B
LJ
BwaB
u
,S
.
i s c l o s e d under i n v e r s i o n , and s i n c e
u
BwcBB u BwaBwaB C S
CS
.
We f i r s t show t h a t
waBwa
i s a group and
B
Since
BBwG B
then there e x i s t s
t e kS
u
such t h a t
= x , ( t -1) w a ( - t -1)xa(tml) e B w a B .
a
x a X P-{a)
(since
I
wa
Lemma 25:
(a)
HW-I
= w X w-I
c
a a a wa
preserves
If
If
P-{a)
w e 14 and
wa
>
0
a
y =x
Hence
(b)
I n any case
CS
-a
it s u f f i c e s t o
.
If
1# y
B
C S . Nowlet P
-a Then waBwa = waBwa-1
=
HW;'
' .x-
p-iaj H
-a
by Appendix 1.11) C SB = S
i s a simple r o o t , t h e n :
(i.e. if
N(wwa) = N ( w )
+
1
BwB-BwaB C
- - BwwaB
BwBmBwaB C BwwaB u BwB
-
-a
(t)
-1
~ - I a ] ~- Wa
o
( s e e Appendix 1 1 . 1 7 ) ) t h e n
.,.
BwaBwaB
-a
be t h e c o l l e c t i o n of a l l p o s i t i v e r o o t s .
= w
.
is a simple r o o t then
a
q(wa) = r n ( w a ) - l
show
i n p l a c e of
N,/H)
i s a group.
Let
s2 C- BB
(under .rp : W ->
w e W
represents
.
.
(El
(E)
Proof:
BW
Ya W-'W
(for
w
0
,
wyawm1
B
(b)
If
wa
.
Then
0
Bw w B - Bw,B
a
=
>
u
BWW~B
0
( a ) gives the result.
B
and
0
0
minimal l e n g t h of
I
J
If
By ( a )
BW,B)
.
B)
wa
<
0
set
BwB*BwaB=
( b y Lemma 24) =
.
BwwaB
w e tJ and
.
0
w=wwa
and
Bw B-BwaB-BwaB = Bw B ( B
u
c1Hwa
yp - I a ~wa
0
Bw waB = BwB
C o r o l l a r y : If
a,
wa
w a > 0
P
0
Hw B =
&{a]
a W-l
CL
p - i a ] waWa- ~ H W ~ B=
=T
Bw B
Ya
BwB*Bw,B = B w
(a)
w =w w
...
i s an e x p r e s s i o n of
a B
a s a product of simple r e f l e c t i o n s t h e n
w
.
BwB = BwaBBw B o g *
P
Lemma 26:
Then
Proof:
Let
G
be t h e Chevalley group
Fa ,
i s g e n e r a t e d by a l l
G
We have mu%
W
for
a
<
a
la11 a
U J - ~
a
=
t h e r e s u l t follows.
Theorem 4:
)
a simple root.
.
X.
>
.
(E)
S i n c e t h e simple r e f l e c t i o n s
wa@
and e v e r y r o o t i s c o n j u g a t e under W t o a simple r o o t
P
geherate
wa
(G =
(Bruhat , C h e v a l l e y )
P
( b ) ,BwB = Bw B
=>
w = wq
.
Thus any system of r e p r e s e n t a t i v e s f o r
a system of r e p r e s e n t a t i v e s f o r
B\G
/
B
.
N/H
is alsd
I
.
U BwB i s c l o s e d u n d e r m u l t i p l i c a t i o n by t h e s e
we W
g e n e r a t o r s ( b y Lemma 2 5 ) and r e c i p r o c a t i o n it i s e q u a l t o G
for
G
Since
.
(b)
Suppose
v
BwB = Bw B
We w i l l show by i n d u c t i o n on
N(w)
~ ( w>
) 0
Then
I
II
0
wwa e Bw BBwaB
wwG
by i n d u c t i o n
wa
and choose
=
5 Bw
w
0
The g r o u p s
0
.
0
w = 1 so
N(wwa)
9
0
-
9
wwa - w wa
form a
J. T i t s (Annals of Math. 1 9 6 4 ) .
.
But
0
Hence
wwa = w w
a
B
-
w
0
e B
.
= 1 ( s e e Appendix 1 1 . 2 3 ) .
simple so t h a t
a
N(w)
(Here
B u Bw waB = BwB u Bw waB
or
B, N
w
.
e W
w =w
that
and
= 1 which i s i m p o s s i b l e .
Remark:
9
N(w) = 0 t h e n
i s a s i n t h e Appendix 11.) I f
v
v
Then w Bw ' - l = B
so w P = P
Assume
w, w
with
N
wwa
=
so
.
<
.
N(w)
Hence
w implies
w =w
0
.
p a i r i n t h e s e n s e of
We s h a l l n o t a x i o m a t i z e t h i s
concept b u t a d a p t c e r t a i n a r g u m e n t s , such a s t h e l a s t one, t o t h e
present context
.
0
For a f i x e d w e W choose w w r e p r e s e n t i n g w i n
-1
Q = P n w (-P) , R = P n w-l P ( a s before P denotes t h e
Theorem 4 :
Set
s e t of p o s i t i v e r o o t s ) .
Write
UW
I
W
for
Y Q . Then:
.
(a)
B w B = B wW
(b)
Every element of
i n t h i s form.
U
(E)
BwB
h a s a unique e x p r e s s i o n
N
.
Proof:
BwB = Bw X R X Q ~( b y Lemma 17 and Lemma 21)
(a)
_C__
4 w - Xl Q
~ =~B W X ~ H ( s i n c e wjERw-'
= ~
(b)
0
0
bw x = b owx
If
W
P -l
c
B) = B
ww
x ~
then
.
b - l b q = w x x w -1
w
w
R e l a t i v e tQ a n a p p r o p r i a t e b a s i s t h i s i s b o t h
s u p e r d i a g o n a l and s u b d i a g o n a l u n i p o t e n t ' a n d hence = 1
Thus
b = b
I
I.
and
I
Theorem 4:
I
.
B
Examples:
k
,
v
x = x
( a ) Prove
Exercise:
a l s o of
0
(b)
.
.
B
is t h e normalizer i n
N
Prove
G
i s t h e normalizer i n
of
G
and
U
H
of
if
h a s more t h a n 3 elements.
Let
.
X=stn -
SO
that
,
G=SLn
B , H, N
and
a r e r e s p e c t i v e l y t h e s u p e r d i a g o n a l , d i a g o n a l , monomial.subgroups,
1
W
may be i d e n t i f i e d w i t h t h e group of p e r m u t a t i o n s of
t h e coordinates.
'
Going t o
t h e permutation m a t r i c e s
(*)
representatives f o r
of t h i s .
Here
f o r c o n v e n i e n c e , we g e t from
G = GLn
k
Sn
form a system of
.
We s h a l l g i v e a s i m p l e d i r e c t proof
Assume g i v e n x e G
can be any d i v i s i o n r i n g . / Choose b e B t o
B\G/B
I
maximize t h e t o t a l number of z e r o s a t t h e b e g i n n i n g s o f a l l of
I
l e n g t h s s i n c e o t h e r w i s e we c o u l d s u b t r a c t a m u l t i p l e o f some row
(
II
I
t h e rows o f
bx
.
These b e g i n n i n g s must a l l be of d i f f e r e n t
from a n e a r l i e r one, i. e.
number of z e r o s .
, modify
b
,
and i n c r e a s e t h e t o t a l
It f o l l o w s t h a t f o r some
s u p e r d i a g o n a l , whence
x e BW-~B
.
w
Now assume
6
Sn
,
wbx
P
BwB = Bw B
is
.
w, w
with
9
w, w
Since
e Sn
.
Then
w-lbw
9
i s s u p e r d i a g o n a i f o r some
b e B
P
a r e p e r m u t a t i o n m a t r i c e s and t h e m a t r i c p o s i t i o n s
where t h e i d e n t i t y i s nonzero a r e i n c l u d e d among t h o s e of b
,
-1 q
P
we conclude t h a t w w
i s s u p e r d i a g o n a l , whence w = w ,
.
)
which p r o v e s
,
Next we w i l l g i v e a g e o m e t r i c i n t e r p r e t a t i o n
of t h e r e s u l t j u s t proved.
Let
be t h e u n d e r l y i n g v e c t o r s p a c e .
V
A f l a g i n . .V, i s a n i n c r e a s i n g sequence o f s u b s p a c e s
Associated with t h e
V1 C V2 C - * * C Vn , where dim Vi = i
d e f i n e d by
Fi
G
V
a c t s on
... , v n ] of
= < v l , ..., v >
-i
{vl,
chosen b a s i s
B \G
with t h e s e t of
pl,
i s i n one-to-one
... ,
[ pl,
..., pn > = n .
A flag
i n c i d e n t w i t h t h i s s i m p l e x if
T 6
Sn
.
pn ]
Hence t h e r e a r e
n!
V1 C
<
1951), t h a t
i n c i d e n t with both.
)
tion.
Now
correspondence
a * -
D e f i n e a simplex
such t h a t
C Vn
i s s a i d t o be
... , P r i >
f o r some
f l a g s i n c i d e n t w i t h a g i v e n simplex.
n
(see Steinberg,
g i v e n any two f l a g s t h e r e i s a s i m p l e x
($:)
Thus a s s o c i a t e d t o each p a i r o f f l a g s
t h e r e i s a n element of
one t o t h e o t h e r .
.
pTl,
It can be shown, by i n d u c t i o n on
T.A.M.S.
V
of
.
Vi =
C Fn
*
i s t h e s t a b i l i z e r of t h e
B
G - o r b i t s of p a i r s of f l a g s .
t o be a s e t of p o i n t s
<
/B
F1 C
called thestandard flag.
and hence on f l a g s .
standard f l a g , so
dim
there is a f l a g
V
Hence
Sn
,
t h e p e r m u t a t i o n which t r a n s f o r m s
B\ G / B
corresponds t o
Sn
.
Thus
i s t h e g e o m e t r i c i n t e r p r e t a t i o n o f t h e Bruhat decomposi-
.
(b)
[X e
=
simern/~~ + AX^
01
.
.
X -> A(-xt)A-')
of t h e automorphism
skew t h i s g i v e s a n a l g e b r a of t y p e
If
rn
if
C
n
symmetric t h i s g i v e s a n a l g e b r a o f t y p e
and
=
i s f i x e d and n o n s i n g u l a r ( i e . , c o n s i d e r t h e i n v a r i a n t s
A
where
x
Consider
m
= 2n
= 2n
is
A
and
A
and
is
j
[fl
and i f rn = 2 n + l
n '
i s symmetric t h i s g i v e s a n a l g e b r a o f t y p e Bn
If we
A
t a ~ e A
=
D
.
i n t h e first case
-1'
and
Y
A = I1
-1
i n t h e second and t h i r d c a s e s a n element
i s s u p e r d i a g o n a l i f and o n i y i f a d y p r e s e r v e s
E
( w i t h t h e u s u a l o r d e r i n g of r o o t s ) .
X
i n v a r i a n t s of
c a s e we g e t
GC
z>AX- It
Spm
A-l
(that is
serves the basic ingredients
t i o n of
'Om
SLm
.
X A X ~= A )
The autornorphisrn
B ,H
,
I n t h e first
G
C SO,
( F o r t h e proof t h a t e q u a l i t y
N
cr above pre-
of t h e B r u h a t decomposi-
From t h i s a B r u h a t d e c o m p o s i t i o n f o r
can b e i n f e r r e d .
.
i n t h e s e c o n d and t h i r d c a s e s
1957.)
xa
E x p o n e n t i a t i n g we g e t t h e
( r e l a t i v e t o a f o r m of maximal i n d e x ) .
h o l d s s e e Ree, T.A.M.S.
X0
a>
Spm and
By a s l i g h t m o d i f i c a t i o n o f t h e p r o c e d u r e ,
we c a n a t t h e same t i m e t a k e c a r e of u n i t a r y g r o u p s .
If
(c)
i s of t y p e
of a s p l i t C a y l e y a l g e b r a .
it i s t h e d e r i v a t i o n a l g e b r a
G2
The c o r r e s p o n d i n g g r o u p
is the
G
g r o u p of autornorphisrns o f t h i s a l g e b r a .
S i n c e t h e r e s u l t s l a b e l l e d (E) depend o n l y on t h e r e l a t i o n s
( R ) (which a r e i n d e p e n d e n t of t h e r e p r e s e n t a t i o n c h o s e n ) we may
e x t r a c t from t h e d i s c u s s i o n s o f a r t h e f o l l o w i n g r e s u l t .
Proposition:
0
(arE
x,(t)
9
U , H
P
,
Let
z, t
... b e
G
E
v
b e a g r o u p g e n e r a t e d by e l e m e n t s l a b e l l e d
such t h a t t h e r e l a t i o n s ( R ) h o l d and l e t
k)
defined a s i n
G .
(1) E v e r y e l e m e n t of
xa(ta)
a> 0
(2)
F o r each
w
,
W
E
can be w r i t t e n i n t h e
.
v
TT
form
v
U
w
write
a p r o d u c t of r e f l e c t i o n s .
(where
v
9
wa = wa(l) )
0
can be w r i t t e n
(where
u
C o r o l l a r y 1:
Suppose
of
G
G?
onto
v
U
E
G
v
such t h a t
.
Y
0
9
Define w w = u a m B
Then e v e r y e l e m e n t of
7
0
w w
a P"'
=
.*.
G
v
V
u h wWv
P
h
?
E
H
v
,v
i s a s above and
0
v
0
E
cp
cp(xa(t))= x a ( t )
UW)
.
i s a homomorphism
for all
a
and
t
Then:
( a ) U n i q u e n e s s of e x p r e s s i o n h o l d s i n (1) a n d ( 2 ) above.
.
(b)
ker
Proof :
(a)
get
x
c e n t e r of
cp
rr
Suppose
t
fT
=
P
q
t
c p I ~ i s a n isomorphism.
by a p p l y i n g
By Theorem 4
9
Hence
we g e t
cp
0
-0
and
<
(b)
Let
x
v
0
0
P
0
0
P
.
Y
P
u h ww v
D
0
cp(u ) =
,q
=v
P
so
O
P
O
= u h wwv
m("u
P
P
-9
h ww= h w
.
o k e r cp
P
u
= v
0
0
= 1 so
x
Y
= h
.
V m P
= u h ww v
rJ
F
cp(v ) = cp(?
0
so
W
h
Y
v
)
rnP
= h
0
w = 1
by ( R B ) .
u)
,,YN?
a
.
.
Then
P
Hence
for a l l
o(Gq
and
)
P
we
cp
)rn(xq)ww ' ~ ( v
P
1 = m(u )m(h ) wwrp(v ) o U H o w U w ; so
m(v ) = 1
N
ta - ta
P
Now i f
Applying
v ( u ) y ( h ) u wcp(v ) =
and Lemma 2 1
u =u
b
and by Lemma 1 7
0
Hence
.
xa ( ta ) = fT xa ( ta )
Tu
x
c H' .
G'
0
, ww = 1,
=rr
0
hc(ta)
Applying
qo
0
rp(u ) = 1
.
,
Then
we s e e
a
.
that
B
all
0
1
t <?,a> = 1
Hence x
commutes w i t h
xR(u) for
a a
and u , so i s i n c e n t e r of G '
To complete t h e proof
i t i s enough t o show t h a t c e n t e r of
P
k e r m C H ).
a
>
0
such t h a t
which c o n t r a d i c t s Theorem 4
wo
Then
h
be t h e element of
x =
xail
Wo
0
is diagonal,
G C H
x = u h w w v e c e n t e r of
If
then t h e r e e x i s t s
Let
.
9
u
0
.
Hence
wa
( f o r we have shown
G
<
0
and
.
w # 1
Then
w = 1 so
x = uh
=xa(l)x
.
W making a l l p o s i t i v e r o o t s n e g a t i v e .
i s b o t h s u p e r d i a g o n a l and subdiagonal.
i s d i a g o n a l , and a l s o u n i p o t e n t .
Since
Hence
u = 1 and
x =h e H
C
.
.'
Corollary 2:
Center
Corollary 3:
The r e l a t i o n s ( R ) and t h o s e i n
G
H
form a d e f i n i n g s e t of r e l a t i o n s f o r
If tKe r e l a t i o n s i n
Proof:
H
G
on t h e
H
h,(t)
.
a r e imposed on
H
0
m in
then
C o r o l l a r y 1 becomes a n isomoqphism by ( b ) .
C a r o l l a r y 4:
-
If
G
0
is constructed a s
from
G
,
k,
but using a perhaps d i f f e r e n t r e p r e s e n t a t i o n space
of
,
V
t h e n t h e r e e x i s t s a homomorphism
0
such t h a t
a
and
8
0
->
H
y
Clearly if
exists.
e x i s t s then
H
D
onto
0 h a ( t ) = h ( t)
G
for a l l
C:
exists.
8
Matching up t h e g e n e r a t o r s of
that the relations i n
0
i n place
0
such t h a t
.
t
Proof:
9 : H
G
Y
i f and o n l y i f t h e r e e x i s t s a
y(x,(t) ) = x,(t)
homomorphism
of
V
...
H
Conversely assume
P
and H
,
form a s u b s e t of t h o s e i n
we s e e
H
.
By
C o r o l l a r y 3 and t h e f a c t t h a t t h e r e l a t i o n s ( R ) a r e t h e same
.
for
G
P
and
G
,
t h e r e l a t i o n s on
a s u b s e t of t h o s e o n
xa(t),
...
in
3'
xa(t),
G
.
.
.
Thus
in
G
Y
form
exists.
H
So f a r t h e s t r u c t u r e of
i n t h e proceedings.
h a s p l a y e d a minor r o l e
To make t h e p r e c e d i n g r e s u l t s more p r e c i s e
we w i l l now d e t e r m i n e it.
We r e c a l l t h a t
ha(t) (a
X, t e k)
E.
a s m u l t i p l i c a t i o n by
Theorem 3 ( e ) ,
2
H
i s t h e group g e n e r a t e d by a l l
and
("*)
t<
' ' Y>~.
l i n e a r function
h,(t)
a c t s on t h e weight s p a c e
>
a
0
.
,u,
on
%
<
hi, a
j
>
= Gij
i s t h e h i g h e s t weight of
<
C l e a r l y , it s u f f i c e s t h a t
f o r a l l simple r o o t s
.
ai
.
Define
We s e e t h a t
,cr.
A l s o , we r e c a l l t h a t by
some i r r e d u c i b l e r e p r e s e n t a t i o n provided
for a l l
V
hi
hi,
> = ,!a(Ha) e Z +
< ::, a i > o Z+
I:, a
..., L
i = 1, 2 ,
by
o c c u r s as t h e h i g h e s t weight
of some i r r e d u c i b l e r e p r e s e n t a t i o n , and we c a l l
hi
a fundamental
weight.
Lemma 27:
(a)
The a d d i t i v e group g e n e r a t e d by a l l t h e w e i g h t s of
a l l r e p r e s e n t a t i o n s forms a l a t t i c e
L1
having
{hi]
as
a basis.
(b)
The a d d i t i v e group g e n e r a t e d by a l l r o o t s i s a sub-
lattice
Lo
of
L1
.
Moreover, (< a i , a
i s a relation matrix f o r
(c)
L~'LO
,
j
>)
i , j = 1, 2 ,
which i s t h u s f i n i t e .
The a d d i t i v e group g e n e r a t e d by a l l w e i g h t s o f a
f a i t h f u l r e p r e s e n t a t i o n on
Lo and
Ll
.
V
forms a l a t t i c e
L
v
between
...,
4,
Proof:
P a r t ( a ) i s immediate from t h e d e f i n i t i o n of t h e f u n d a m e n t a l
weights.
<
then
(b)
a i y ak
a
>
= cik
i
and
;r
0
with
.
Lo C_ LTi 2 L1
Remark:
#
ai = X
and
<
ai, a
j
> XJ. .
Xa # 0 t h e r e e x i s t s
is a root, then since
weight
ai = C c i j h . ( c i j e
J
i s a s i m p l e r o o t and
If
Xav
E
V,,,+a
.
0
a =
Hence
#v
(/s +
E
c)
(c)
If
V
f o r some
a)
,I
l
-
p
a
E
-,
A l l l a t t i c e s between
Lo
and
can be r e a l i z e d as
L1
i n Lemma 27 ( c ) by an a p p r o p r i a t e c h o i c e o f
V
.
In particular,
hJ = Lo
if
V
c o r r e s p o n d s t o t h e a d j o i n t r e p r e s e n t a t i o n , and
LV = L1
if
V
c o r r e s p o n d s t o t h e sum o f t h e r e p r e s e n t a t i o n s
h a v i n g t h e fundamental w e i g h t s a s h i g h e s t w e i g h t s .
Lemma 2 8 ( S t r u c t u r e of H ) :
( a ) For each
(b)
H
,
a
ha(t)
i s m u l t i p l i c a t i v e a s a f u n c t i o n of t
i s an Abelian group g e n e r a t e d by t h e
(with
hi(t)
=
h
a4
hi(t)9s
.
(t))
&
(c)
i =1
hi(ti)
= 1 i f and o n l y if
4, < p , a i >
hi(ti)]
ti
= 1
i= 1
i =l
hence i s f i n i t e .
t.
(dl
The c e n t e r o f
for all
Proof:
1
p
E
G = {
L")
( a ) , ( b ) , and ( c )
f o l l o w f r o m (::)
.n
above.
( a ) and ( c )
11. ( Ha )
kr(XniHi )
< /.(,a> =
=t
immediate and ( b ) r e s u l t s from t
Eni < ! h a i >
For ( d ) , we n o t e
if Ha = Xn. H
1
i *
ti
are
.
rr
4,
that
<
4,
hi(tik,)
i=l
commutes w i t h
x (u)
i f and b n l y i f
B
rr
@,ai>=1
ti
i=l
by Lemma 1 9 ( c ) .
Corollary:
(a)
If
LV = Ll
,
t h e n every
(b)
If
LV = L o
,
then
4
C o r o l l a r y 5 (To Theorem
G
u s u a l and l e t
t h e same
0
,
k
G
as
cp(xa(t)) = x a ( t )
P
where
G
xa(t)
.
be a C h e v a l l e y g r o u p a s
but using
V
0
i n p l a c e of
a, t
for a l l
corresponds t o
and
xa(t)
in
G
0
m:
G
ker o
L
t h e n ' t h e r e e x i s t s a homomorphism
P
that
1
has center
Let
):
can be w r i t t e n u n i q u e l y
be a n o t h e r C h e v a l l e y g r o u p c o n s t r u c t e d from
and
L v 9 2 LV
P
G
h e H
v
.
If
P
->
H
->
V
.
If
such
G
C e n t e r of
G
P
,
L
o i s a n isomorphism.
then
Proof:
8 : H
T h e r e e x i s t s a homomorphism
such t h a t
0
@hi(t) = h i ( t )
by Lemma 2 8 ( c ) .
a
If
i s any r o o t and
m
, ni
Ha = I: niHi
for
.
h i ( t)
9 exists.
e
Hence
then
ha(t ) =
v
8ha(t) = ha(t )
.
hi(t )
$om
I
= i d G ? : go$ =
->
idG ,
: G
G
v
and
and s i m i l a r l y
By C o r o l l a r y
By C o r o l l a r y 1, k e r cq C C e n t e r of
we have a homomorphism
Hence,
11
2 ,
suTh t h a t
G
P
.
4 t o Theorem 4
If
L. = L
v
P
v' '.
i ( x a ( t )) = x a ( t )
i s a n isomorphism.
?
.
,
We c a l l t h e C h e v a l l e y g r o u p s
Lo
t o the lattices
and
If
group r e s p e c t i v e l y .
t o the l a t t i c e
homomorphisms
We c a l l
ker a
LV
G = GV
if
G
i s a Chevalley group c o r r e s p o n d i n g
-> Gv
of 5 , and
a : G1
such t h a t
t h e fundamental g r o u p
The c e n t e r o f t h e u n i v e r s a l group, i . e . ,
k =
,
->
t h e fundamental
H O ~ ( L ~ / L k~ *, ) .
C e n t e r of
GV
.
L ~ / L ~
L ~ / L, ~ and t h e
N
GV = L1/L;,
I n t h e f o l l o w i n g t a b l e , we l i s t some i n f o r m a t i o n known
a b o u t t h e l a t t i c e s and C h e v a l l e y g r o u p s o f t h e v a r i o u s
algebras
x:
G
we s e e
t h e l a s t group i s isomorphic w i t h
Also i n t h i s c a s e t h e
fundamental group o f
p : GV
and
.
G
group of t h e a d j o i n t group i s isomorphic t o
E.g.,
and
t h e n by C o r o l l a r y 5, we have c e n t r a l
and , p
a
k e r p = c e n t e r of
Exercise:
,
L1
corresponding
1
t h e a d , j o i n t group and t h e u n i v e r s a l
Go
Lie
Type of
Spin
4n
0 -
GV i s a C h e v a l l e y group o t h e r t h a n
Here
i s t h e c y c l i c group o r o r d e r
group,
PG
Spinn
n
i s t h e s p i n group,
,
G
.
To o b t a i n t h e column headed by
(< ai, a
relation matrix
example, t h a t
j
,,; .
g,
<*,
An-1
)
SLn
:. .
.1.'.,',
we l e t wi
=hi (E
jj
t o d i a g o n a l form.
To show, f o r
2 = s&,
+
**.
- Ej+l,j+l
+W
of t y p e
... , a n ) ->
be t h e weight oi : d i a g ( a l ,
Then if hi - cu + m 2
1
hi(Hj)
>)
one r e d u c e s t h e
L1/LC
i s t h e u n i v e r s a l g r o u p of
.7,
,,?-4t
j
.
) = Fij
L ~ / L ~i s isomorphic t o
I
I
If
Exercise:
of
G
,
(a)
Each
(b)
G
(c)
it follows t h a t
G1,
G2,
..., Gr
G
and
G = G1G2
'.*
2.
subgroups
,
Gr
then:
.
i s u n i v e r s a l ( r e s p e c t i v e l y a d j o i n t ) i f and o n l y i f
G,
is.
I n each c a s e i n ( b ) , t h e p r o d u c t i n ( a ) i s d i r e c t .
Corollary 6:
: SL2
i s normal i n
Gi
Since t h e
i n t h i s case.
i s a Chevalley group,
G
n
c o r r e s p o n d i n g t o indecomposable components of
each
va
Z
If
a
.
Hence t h e f u n d a m e n t a l w e i g h t s
is g e n e r i c a l l y c y c l i c of o r d e r
SLn
ai
l S i < , n - 1 , we have
i
a r e i n the l a t t i c e associated with t h i s representation.
c e n t e r of
,
G1
is t h e s p e c i a l orthogonal
'On
i s t h e s i m p l e c t i c group, a n d
Spn
d e n o t e s t h e p r o j e c t i v e group of
and
Go
is a r o o t , t h e n t h e r e e x i s t s a homomorphism
=x
-a
(t4
k e r aa = 11)
either
SL2
Proof:
Let
[X, Y] = H
.
.
yI=-
[ -1
O
0
{+ 1 )
or
or
a
PSC2
,
.
[H, X] = 2X
v e c t o r space
x ->
V
9
i s isomorphic t o
so that<xa,
and
[H, Y] =
i],
Y
V
-
-
and a r e p r e s e n t a t i o n
on t h e same v e c t o r space
X, Y
1 spanned by
be of rank
representation
X
2Y
.
H
-
->
ker u C
Exercise:
{t 1)
If
Now
9 4 S i m p l i c i t y of
2)-
has a
s t 2 on a
-:]as
, Y ->
XG
with
X -a
, H ->
a s t h e o r i g i n a l r e p r e s e n t a t i o n of
Ha
x.
e x i s t s and
CQ
by C o r o l l a r y 5.
i s u n i v e r s a l , each
G
H
and
SL2 i s u n i v e r s a l , t h e r e q u i r e d homomorphism
Since
has
'
.
G
aCl i s an isomorphism.
The main purpose of t h i s s e c t i o n i s t o
prove t h e f o l l o w i n g theorem:
Theorem 5 ( C h e v a l l e y , D i c k s o n ) : . Let
be an a d j o i n t group and
G
i s simple ( X indecomposable).
assume
i s n o t of t y p e
.
A1,
B2,
Then
G
or
G2
.
If
If
(kl = 2
(kt = 3
,
,
assume
assume
A1
Remark:
The c a s e s excluded i n Theorem 5 must be excluded.
,
then
G
has
is not
i s simple.
of type
lkl = 2
2?
a3, Q6 ,
SU ( 3 )
3
If
a s a normal subgroup
:
%&"',
9 4 S i m p l i c i t y of
.
G
97
The main purp6se of t h i s s e c t i o n i s t o
prove t h e f o l l o w i n g theorem:
Theorem 5 ( C h e v a l l e y , D i c k s o n ) :
assume
2 i s simple
i s n o t of t y p e
.
(I)
A1,
B2,
Then
G
Let
be a n a d j o i n t group and
G
indecomposable).
or
G2
.
If
1 kl
1 kl
If
= 3
,
= 2
,
assume
assume
of t y p e
A1
Remark:
The c a s e s excluded i n Theorem 5 must be excluded.
lkl = 2
,
then
G
has
2
is not
i s simple.
a3,
,
SU3(3)
If
a s a normal subgroup
of i n d e x
2
(kI = 3
of
if
and
xis
of i n d e x 2.
G
i s of t y p e
A1,
B2,
respectively.
G;
a4
If
of t y p e
A1,
Here
d e n o t e s t h e a l t e r n a t i n g group.
a
then
i s a normal subgroup
A proof of Theorem 5 e s s e n t i a l l y due t o Iwasawa and T i t s
w i l l be g i v e n h e r e i n a sequence of lemmas.
Lemma 29:
Let
be a Chevalley group.
G
w
If
E:
W , w
=w w
a B
...
i s a minimal e x p r e s s i o n a s a product of simple r e f l e c t i o n s , t h e n
-1
w, , w g ,
e G1 , t h e group g e n e r a t e d by B and wBw
...
Proof:
-
.
We know
<
w-l a
0
by t h e rninimality of t h e e x p r e s s i o n
p
=
~ h u s , wa e G~
.
( s e e Appendix 11.19 and 1 1 . 2 2 ) .
1WP -
G~ 2 W x B W-
wawBw -1wa-1
X -a
.
Hence i f
and s i n c e l e n g t h
GI
waw
<
-
w-l a
>
0
, then
Since
length w
,
we may complete
t h e proof by i n d u c t i o n .
Lemma 30:
If
G
a g a i n i s any Chevalley group, i f
T
i s a subset
WT i s t h e group g e n e r a t e d by a l l
= 'd
BwB , t h e n
we
of t h e s e t of s i m p l e r o o t s , i f
w
a e
a'
,
and i f
G
7~
i s a group.
(a)
GT
(b)
The 2'
c )
Every subgroup of
Proof:
(b)
T
groups so obtained a r e a l l d i s t i n c t .
G
P a r t ( a ) f o l l o w s from
Suppose
T
,
0
7~
containing
BwB
B
i s e q u a l t o one of them.
BwaB C
- B wwaB W BwB
.
a r e d i s t i n c t s u b s e t s of t h e s e t of simple
r o o t s , say
w
if
a
.
WT
B
Thus
independent.
#
waa
wa
Hence,
.
4 1~
rT a
E
NOW
a
and
wa = a
+
cg8
s i n c e simple r o o t s a r e l i n e a r l y
,
WT
t$
-
Be*
,
wa
a =
W,
W*,
# W, ,
and
#
GT,
GT
since
d i s t i n c t e l e m e n t s o f t h e Weyl group c o r r e s p o n d t o d i s t i n c t double
(c)
cosets.
Let
b e any subgroup c o n t a i n i n g
A
= [a la s i m p l e , we
r
A
3 G,
w
B
w
=
,
A
Since
w e
w
...,
W' ,
E
.
A]
We s h a l l show
BwB and A
weW
t o g e t A C G,
G =
implies
w w
a B
reflections.
B
G,
By Lemma 29,
w
and
B
w,,
wg,
G
.
.
Clearly,
Let
w
B
A
,
w a s a product of simple
B
A
.
Hence,
a , p,
... r
GT ,
A g r o u p c o n j u g a t e t o some
subgroup o f
...
A = G*
Set
2 B , we need o n l y show
.
a minimal e x p r e s s i o n of
.
B
GT
is called a parabolic
We s t a t e w i t h o u t proof some f u r t h e r p r o p e r t i e s
of p a r a b o l i c subgroups which f o l l o w from Lemma 29.
(1) No two
a r e conjugate.
G,'s
(2)
Each p a r a b o l i c subgroup i s i t s own n o r m a l i z e r .
3
G7T
n G*'
=
(4) B U B w B (w
B
G * ~ * q
W)
i s a group i f and o n l y i f
w = 1 or
w
i s a simple r e f l e c t i o n ,
Example:
If
G = SLn
,
then
r
corresponds t o a p a r t i t i o n
of t h e n x n m a t r i c e s i n t o b l o c k s w i t h t h e d i a g o n a l b l o c k s b e i n g
s q u a r e m a t r i c e s , C l e a r l y , t h e r e a r e 2n-1 p o s s i b i l i t i e s f o r such
,
prtitions.
i s t h e n t h e s u b s e t of
GTT
SL,
of m a t r i c e s whose
subdiagonal blocks a r e zero.
?
c
~emma31: Let ,
. b e s i m p l e and l e t
f 1 is a
group.
If
proof:
We f i r s t show N
I
N
~ . = u h ,U E U ,
w
E
t
v
#
t
,
.
(B
.
H
If
t, t
E
k
T
, x a ( t -t) f 1 ,
N
E
,
IT
.
We must show
does n o t .
Since
N
bwa
a
we s e e
E
N
with
,
NOW
I
I
<
w w w
a, $
that
w w w
a P a
= W
P U P
>f
0
Then
,
N(w w w )
$ a @
.
Hence e i t h e r
w w
a P
where
Y
y
E
B
=
wBa
Z .
then
N ,
#
h
1
.
v
Hence
.
.s f @
xa(t))
(h,
NB =. G
blwab2
w
E
= a
by Appendix 11.20.
N , bi
E
bw w - l ~ N
a p
P
W
with
-<
Suppose i t
Z
is
a,
p> p
N(wgwawg)
Hence
a r e b o t h e x p r e s s i o n s of minimal l e n g t h .
a
E
s
, B
B
E
,
& q
then
n (BwawP BVBwgwawg
w w w
B a P
or
T
for
'T
Also s i n c e
a,f3
i s n o t a s i m p l e r o o t and
>
-3
E
.
hxa(t)h-I = x,(t )
and
By Lemma 3 0 ( c ) ,
Let
.
b = b2bl
by Lemma 2 5 ( b ) .
.
P
not orthogonal t o
u = 1,
If
indecomposable, we c a n f i n d s i m p l e r o o t s
and
1f x
contains a l l simple r o o t s .
n-
d_ B ,
E
and
NB = G
and w e a r e back i n t h e f i r s t c a s e .
We now p r o v e t h e lemma.
some
a
then
t h e n f o r some
'
1
center
v
c- B
f 1,
u
,
G
N
Suppose
a contradiction.
f o r some
v
x a ( t -t)
=
E
normal subgroup of
i s a d j o i n t , it h a s
G
Since
.Irith
4B
w xwa1
W
h
be t h e a d j o i n t Chevalley
G
w w
a P
E
.
W
n-
.
Since
f 1,
and
By Lemma 29,
so
B)
51
% ,a
a 32:
contradiction.
and
If
ed group of
G
G
Thus,
i s t h e s e t of a l l simple r o o t s
?r
a r e as i n Theorem 5 , t h e n
G =G
P
,
the
.
Before p r o v i n g Lemma 32, we f i r s t show t h a t Theorem 5
w s from Lemmas 3 1 and 32.
.
By Lemma 31,
e r i v e d group and
.-,5c,.
:$;?r
. ., .
. .:
,
, .:,
,,,*,..
B/B
so
nN
G/N
#
1 be a
2
B/B
nN
i s solvable.
a normal subgroup
.
Now
Hence
G/N
equals
G/N = 1
I n s t e a d o f p r o v i n g Lemma 32 d i r e c t l y , we prove t h e
(.
...>:
.-
NB = G
N
Let
..
,
s t r o n g e r s t a t ernent :
; following
; lemma
i s a s i n Theorem 5
32 : If
group
0
then
G
h o l d s i n any
= G
i n which t h e r e l a t i o n s ( R ) h o l d , i n f a c t i n which t h e
G
relations:
hold.
Proof:
Since
every
a
i s g e n e r a t e d by t h e j f a v s we must show t h a t
G
G
0
.
We w i l l do t h i s i n s e v e r a l s t e p s , e x c l u d i n g a s
we proceed t h e c a s e s a l r e a d y t r e a t e d .
The f i r s t s t e p t a k e s u s
almost a l l t h e way.
(a)
Assume
lkl
>4 .
We may choose
t
E
kQ
,
t2 # 1
.
Then
(ha(t)
2
2
x,(u)) = ~ , ( (-1)~)
t
: Since
and .u
a
g, -C G .
0
a r b i t r a r y , every
By ( a ) we may henceforth assume t h a t t h e rank
at least
and t h a t
2
are
(k( = 2
.
or 3
is
t
By t h e c o r o l l a r y t o
mq ,
Lemma 1 5 , we may w r i t e t h e r i g h t s i d e of ( A ) as x
t h e f a c t o r with
B + Y ( ~ B , Yt u )
i = j = 1 having been i s o l a t e d .
We w i l l use t h e
fact
No
that
(*)
=
. ,Y
+
-
(r
+
1)
with
r = r(j3,y)
as in
Theorem 1, t h e maximum number of t i m e s one can s u b t r a c t
from
y
and s t i l l have a r o o t .
(b) Assume t h a t
a
i s a r o o t which can be w r i t t e n
B and
so t h a t no o t h e r p o s i t i v e i n t e g r a l combination of
.a r o o t and
( A ) with
N
#
B,Y
=1
0
n9 .
.
Then
a C G'
,
+
f3
y
A4,
T h i s covers t h e f o l l o w i n g c a s e s :
DL, E L
.
,
a
(2)
BC(*>3)
(3)
CL ( & 2 3 ) , a
(4)
Fq
(5)
G2
,
a
long ; B2, a
short; or
a
long,
longand
lkl = 3
.
lkl = 3
.
long.
To s e e t h i s we use t h e f a c t t h a t a l l r o o t s of t h e same
l e n g t h a r e congruent under t h e Weyl group, imbed
a
i n an
a p p r o p r i a t e r o o t system based on a p a i r of simple r o o t s , and
use ( * ) .
is
a s f o l l o w s a t once from
(1) If a l l r o o t s 'have t h e same l e n g t h :
types
y
I n a l l c a s e s b u t t h e second c a s e s i n ( 2 ) and ( 3 )
f3
A2
t h i s system can be chosen of t y p e
B2
type
P
with
, while
a
of t h e same l e n g t h a s
and
p
with
and
roots
y
i n t h o s e c a s e s it can be chosen o f
short roots.
y
Because oi t h e e x c l u s i o n s i n t h e theorem, t h i s l e a v e s
the following cases:
( 6 ) B t ( t >-- 2 )
( 7 ) G ,-l ,
(c)
If ( 6 ) o r ( 7 ) h ~ l l d st ~
hen
ip + jy
0 :
Then
X
hence so d c e s
(d)
with
because
/kl
long,
(i,j
us
,
lonz,
+
y
=-
2
.
x a -c
5,
G
Y
~ n b o t hof
so t h a t
y
a =p
+
,
y
l o n g and
belongs t o
G
P
(7)
y short, i n
by c a s e s a l r e a d y t r e a t e d ,
by ( A ) .
C
s h o r t , and
G
P
.
a = p + 2y
8, y
Choose r o o t s
.
Since
p+Y
i s s h o r t , o u r a s s e r t i o n w i l l f o l l o w from
"f
all
p o s i t i v e i n t e g e r s ) a r e l o n g , and
If ( 8 ) h o l d s , t h e n
C
C12
Gy
#
0
( A ) , hence from t h e n e x t leirma.
Lemxa 33 :
$
a
?
i n ( 6 ) we can choose
both s h o r t .
in
3)
c a s e s we can f i n d r o o t s
other roots
#
jkl = 3 .
sh3rtJ
ct i t 2
.,?ese
BY
short.
(8)
.L 7
N
o:
a
l o n g and
p
If
y
and
Y
form a simple system of t y p e
short, then
i xp(t),
D2
with
x y ( u ) ) = ~ @ + ~ ( tf u- ) xP+2y('
tu')
.
By Lemma 1 4 , we have
x Y iu)Xg ?(u)
=X
Here
+uN
B
Ny , =
-+ 1
-1
=
-7
Y ,fl iip+y
and
exp ( a d u X y ) X P
iu
N
2
=
N ~ B ',~ Y , B+Y /2
+
V,P+y we m u l t i p l y t h i s e q u a t i o n by -t
p
it a roo
2
since
,
exponentiate, observe t h a t t h e
t h r e e f a c t o r s on t h e r i g h t s i d e cormute, and t h e n s h i f t t h e first
of them t o t h e l e f t , we g- e t Lemma 33,
The proof of Theorem 5 i s now compl
I n t h e c o u r s e o f t h i s d i s c u s s i o n , we have e s t a b l i s h e d
the following r e s u l t ,
If
Corollary:
i s indecomposa.ble and of
C
9 and
is any r o o t , t h e n t h e r e e x i s t r o o t s
integer
I
n
such t h a t
( A ) of Lemiia 32
7
a = R
,
k
nv
and
c,-
1 and i f
and a p o s i t i v e
y
#
>
0
i n the relations
.
C o r o l l a r ~(To Theorem 5 ) :
based on
+
rank
If
[kl >_ 4
and
G
i s a Chevalley group
thel, evt;, y d l v a b l e no:-ilLdl subgroup of
G
is central
and hence f i n i t e .
Proof:
S i n c e t h e c e n t e ~of a Chevalley group i s always f i n i t e
by Lemiia 2 8 ( d j j we need o n l y prove t h e f i r s t s t a t e m e n t .
assume
G = Gg
Theorem 4
v
,
,
Also we may
t h e a d j o i n t group, s i n c e by C o r o l l a r y 5 t o
t h e r e i s a homomorphisrn
Q
of
G
onto
G
0
.with
er y
c e n t e r of
= G1 G 2 *
*
G
and
where
Gr
Go
h a s c e n t e r 1.
Gi
i = 1, 2
Now we may w r i t e
. ,r
is the adjoint
rcup c o r r e s p o n d i n g t o a n indecomposable subsystem of
h e o r e n 5, each
Gi
i s simple.
C
.
By
Thus any normal subgroup o f
A
a product of some of t h e
:$
Gifs
.
If i t a l s o i s s o l v a b l e , t h e
product i s empty and t h e subgroup i s 1.
G
is
56
>
'
I
g5.
5
Chevallzy g ~ o u p sand a l g e b r a i c groups.
3
The s i g n i f i c a n c e of t h e r e s u l t s s o f a r t o t h e t h e o r y of semisimple a l g e b r a i c groups w i l l now be i n d i c a t e d .
Let
k
A subset
be a n a l g e b r a i c a l l y c l o s e d f i e l d .
?C
-
s s a i d t o be a l g e b r a i c i f t h e r e e x i s t s a s u b s e t
p].
p e
.. ,vn)
V = [v = (vl,.
such t h a t
E
.
k[%,
...,xn ] I p ( v l , . ..,vn)
Let
r = nz + 1
D(x) = 1
- xo
i s a subgroup of
closed f i e l d
ko
.
.
k
.
kr
GL,(k)
, and
G
i s a s u b f i e l d of
kn
v c kn
= 0
for a l l
D(x)
E
GL,(~)
Then
.., v n )
f o r some
(c)
k
,G
Diagonal subgroup,
(d)
= G
h)
The groups i n ( a )
E
k[x0;xijll
=
[v E krlD(v)
= o ] i s an
G
n
.
(b)
YLn( k )
..,vn) V) .
< i, <- by
(vl,.
and some a l g e b r a i c a l l y
2
kn +l
i s a n a l g e b r a i c s u b s e t of
,
(a)
for a l l
= 0
i s a rnatric a .e b. r a i c group if
G
.,%I
,
s e t Ik(V] =
- -ko
i s d e f i n e d over
has a b a s i s of polynomials w i t h c o e f f i c i e n t s i n
Examples :
k[xl,.
a r e t h e c l o s e d s e t s of
For
Define
det(xij)
a l g e b r a i c s u b s e t of
If
E
The a l g e b r a i c s u b s e t s of
t h e Z a r i s k i t o p q ~ o ~ony kn
(p
n
k Ip(vl,.
C
- kn
V
:I)
ko
if
Ik(G)
.
Superdiagonal subgroup,
= Ga = a d d i t i v e g r o u p ,
n =multiplicativegroup,
(f)
Sp2,
,
any f i n i t e subgroup,
-
( e ) a r e defined over t h e prime f i e l d .
Whether
S P ~ son
~ , a r e o r n o t depends on t h e c o e f f i c i e n t s of t h e d e f i n i n g
forms,
The groups i n ( h ) a r e n o t connected i n t h e
Zariski topology, t h e others a r e .
57
A map of a l g e b r a i c groups
G
cp:
->
i s a homomorphism
H
if it i s a group homomorphism and each of t h e matric c o e f f i c i e n t s
rp(g)ij
->
ep: G
y/:
i s a r a t i o n a l f u n c t i o n of t h e
H>-
H
i s a n isomorph*
G
such t h a t
phism cp : G
->
cpy
H homomorphism
i f t h e r e e x i s t s a homomorphism
= i d H and
i s defined over
H
.
gij
Y/rp = i%
A homomor-
i f each of t h e r a t i o n a l
k0
f u n c t i o n s above has i t s c o e f f i c i e n t s i n
.
ko
Except f o r t h e l a s t a s s e r t i o n , t h e following r e s u l t s a r e proved
i n ~ e s i n a i r eChevalley (1956-8) , ~ x ~ o s e3'.
( i ) Let
G
be a m a t r i c a l g e b r a i c group,
Then t h e f o l l o w i n g
a r e equivalent :
%'
. Lr Ar
(a)
G
i s connected ( i n t h e Z a r i s k i t o p o l o g y ) .
(b)
G
i s i r r e d u c i b l e ( a s an a l g e b r a i c v a r i e t y ) .
(c)
Ik(G)
d-.
<+
i s a prime i d e a l .
(ii) The image of a n a l g e b r a i c group under a r a t i o n a l homo-
morphism i s a l g e b r a i c
.
(iii) A group generated by connected a l g e b r a i c subgroups i s
a l g e b r a i c and connected
nected).
(e.g.
(a)
-
(g)
a r e con-
It i s d e f i n e d over t h e p e r f e c t f i e l d
ko
i f each of t h e subgroups i s .
If
G
i s an a l g e b r a i c group, t h e r a d i c a l of
t h e maximal connected s o l v a b l e normal subgroup.
if
(1) r a d G = (1) and
(2)
G
G
G
i s connected,
(rad G)
is
is semisirpple
For t h e remainder of t h i s s e c t i o n we assume t h a t
b r a i c a l l y closed,
;
k
,
-
grciup
b a s e d on
,
t . .
.
lc
i s t h e " prime f i e l d ,
0
M the lattice.
and
.
G
k
i s alge-.
i s a Chevalley
(Since a change of b a s i s
is g i v e n by polynomi.als with i n t e g r a l c o e f f i c i e n t s we may
in M
speak of a b a s i s over
?heorem 6:
M ,)
With t h e preceding n o t a t i o n s :
(a)
G
i s a semisimple a l g e b r a i c group r e l a t i v e t o
(b)
B
is a maximal connected s o l v a b l e subgroup (Bore1
M
.
&-
FOUP
,
(c)
i s a maximal connected d i a g o n a l i z a b l e subgroup
H t
i
(maximal t o r u s ) .
(d)
i s t h e normalizer of
N
( e ) G , B, H,
and
N
H
and
N/H 2 W
a r e a l l d e f i n e d over
.
ko
relative t o
M .
Remark:
1
and
B
(a) B
(b)
H
a r e determined by t h e a b s t r a c t group
G;
i s maximal s o l v a b l e and has no subgroups of f i n i t e
H i s maximal n i l p o t e n. t and every subgroup of f i n i t e l a d e x
.
i s of f i n i t e index i n i t s normalizer.
Proof of Theorem 6:
(a)
Map Ga
This i s a r a t i o n a l homomorphism.
->
#,
So s i n c e
by
Ga
x
,
:
t
->
x,(t)
i s a connected
.
Xa .
, ~ g e b r a i c group s o i s
Let
R = rad G
.
Since
Hence
i s s o l v a b l e and normal it i s f i n i t e
R
by t h e C o r o l l a r y t o Theorem 5.
R
s
1
, and
hence
( b and c )
G
Since
i s a l s o connected
R
i s semisimple.
4,
Gm
is t h e image of
H
i s a l g e b r a i c and connected..
G
(t1,-*-,tC
->)
under
4
fl h i ( t i )
is
and hence i s a l g e b r a i c and connected; s o B = U H
i-1
connected, a l g e b r a i c , and s o l v a b l e . Let G1 3 B
Then
-
(some simple r o o t
G1 3 B wa B
, so
a)
hence by C o r o l l a r y 6 of Theorem 4'
( b ) holds.
#
Hence
(d)
is c l e a r .
Lemma 34:
.
nontrivially)
Let
(a)
%,
Fa
=
ko
an isomorphism over
ko
( b ) 3, i s d e f i n e d over
ko
Proof:
-
Let
Choose
vi
so that
Xavi
choose
v
so that
cij
j
i s of weight
( c ) holds.
and ha= [ h a ( t ) l t
[ x a ( t ) l t E kl
i s d e f i n e d over
{vi]
30
G
To prove ( e ) it s u f f i c e s by (iii)t o prove:
homomorphism over
j
U
i s a maximal connected d i a g o n a l i z a b l e subgroup of
H
Then:
, and
#-a>
i s a maximal connected d i a g o n a l i z a b l e subgroup of
H
( b y a theorem i n Chevalleyts jgminaix-e);
1
<xa,
i s n o t s o l v a b l e and hence
G1
( f o r any l a r g e r subgroup must i n t e r s e c t
B
v
>-
G1
.
p + a .
it follows t h a t if a i j
#
#
0
0
M
, then
.
Since
is the
If
xa:
.
and
ha:
Ga
->
is
$a
Gm ->h
a
.
ko
be a b a s i s o f
and
k"]
E
is a
formed of weight v e c t o r s .
write
vi
x,(t)
( i ,j )
Xavi = 2 c .
i s of weight
.V
15 j
p
-
, and
, then
2 2
= 1 +t X a + t x a / 2 +
m a t r i c coordinate
* * =
(i# j)
function then
x,(t)
i~
.
k
in
a i j ( % ( t ) ) = c .t
a r e polynomials o v e r
0
A l l o t h e r c o e f f i c i e n t s of
, hence
t
)(a
This s e t of polynomial r e l a t i o n s d e f i n e s
.
k,
1
b-a
i j : x a ( t ) ->
Now
t
also i n
aij
a s a group over
ia
i s a n i n v e r s e of
ij
x,
map
i s an isoinorphism over
.
ko
,
so the
The proof of ( b ) i s l e f t
a s an excercise.
We can r e c o v e r t h e l a t t i c e s
a s follows.
~
t
Let
.~
~
.
L
H
mined by
by
.
H
A
a(h) =
.
, the
h = mhi(ti)
ntia(Hi)
.
A
then
There e x i s t s a
h
L
0'
.
H
) =
.
ko
$1
X,'s
The
G
$(wi(ti)
by
Gm
generates
a r e deter-
!J
---->
A
c-;
A
h x,(t)hel
xa(e(h)t)
=
.
is c a l l e d a global
a
->
-->
H
c h a r a c t e r group of
A
t h e l a t t i c e g e n e r a t e d by a l l
Lo
:
from t h e group
a s t h e unique minimal u n i p o t e n t subgroups normalized
If
Sxercise:
A
3efine
L
T(h i s ~i s ia c)h a r a c t e r d e f i n e d over
A
a lattice
e L
;i
and
=o
.
Then
A
a
h
->
a
.
h
Lo
W-iscmorphism:
, and
i s g i v e n by t h e a c t i o n of
a
CL
L
Define
where
A
Lo
--
.
A
->
L
such t h a t
(The a c t i o n of
A
W
on
L
with
L
on t h e c h a r a c t e r g r o u p ) .
N/H
We summarize our r e s u l t s i n :
E x i s t e n c e Theorem:
-.-Lo
C
L
C L1
(where
Given a r o o t system
Lo
and
L1
C
,a
lattice
a r e t h e r o o t and weight l a t t i c e s ,
r e s p e c t i v e l y ) , and an a l g e b r a i c a l l y c l o s e d f i e l d
e x i s t s a semisiinple a l g e b r a i c group
L0
and
L
G
lc
d e f i n e d over
, then
k
there
such t h a t
a r e r e a l i z e d a s t h e l a t t i c e s of g l o b a l r o o t s and char-
a c t e r s , r e s p e c t i v e l y , r e l a t i v e t o a maximal t o r u s .
Furthermore
61
ga,...
G,
can b e t a k e n o v e r t h e prime f i e l d .
The c l a s s i f i c a t i o n t h e o r e m , t h a t up t o
k-isoixorphism
every
'%
!!- semisimple a l g e b r a i c group o v e r
k
.&
... ,
h a s been o b t a i n e d a b o v e , i s
<???
$!
+
.g
. ,.
(See Grninair ChevaJJey, 1956-8).
much m o r e d i f f i c u l t .
"
.g,
-
?kF- %- n ez
\
#
*jq,
:? .,
,.'+,,
.+,
.+..:, .
...
,:
We r e c a l l t h a t
{H
=
72
%[p(H) E
E
for a l l
p
Lj
E
.
-<
:
....
..
,.....
.., ...
...
;;,.;
I..
Lemma 35;
over
H
fl(Hi)
=
t
t
H1
v
1
k be a l g e b r a i c a l l y c l o s e d ,
Let
a basis for
.
v E V
for
I-'-
"
1
> t 4-)
>
( tl,
Then t h e map
a C h e v a l l e y group
t
Define
rp:
4,
Gm
by
hi
->
H
i s a n isomorphisin over
)
j=
z
.
G
t
hi(v)
g i v e n by
of a l g e -
k
0
b r a i c groups.
Proof:
-
T
Hi = X n 1
. .H
Write
ni j
I1
{ti]
tT.=
J
such t h a t
divisible).
Tti
-!
I
t
.
t
itj]
Given
we can f i n d
L
.
iJ
1
n h .(t.)
i J J
J
Then
E
d e t ( n . . ) # 0 and kq. i s
1J
a c t s on V
as m u l t i p l i c a t i o n by
(for
P
$(Hi)
pi , i . e . a s n h i ( t i ) . T h i s shows t h a t cp
o n t o i! . C l e a r l y rp i s a r a t i o n a l ii1agi2ing d e f i n e d
. L e t { p i ] b e t h e b a s i s of L d u a l t o { H i ] l ( i . e .
=
J
maps
G~m
over
k
1
t
;;. (H . )
J
1
0
= 6. .)
13
I
=
ti
2
SO
Theorem 7:
CP
.
,
-..1
Let
prime s u b f i e l d .
Write
pi=
.Z n p .
,LELP
e x i s t s and i s d e f i n e d over
k
Let
,P(Hj)
,
lc
0
.
j
j
be a n a l g e b r a i c a l l y c l o s e d f i e l d and
G
np
1
ko
be a C h e v a l l e y group p a r a n e t r i z e d by
and viewed a s a n a l g e b r a i c group d e f i n e d o v e r
Then;
n(n"t
Then
ko
a s above.
the
k
62
(a)
U-HU
(b)
If
i s a n open s u b v a r i e t y of
v'
xa(ta)nhi(ti)
a
0
a
of v a r i e k i e s over k
(a)
t
His
1
Xas
and
xYi
set
xa(ta)
i s an isomorphism of
.
We c o n s i d e r t h e n a t u r a l a c t i o n of
(Yl , Y 2 , .
.., Y r )
over
such t h a t
Y1 =
Ax,(u >
relative t o a basis
of
YO
0
Proof:
.
ko
n i s t h e number of p o s i t i v e r o o t s , t h e n t h e map
kn
k*.L
x kn ->
U-HU
d e f i n e d by
CP:
-
d e f i n e d over
G
a . . (x)Y
= Z,
.
and t h e n
j
1~
k
made up of p r o d u c t s
0
.
0)
,a
d = afl
An
on
G
x
For
E
f u n c t i o n on
we
G
over
G
.
We c l a i m t h a t x E U-HU = U-B i f and o n l y i f d ( x ) # 0
ko
Assume x E U-B
S i n c e B f i x e s Y1
up t o a nonzero m u l t i p l e
.
and if
that
u
d(x)
#
uX,
then
U-
E
0
.
x
If
E
Xa +
U- w B
E
%+
w
with
d(x) = 0
c o n s i d e r a t i o n s show t h a t
x< h t ( a )kXg
ht(p)
.
If
t i v e r o o t s n e g a t i v e t h e n by t h e e q u a t i o n
Theorem 4
1
W, w
E
wo
E
#
1
, it
, the
follows
same
E makes a l l p o s i -
woU'-
w B = B wow B
and
t h e two c a s e s above a r e e x c l u s i v e and e x h a u s t i v e ,
whence ( a ) .
(bj
Y'=
and
The map cp
(y1y\Y2,
8:
U- x H x U
morphisms over
sider
Y3) :
y3 .
ko
Let
i s composed of t h e two maps
(tala
->
.
>
U-HU
For
[vi]
0 x (ti)x
.
y2
( t a l a > 0 ->
U- x H x
u
We w i l l show t h a t t h e s e a r e i s o t h i s f o l l o w s from Lemma 35.
be a b a s i s f o r
V
, the
s p a c e , made up of weight v e c t o r s i n t h e l a t t i c e
Con-
underlying vector
i
, and
fij
the
,
?.
norresponding c o o r d i n a t e f u n c t i o n s on
-qm
choose
34.
i = i ( a ) , j = j(a) ni
.
1.1.-
End V ,
n(a)
=
For each r o o t
a
a s i n t h e proof of Lemma
xg(tg)
Choosing an o r d e r i n g of t h e p o s i t i v e
8 > 0
r o o t s c o n s i s t e n t w i t h a d d i t i o n , we s e e a t once t l i a ' ~
f i ( a ) , j ( a )( X I =
n ( a ) t a + an i n t e ~ r a lpolynomial in t h e e a r l i e r t t s and t h a t
x
set
fij(x)
Thus
=
i s an i n t e g r a l polynomial i n t h e
v3
To prove
i s an isomorphism over
k
0
t f s for a l l
, and
vi
c o n s i s t r e s p e c t i v e l y of s u b d i a g o n a l u n i p o t e n t
:, .<.,<....,
. .. .
,
.:._
-c
. ..
,
.
..
I
I
,
U-,H;U
so t h a t
d i a g o n a l , super-
d i a g o n a l u n i p o t e n t m a t r i c e s ( s e e Lemma 18, Cor. 3 ) , and t h e n w e
may assume t h a t t h e y c o n s i s t 'of a l l of t h e i n v e r t i b l e m a t r i c e s of
these types.
I
Yl .
sinilarly for
i s a n isomorphism we o r d e r t h e
0
.
i,j
e n t r i e s of
u
e n t r i e s of
u
spectively.
case
i
5
x
Let
-
=
u-hu
be i n
U-HU
, t l i e d i a g o n a l e n t r i e s of
be l a b e l l e d
ti
with
i
We o r d e r t h e i n d i c e s s o t h a t
k, j
5
*
and
i j
#
kt
.
and l e t t h e s u b d i a g o n a l
h , the su~erdiaaonal
>
, the
t f s a s r a t i o n a l forms over
j u s t i f i e d by t h e f a c t t h a t t h e y a r e nonzero oil
ko
Call
1,
/:::21
re-
kg
in
i n the
t..
JJ
UoHU
I s
.
being
Thus
and ( b ) f o l l o w s .
c o n s i s t s of a l l
minors
j
, i n c r e a s e d by a n
. We may now induc-
d i v i s i o n by t h e forms r e p r e s e n t i n g t l i e
i s a n isomorphism over
<
Then i n t h e t h r e e c a s e s above
,
fls
i
precedes
ij
f . .(x) = t . i t . .
resp.
t . .t . .
11 1J
1J
1J JJ
t i j ~
i n t e g r a l polynomial i n t ' s p r e c e e d i n g t i j
t i v e l y s o l v e f o r tlie
i = j
j
(aij )
such t h a t t h e
are n~nsing~lar.
8
I
64
Remark : It e a s i l y f o l l o w s t h a t t h e L i e a l g e b r a of
dk
is
G
We can now e a s i l y prove t h e f o l l o w i n g i m p o r t a n t f a c t ( b u t
w i l l r e f e r t h e r e a d e r t o ~ g m i n a i r eBourbaki, Exp
Let
Q,
b e a Chevalley group over
G
a l g e b r a i c maLric group over
22
r e s p o n d i n g i d e a l over
which v a n i s h oil
.
G)
,
$
viewed a s above a s a n
t h e prime f i e l d , a n d
I
t h e cor-
( c o n s i s t i n g of a l l polynomials over
I
Then t h e s e t of z e r o s of
braically closed f i e l d
219 i n s t e a d ) .
k
i n any a l g e -
i s j u s t t h e Chevalley group over
k
of t h e same t y p e (same r o o t s y s t e m and same weight l a t t i c e ) a s
G
.
Thus we have a f u n c t o r i a l d e f i n i t i o n i n t e r m s of e q u a t i o n s
of a l l of t h e semisimple a l g e b r a i c groups of any g i v e n t y p e .
C o r o l l a r-y -.
1:
-
Let
V'
group construcZzd u s i n g
Assume t h a t
LV
> LVT
.
I
1
Proof:
that
T
of
1
h
are
.
1
.
1
-L
P (Hi)
i
are
ti
k
0
.
Now f o r
coefficients i n
cp J(JU
: -B
clude t h a t
ko
w
t
1
(
E
AHi)
(ii
LV1)
.
LV)
E
LVl
.
i s r a t i o n a l over
->
I
G
i s a homomorphism
i s r a t i o n a l over
we need o n l y show
C
- LV) ,
.
and hence i s r a t i o n a l
can be chosen w i t h
( 1 )( - 1 ) a ( 1 ) ), s o t h a t
wa(l) =
ko
7
Each of t h e former i s a
W ,uW ( r e s p . w ),
(for
G
The nonzero c o o r d i n a t e s
1
E
~ 3 :
The nonzero c o o r d i n a t e s of
monomial i n t h e l a t t e r ( b e c a u s e
over
1
TT
and
b e a Chevalley
b u t w i t h t h e same
By Theorem
ko
1
hi(ti)
.
cp U-HU
i s r a t i o n a l over
cp [ H
V
1
.
ko
I
Consider f i r s t
a
for a l l
of a l g e b r a i c groups over
I
i n s t e a d of
G
Let
Then t h e homomorphir;ia
---3 x,(t)
x,(t)
taking
be as above.
k,ko , G ,V
Since
ko
B.LB C w w-'u-B
, we
con-
.
65
C o. r o l l a r y 2 :
The homomorphism c p
.
4
C o r o l l a r y 6 t o Theorem
CorollaryL:
versal,
k
Proof:
C 1;
with
u
-=
t a , ti
>
and
cp-'
uhu-
E
Then
G
1c
= uhw,v
prime f i e l d .
0
Gg
and
a r e f i e l d s and
Clearly
x
Z,V,
Assume
Chevalley g r o u p s .
where
(of
) i s a homomorphism of a l g e b r a i c groups
2
T h i s i s a s p e c i a l c a s e of C o r o l l a r y 1.
Proof:
7
Then
--><Ka,)(-a>
SL
.:
.
lco
over
1
a
Gk = GK
E
K
.
.
Write
Applying
(ti) X ( t a l a < 0
i s d e f i n e d o v e r ko
.
I
a
Since
,
Gk
E
GKnGLMjk
.
d e f i n e d over t h e
, i.e.
uhu-
E
Gk
> oAa ('17h i ( t i I a F1
x,(%)
0
of Theorem 7 , we g e t
cp-'
X
x
Ww
E
=
i s uni-
.
with
uhu-
V
a r e t h e corresponding
Suppose
xu,'
We must show t h a t
GK
n GLM,
4)
( s e e Theorem
-..1
and
Gk
C
- GK n GLM,
UWvwW
are fixed, t h a t
4
all
i s defined over
uhu-
t,, ti
E
k
.
k
Hence
.
Remark:
Suppose
Then
h a s t h e s t r u c t u r e of a complex L i e g r o u p , and a l l t h e
G
k =
(t
and
G
i s a Chevalley g r o u p o v e r
k
p r e c e d i n g s t a t e m e n t s have obvious m o d i f i c a t i o n s i n t h e l a n g u a g e
of L i e g r o u p s , a l l of which a r e t r u e .
For example, a l l complex
semisimple L i e grougs a r e i n c l u d e d i n t h e c o n s t r u c t i o n , and
i n Theorem 7 i s a n isomorphism of complex a n a l y t i c m a n i f o l d s .
.
66
1
I
36.
G e n e r a t o r ~ m dr e l a t i o n s
I n t h i s s e c t i o n we g i v e a p r e s e n t a t i o n of t h e u n i v e r s a l
I'
!
Chevalley grou;3 i n terms o f g e n e r a t o r s and r e l a t i o n s .
i; a r o o t System and
k
a f i e l d we c o n s i d e r t h e group g e n e r a t e d
by t h e c o l l e c t i o n of symbols
t h e following r e l a t i o n s
I
C
If
[ x a ( t ) la e Z, t e k ]
, taken
subject t o
from t h e corresponding Chevalley
groups :
xa(t)
is a d d i t i v e i n
If
and
u
p
a r e r o o t s and
lT Xis+ j p ( c i jt iu j )
and t h e
, where
t
E,
k'"
#
and
j
i
-2
0
, then
(x,(t) , x g ( u ) ) =
are positive integers
t
t
for
u)
i s multiplicative i n
h,(t)
a + B
a r e a s i n Lemma 15.
cij
wa(t)x,(u)wL(-t) = x-,(-t
-1
x,(t)x,(-t
)\(t)
for
for
.
t
t
k"
, where
w,(t)
=
.
k':
E
E.
, where
hL(t} = wu(t)wa(-l)
.
The r e a d e r i s r e f e r r e d t o t h e l e c t u r e r f s paper i n Colloque
thkorie des groupes al,q/ebriqies, B r u x e l l e s
I
Theorem 8 : Assume t h a t
(a)
(R)
(B)
rank C
if
rank C = 1
*I
I
, 1962.
i s o r t h o g o n a l l y indecomposable.
The r e l a t i o n s
and
(b)
C
(see
>
-
2
3~ )
la
a r e consequences of
and of
( 8 ) and
(B1 )
Then:
(A)
if
.
I n e i t h e r c a s e i f we add t h e r e l a t i o n
(C]
we o b t a i n a
. complete s e t of r e l a t i o n s f o r t h e u n i v e r s a l Chevalley
group c o n s t r u c t e d from
C and
k
.
The proof depends on a sequence of lemmas.
Throughout we l e t
1
'
Ix,(t) la
rank
C
C, t
E
>
-
be t h e g r o u p g e n e r a t e d by
k]
E
(B')
r a n k C = 1, G
if
v e r s a 1 Chevalley group c o n s t r u c t e d from
n:
G
t
->
a
for a l l
b e t h e homomorphism d e f i n e d by
C,
t
Let
(a)
a
(b)
a, p
Let
lxA(t)la
E
3, t
E
b e the. u n i -
, and
1
n ( ~ a () t=)x,(t)
S
a + /3
and
.
k]
E
G
a +
implies
C
E
Then
t h e c o r r e s p o n d i n g group i n
.
$ S
-a
implies
E
if
be a s e t of r o o t s such t h a t :
S
S
k
(B)
.
k
E
b e t h e subgroup of
XIs
and
C
G
E
I.,emma 36:
(A) and
subject t o relations
(A) and
or
2
t
G
t
P
E
S
.
g e n e r a t e d by
yts
maps
n
isomorphically onto
.
G
9
Proof:
Using
and
(11)
t
t o t h e form
a c 3
e v e r y element of
Lemma 3'7.:
rank C
>
-
we can r e d u c e e v e r y element of
(B)
x a ( t F , ) (t,
gs
a n d we know by Lemma 1 7 t h a t
k)
E
can b e w r i t t e n u n i q u e l y i n t h i s form.
The f o l l o w i n g a r e consequences of
2
a n d of
(A)
t
0
1
1
t
?
and ( B ' )
1
wa(t)xB(u)wa(-t) = x r ( c t
(b)
w a ( t ) w (u)wa(-t)
P
and
u
Y
=
,
and
w,B,
-
(A)
and
(B)
if
rank C = 1 :
if
(a)
where
ls
'La>u)
P
=
wa(ct-
< B j a > ~ )
c = c(a,p)
c(a,p)
=
=
2
c(a,-B)
i s independent of
.
t
I
'
( d ) , h i ( t ) ~ g ( u ) h ; ( t ) -=~ x ( t<kba>ul
B
t
<B,a>u)
(el hi(t)wg(~)h:(t)-l = wg(t
'
I
( f ) h i ( t ' ) h L ( ~ ] h ; , ( t ) -=~ h t (t<Pya>u)hT
B ( t<p ? a>]-l
Proof:
-
(a)
assume
ia + j p
form
X's
(B)
t
wa(t)
#
a
where
.
+p
and
i
t
'
be t h e s e t of r o o t s of t h e
and
g S.
t
I
S
j a r e i n t e g e r s and
i s normalized by
Thus
Let
w,(t)xB(ujwa(-t)
T
E
o n l y prove t h a t r e l a t i o n ( a ) h o l d s i n
from t h e r e l a t i o n s
.
(R)
.
G
Now assume
xt,
j
>
0
.
By
and hence by
By Lemma 36 we need
.
But i n
a =
G
and
(a) follows
rank C
2
2
.
I n t h i s c a s e we u s e t h e f a c t ( s e e t h e C o r o l l a r y t o Lemma 33)
There e x i s t r o o t s
(
a
that
t
(
t
X
9
jr
+
= 6
6
t
!
Xm*+nr
Y *
7
and a p o s i t i v e i n t e g e r
such
j
and
n , >
~o
T = {mwa6 + nwarlm,n
Set
and
/
(c
tmun)
m,n
and
clj#
positive integers]
s i d e s of t h e above e q u a t i o n by
0.
Transforming b o t h
and a p p l y i n g t h e c a s e of
wA(t)
( a ) a l r e a d y proved we s e e t h a t t h e t r a n s f o r m of every t e r m except
KT .
I
1
XT ,
1
1
Hence ~ , ( t ) x g + ~ ~ u(jc)w,(-t)
~ t
&
X*+j7 13.t uJ) E
s o by t h e e a r l i e r argument, w i t h T i n p l a c e of S , ( a ) h o l d s .
(C
If
a = p
t
wa(t)-'
=
and
1
wa(-t)
(b)
and
t
h,(t)
I
r a n k C = 1 t h e n ( a j h o l d s by (B )
, the
case
a = -B
.
Since
f o l l o w s from t h e c a s e
-..(f) f o l l o w from ( a ) and t h e d e f i n i t i o n s of
a =
.
I
w,(t)
.
Part ( a ) of Theorem 8 f o l l o w s from p a r t s ( a )
-
( d ) o f Lemma 37.
69
Lemma 3 8 :
%(.
= h'
1
a
, and
1
T
ka
Each
Proof:
H
i s normal i n
f o l l o w s from Lemma 37 ( f )
(b)
Let
w
M
E
be any r o o t , and w r i t e
.
Let
=
..
w,.
B
I=
Set
t
T
t
%$*hi.
By i n d u c t i o n on t h e l e n g t h of
- ---
P r o o f of T h e o r e m u :
I1
{x,(t) l a
rank C
E
>
It
2, t
E
1 or
k]
Let
rf
w a y h ( t ) , .
.
.O
E
ker n
so
1=
: G
it
->
i9
Y =
17. h i ( t i ) .
,
(b) follows,
( A ) , ( B ) if
,
and ( C )
.
Let
11
xa(t)
.
(ti
Since
i s u n i v e r s a l each
G
Let
G
nhi(ti)
E
ki']
Applying
.
.
It
E
H
91
we
3
ti = 1
,
x = l ,
F f --.
imlrl-s:
(a)
erators
x,(t)
- 7 2
w
i s an isomorphism.
S
By C o r o l l a r y 1 of t h e p r o p o s i t i o n i n 3 3 , x
ByLemma 36 and ( C )
obtain
by ~ e m a
b e d e f i n e d a s u s u a l i n t e r m s of t h e
'IT
.T,
Then
b e t h e group g e n e r a t e d by
G"
rank C = 1
We wish t o p r o v e t h a t
.
ya>)w~(-l)
subject t o t h e relations
( B f ) if
ai
with
(e)
h A ( t ) = h r ( c ( - l ) - < B , a > t ) h k ( ~ ( - l )-<B , a > ) - l waT
x
wa$
= w a ( l ) h g ( c ( - l ) - < B y a > t ) h k ( c(-1)
-<B,a>l-lwT(-l)
a
37 ( c ) , a n d hence by Lemma 37
E
w ai
=
b e a minimal e x p r e s s i o n f o r
a s a p r o d u c t of s i m p l e r e f l e c t i o n s .
$(t)
Then:
.
$
w
.
.
1
(a)
s i m p l e and
,
h,(t)
H~ t h e gr,oup g e n e r a t e d by a l l ha
i
L
w
h i b e t h e group g e n e r a t e d by a l l
Let
(a)
1
%
.
.
In
(A)
where
and
a
(B)
it i s s u f f i c i e n t t o u s e a s gen-
i s a l i n e a r combination of 2 s i m p l e
<
r o o t s and t h e r e l a t i o n s
(1)
and
(B)
which can b e w r i t t e n i n
t e r m s of s u c h elements.
It i s s u f f i c i e n t t o assume
(b)
f o r one r o o t i n
(C)
.
each o r b i t under
W
Exercise:
If
i s ' i . n d e c o m p o s a b l e , prove t h a t it i s s u f f i c i e n t
t o assume
(C)
C
f o r any l o n g r o o t
We w i l l now show t h a t i f
f i n i t e f i e l d then
(1)
Lemma 39:
b e a r o o t and
Let
a
and
k
(B)
.
f ( t, u ) = h,(t)h,(u)h,(tu)-'
2
.
a
i s an a l g e b r a i c e x t e n s i o n of a
imply
G
f
(C)
.
a s above.
In
G
1
set
Then:
2
(a)
f ( t,u v ) = f ( t,u ) f ( t, v )
(b)
If
.
.I.
t ,u
g e n e r a t e a c y c l i c subgroup o f
f ( t, u ) = f ( u , t )
If
f(t,u) = f(u,t)
(d)
If
t Y u0
#
Since
f ( t, u )
h,(t)
=
h(t)
(b)
h(t)
n
(c)
Proof:
=
Let
.
E
and
=
f ( t , u2) = 1
, then
t + u = 1
k e r n, f ( t , u )
t = vm, u = v
h ( ~ ) ~ hc( u, )
kq' t h e n
h(v)"d
n
with
with
, then
E
f(t,u) = 1.
c e n t e r of
m,n
E
c, d
E
.
.
center
G
1
Set
Then
G
1
, since
1
I,
I
is a c e n t r a l extension of
G'
f(t,u) = f(u,t)
(c) h ( t )
tively.
h ( i ) , h(u)
2
= h(u)h(t)h(u)-l = h(tu )h(u )
2
c ~ m m u t eand
-
( b y Lemma 37 ( f ) )
.
A b b r e v i a t e x a , x W a , Wa
(d)
Thus
.
f ( t ,u ) = 1
so that
.
G
ha t o x , y , w, h ,
respec-
We have:
2
(1) w ( t ) x ( u ) w ( - t ) = y ( - t u )
(3
Then
1
w ( t ) = y(-t-l)x(t)y(-t-l)
h(tu)h(u)-l
(by
(11,
(2L (3))-
v(tu)w(-u)
by d e f i n i t i o n of h
-1 -2
= x(t)x(-t)w(tu]w(-u) = x ( t ) w ( t u ) y ( t u
)w(-U)
(by (1))
-1 -1
-1 -1
= x ( t ) ~ ~ ( -ut ) x ( t u ) y ( - t u
) y ( t - 1 ~ - 2 j ~ ( - ~ ) ( b y ( 3 r) I
=
-1v.*-1
) x ( ~ u ) ~ ( u - ~ ) w () b- yu )( A )
= x(t)y(-i
y ( - t -1u -1) y ( t-1u -2 ) = y ( t -1u -2 ( 1 - u ) ) = y ( u-2 ) )
-1 --1
= x(t)y(-t u
) w ( - u ) y ( - t ~ - ~ ) x ( - l ) ( b y (1) and ( 2 ) )
for
I
=
~(t)~(-t~~]x(t]x(--1)~(l)x(-l)
= w(t)w(-1)
Lemma 4 0 :
I
t,u
such t h a t
Proof:
If
((q+1)/2)#
I
find
In a field
a,b ,c
t
I k[ =
k
and
q
u
=
h(t)
, proving
(d)
of f i n i t e odd o r d e r t h e r e e x i s t elements
a r e n o t s q u a r e s and
there are
(q+1)/2
t + u = 1
squares.
.
Since
t h e s q u a r e s do n o t form a n a d d i t i v e g r o u p , s o we can
so that
.
a + b = c
where
a
and
b
a r e squares
,
and
is not.
c
Theorem 9:
t = a / c , u = b/c
Then t a k e
Assume t h a t
i s indecomposable and t h a t
C
a l g e b r a i c e x t e n s i o n of a f i n i t e f i e l d .
and
(B)
(By) i f
(or
I.
.
t , u e k"'
Let
a s i n Lemma 39.
square
i s an
(A)
Then t h e r e l a t i o n s
i . e . t h e y imply t h e r e l a -
.
(C)
Proof:
-
k
r a n k C = 1) s u f f i c e t o d e f i n e t h e c o r -
responding u n i v e r s a l Chevalley group,
tions
.
We must show
f ( t, u )
=
By Lemma 3 9 ( b and c ) i f e i t h e r
f(t,u) = 1
.
1 where
f
t
is a
or
u
Assume t h a t b o t h a r e n o t s q u a r e s .
40 ( a p p l i e d t o t h e f i n i t e f i e l d g e n e r a t e d by
t = r 2t l , u =
t
and
is
By Lemma
u)
4
.
w i t h r 7 s ~ k * , t l + u l = 1 , t l a n d ul
u1
n o t s q u a r e s . T h e n f ( t , u ) = f ( t , s 2 ul) = f ( t , u l ) = f ( r 2 t l , u l )
= f(t17u1)
Example:
x i j ( t ) (1
=
1 by Lemma 3 9 ( a a n d d ) .
If
>
n
i
3
and
n, i
j
(a)
xij(t)xij(u)
(B)
(xi j ( t )
yxj
#
k
i s a f i n i t e f i e l d , t h e symbols
j
t
k)
subject t o t h e relations:
= xij(t+u)
k ( ~) ) = x i k ( k )
( x i j ( t) ,xkL(c))
d e f i n e t h e group
E
SLn(k)
=
.
1
if
i
k
are distinct,
if
j f k, i f & ,
s7.
C e n t r a l at e n s i o n s .
n, G
Our o b j e c t i s t o prove t h a t i f
P
(n , G )
36, t h e n
T
,
and
are as i n
G
i s a universal c e n t r a l extension of
sense t o be defined.
G
in a
The r e a d e r i s r e f e r r e d t o t h e 1 e c t u r e r T s
--
-
p a p e r i n Colloque s u r l a t h e o r i e d e s groupes a l g 8 b r i q u e s ,
B r u x e l l e s , 1962 and f o r g e n e r a l i t i e s t o S c h u r f s p a p e r s i n J. R e i n e
1904, 1907, 1.911.
Angew. Math.
Definition:
where
ker n
.
G
1
A c e n t r a l e x t e n s i o n o f a group
fs
C
- center
a group,
of
t
i s a couple ( n , G )
G
i s a homomorphism o f
n
G
1
onto
and
G,
.
G'
Examples :
(a)
n, G I , G
(b)
n :G
t
as i n
->
36.
t h e n a t u r a l homomorphism o f one Chevalley
G
g r o u p o n t o a n o t h e r c o n s t r u c t e d from a s m a l l e r weight
lattice.
n : Spinn
and
(c)
E. g .
n :G
f
>-
G
,
x : SLn
-+
PSI,,,
-+P
a t o p o l o g i c a l c o v e r i n g o f a connected
,
i s a l o c a l isomorphism,
x
c a r r y i n g a neighborhood of
G.
,
.
SOn
t o p o l o g i c a l group; i . e.
of
n : Spn ->PSpn
We n o t e t h a t
n
1 i s o m o r p h i c a l l y o n t o one
i s central since a discrete
normal subgroup o f a connected group i s n e c e s s a r i l y
central.
N
To s e e t h i s , l e t
subgroup o f a c o n n e c t e d group
t h e map
G>
-
N
g i v e n by
g
->
b e a d i s c r e t e normal
G
EN. S i n c e
, ~ E Gh a,s a
and l e t
gng-'
n
-1
gng
= n
d i s c r e t e and c o n n e c t e d image,
E G-
g
Definition:
A c e n t r a l extension
( n , ~ ) of a g r o u p
1
u n i v e r s a l if f o r any c e n t r a l e x t e n s i o n
cp
e x i s t s a u n i q u e homomorphism
i. e .
,
for a l l
2
, E')
of
G
such t h a t
n
(n
E->
E
Y
is
G
thwe
t
Cp
n
=
,
t h e f o l l o w i n g diagram i s commutative :
We a b b r e v i a t e u n i v e r a l c e n t r a l e x t e n s i o n b y
u. c. e.
We
d e v e l o p t h i s p r o p e r t y Xn a s e q u e n c e o f s t a t e m e n t s .
(i)
If a u. c . e .
If
Proof:.
rp : E->
ncq
1
=
P
.
u. c . e .
(ii) I f
G =
G,
are
?
E
( n , ~ )is a
and
1
n(.p cp)
?;ji
u . c . e.
H
yl(a) = ( a , l )
and hence
and
nr(a,b')
.
=
Thus,
=
let
?
n g ~= n a n d
n.
=
cp
Hence
?
cp y
i n the definition
E~
.
of
E
and hence
G
then
E =
i s t h e derived group of
and y 2 ( a )
if1 = y2
G,
i s t h e i d e n t i t y on
P r o o f : Consider t h e c e n t r a l e x t e n s i o n
E' = E x E / & E
of
by t h e l ~ n i q u e n e s sof
Similarly
where
u.c.e.
be such t h a t
rp'p : E ->E
i s t h e i d e n t i t y on
of a
(n , E )
rpi : E ->E
and
Now
F
t
( n , ~ ) and
E'
n
e x i s t s , it 1s unique up t o isomorphism^
(n
1
,E 1 )
H.
where
n(a), a E E , b E E / O E .
(a,a+B E),
then
E l D E = 1 and
t
Nowif
a p- = n , 1 = 1 , 2 ,
1
E = a E .
If
(iii)
then
E
=
= D G and
G
fi
C,
E
rr i s t r i v i a l .
Proof:
-
If
(iv)
where
Proof:
-
f!G ,
For each
x
&
E
>
-
expression i n
morphism o f
(a)
this, l e t
t
e (x) E E'
F,
possesses a
G
Ee (x),x
G]
(E
n
we s e e t h a t
1
E,
1
t
,
n )
t
then
for a l l
1
1
n. e (x) = x .
>
-
E
DF
and
:F
rp
.
Let
F
be
-
x,y,z
G
.
e x t e n d s t o a c e n t r a l homo-
b e any c e n t r a l e x t e n s i o n of
bince
1
n
t
Choose
G.
is central, the
e(x)ts.
such t h a t
To s e e
G.
Hence, t h e r e i s
rp e ( x )
e ( x ),
1
=
and
.
If
E
then
=
TI;
a l s o denotes t h e r e s t r i c t i o n of
( n , ~ ) covers a l l c e n t r a l extensions uniquely.
By ( i i i ), we have t h a t
( n , ~)
e(x)
G.
such t h a t
n rp = n.
to
.
( n , ~ ) covers a l l c e n t r a l extensions of
a homomorphism
n
u. c . e
e(z)
1
(b)
E = C J ~ Ewhere
subject t o t h e condition
e ( x ) ~ ss a t i s f y t h e c o n d i t i o n on t h e
thus
Hence,
t h e n by u s i n g i n d u c t i o n on t h e l e n g t h o f a n
onto
F
on which
.
we . i n t r o d u c e a symbol
G
x,
G,
& ( c ~ E )= D * E .
e ( ~ ) e ( ~ ) e ( x y ) -commutes
l
with
n : e(x)
If
=
d g ( r r E ) = & G = G.
then
t h e group g e n e r a t e d by
that
QE
B E=
Also,
G =
-
n B E
E
i s a c e n t r a l subgroup o f
C
Moreover,
We have
C =kern.
( s , ~ )i s a c e n t r a l e x t e n s i o n o f
( n , ~ ) covers a l l c e n t r a l extensions.
i s a c e n t r a l extension of
t
cp(x)cp ( x ) - l E
i s a homomorphism of
i s t r i v i a l and
rp
center of
E
G
.
and i f
Thus,'y:
E i n t o an Abelian group.
?
= cp
f
a cp
=
n =
1
71
rp
If
,
1
-1
x--> tp(x)cp (x)
Since
E
=
k ) ~ ,
Remark:
P a r t ( a ) shows t h a t i f
i s any group t h e n t h e r e i s
G
a c e n t r a l extension covering a l l others.
. t
(iv)
If
is a central extensionof
(71,~)
a l l o t h e r s and i f
n :E ->
(v If
sois
F
E =
BE,t h e n
and
Y : F >-
G
, provided
E
yn:E->G
Proof:
-
If
a
ker
.
axa-1x-1 , x E E
n,
Now
( n , ~ ) Is a
E
=BE.
center of
a homomorphism, s o
ep i s t r i v i a l , and a
(vi)
v
if
Exercise: I n
(Ifrn,~) is a
Definition:
is a
A group
u. c. e .
(vii)
U.C.
of
,
( n , ~ ) is a
e.
G
of
u. c . e.
a r e c e n t r a l extensions, then
E,
n a E center of
n ( a , x ) = (na,m) = 1 b e c a u s e
which c o v e r s
g, b e t h e map rg : x >
-
let
cp(x)
G
since
F. Now 7 i s
center of
u. c. e of
(a,x) =
F
E.
if and o n l y
G.
id,^)
i s said t o be centrally closed i f
G.
Corollary:
If
( n , ~ )is a
u. c. e.
of
G,
then
E
is
c e n t r a l l y closed.
(viii)
If
y : F+
E
9 : E -3
(1x1
of
F
,
E splits;
i.e,
fp
such t h a t
( n , ~ )is a
o f t h e form
.
i s c e n t r a l l y c l o s e d , t h e n every c e n t r a l e x t e n s i o n
E
u. c. e.
of
=
t h e r e e x i s t s a homomorphism
id
.
G
i f and o n l y i f e v e r y dlagram
E--nJ
G
' i t
G!
P >
can b e uniquely completed,
of
and
G'
Proof:
-
(n
where
t
,E
9
i s a c e n t r a l extension
)
f i s a homomorphism.
One d i r ' e c t i o n i s immediate by t a k i n g
G'
H b e t h e subgroup o f
U/: H->G
i s g i v e n by
( n , ~ ) is a
0 :E +H
P
wG have
1
=
H
Q = a.
whereyl : H+E
d
-> H
= 9
p
t
u. c. e.
t
1.
is central.
If
Since
rp : E+E'
Now t h e homomorphism
1
7
1
IS g i v e n b y y ( x , e ) = e
1
1
1
frr=nrp
If
?
1
0 ( e l = (rr(e) , y ( e )
be given, by
and
id.
t h e r e i s a unique homomorphism
G,
1
:E
Ifx=n e
Y
x , then
=
is a
t
t
[(x,e
=
r n = p y e = n y/ e = n r p .
satisfies
0'
of
?
(x,e
such t h a t
=Y@ ,
let
u. c. e.
,
G y E'
7
and
G
(y, E )
Conversely, suppose t h e diagram i s g i v e n and
Let
=
= y t g 1 =y18
= p
,
1.
1
1
,
,
,
then
t
S i n c e y/ 9 = z ,
proving t h e uniqueness
D e f i n i t i o n : A l i n e a r ( r e s p e c t i v e l y p r o j e c t i v e ) r e p r e s e n t a t i o n of
a group
G
i s a homomorphism o f
G
GL(v)
i n t o some
(re-
s p h ~ t i v c ~FGL(V)
).
Since
GL(v) i s a c e n t r a l e x t e n s i o n of
PGL(V)
we have
t h e following r e s u l t :
(r) C o r o l l a r y :
If
is a
(71,~)
p r o j e c t i v e representation of
l i n e a r r e p r e s e n t a t i o n of
(x')
-
can r e p l a c e t h e c o n d i t i o n
( n , ~ ) is a
c.e.
of
Gp
t h e n every
can b e l i f t e d u n i q u e l y t o a
G
E.
Topological s i t u a t i o n :
condition
U.
If
G =
u.c.e.
G
i s a t o p o l o g i c a l group one
DG by
by
G
,'
i s connected, t h e
( x , ~ )i s a u n i v e r s a l coveringgroup
i n t h e t o p o l o g i c a l s e n s e , and t h e c o n d i t i o n
G
i s c e n t r a l l y closed
by
i s simply connected i n t h e above d i s c u s s i o n and o b t a i n s i m i l a r
G
results.
Definition:
( n , ~ ) is a
If
-
t h e S c h ~ rm u l t i p l i e r o f
ker n
If we w r i t e
:'
f
M(G )
c a t e g o r y o f groups
G 4G
by ( i X
!
t h e n we c a l l
G,
t o i n d i c a t e t h e dependence on
1
.
Thus
G
M
=aG
i s a f u n c t o r from t h e
t o t h e category of
Abelian groups w i t h t h e f o l l o w i n g p r o p e r t y :
q) i s o n t o ,. t h e n
if
go).
so i s
Remark:
Schur used d i f f e r e n t d e f i n i t i o n s (and terminology) s i n c e
he c o n s i d e r e d o n l y f i n i t e groups b u t d i d not r e q u i r e t h a t
If
G =
d4 G
i s f i n i t e t h e n so
G
M(G).
Theorem 10:
Let
f i e l d such t h a t
If
G
b e an indecomposable r o o t system and
C
>4
]kl
and, i f rank
1, ( c )
7
i s t h e group d e f i n e d b y t h e r e l a t i o n s ( A ) , ( B ) , ( B )
o n l y if r a n k
to
Bemark:
SL ( 9 )
2
a
i s t h e corresponding u n i v e r s a l 3hevaUey group
?
G~
k
] k l $ 9.
'E = 1, t h e n
( a b s t r a c t l y d e f i n e d by t h e r e l a t i o n s ( A ) , ( B ) ,(B
G'
G = &G.
One o f Schurr s
our d e f i n i t i o n s a r e e q u i v a l e n t t o h i s .
r e s u l t s , which we s h a l l not u s e , i s t h a t i f
is
G,
l e a d s t a a corresponding one
such t h a t
G
o f t h e group
G..
ker n = M(G)
t h e n a homomorphism.
M(?) :. M ( G ) ->
e
U.C.
G,
,
then
n
and i f
C = 1)
1
( n , ~ ~
s
Indeed
if
(we u s e (B')
i s t h e n a t u r a l homomorphism from
u.c.e.
a
of
G.
There a r e e x c e p t i o n s t o t h e conclusion.
a r e such.
of 3 6 ) ,
sL2 ( 4 ) 2 PSL ( 5 )
2
and
E.g..
SL2(4)
and
sL2 ( 5 ) i s a c e n t r a l
e x t e n s i o n of
P S L ~ ( ~ ) For
.
~ ~ ~ s( e e9 Schur.
1
It c a n b e shown
f o r which t h e c o n c l u s i o n f a i l s
t h a t t h e number of c o u p l e s ( C , k )
is finite.
Proof:
-
Since
f o r both
and
G
> 4,
lkl
= B G G,t
G
t
=&G
, - and
The c o n c l u s i o n becomes
GI.
c l o s e d , b y t h e above remarks.
u.c.e.
GT
exist
is centrally
We need o n l y show t h a t e v e r y c e n t r a l
(y, E ) of G' s p l i t s , i . e . , t h e r e e x i s t s 9
y 0 = i d , ; i . e . , t h e r e l a t i o n s d e f i n i n g G'
t
: G->
extension
so t h a t
can b e
G
lifted t o
E.
We may assume
extension o f
(1)
E = B E ; but then
by
G
(v
( y,E )
If
1.
We need o n l y show
i s a c e n t r a l extension of
t
r e l a t i o n s ( A ) , ( B ) , ( B ) can b e l i f t e d t o
,
Let
C = ker
(2)
A commutator
the classes
mod C
(ha(a),xa(t))
'PXa(t)
E E (aEZ, t
x
E.
x,y E E
with
so that
xa(ct)
=
(x,y)
then t h e
G,
E.
a c e n t r a l subgroup o f
t o which
a E k"
Choose
is a central
(n*,~)
We have:
depends o n l y on
and y belong.
c
for a l l
=
a2
a
E
-
I f 0.
Z , t E k.
E k ) so t h a t
Then i n
G
We d e f i n e
rpxa ( t ) = x a ( t )
and s o
that
(3)
later
po w
define the
(rpha(a),
1s)
c p x a ( t ) ) = y x a( c t )
i n t e r m s of t h e
and t h e
hts
wfs
rp
(
,
T
A ( B (B )
phts
(and
xTs by t h e same f o r m u l a s which
i n terms of t h e
t h i s choice i s not c i r c u l a r because of
the relations
and t h e n
hold with
(2).
xfs.
Note t h a t
We s h a l l show t h a t
p x f s i n p l a c e of
xts.
E
set
x,
h
= xa(dt)
(t)h-'
.
d E K"
with
Conjugating ( 3 ) by
yh, we g e t
( c p h a ( a ) , y x a ( d t ) ) = c ~ h y x a ( c t ) c (ph ) - I
, and
side equals
cp x a ( c d t ) =
S i m i l a r l y we h a v e
(4')Cp
n
q~ xu
by ( 3 ) .
( h x a ( c t)h - l )
(t)(cpn)-'
=
-
cp(nxa ( t ) n ')
for a l l
the left
E Ny a E E,
n
t E k .
(5)
a root, then
must show
(6)
a
If
and
rp x a ( t )
(5a)
>
and
commute f o r a l l
a+j3 is not
E k.
t,u
Set
C l e a r l y , from t h e d e f i n i t i o n s we have
a f f3
Assume
0,
0,
a+B
is additive i n both positions.
.
(a,~=
) 0,
If
then
by Lemma 2 0 ( c ) and b y (4)w i t h
d
f ( t , u ) = f ( t v ,u) where d = 4
=f(tv2,u)(vf0)
(a,$)
rp x g ( u )
and
f ( t , u ) = 1.
f
are roots,
3j
then
f(t,u)
h =
ha(v).
If
- <a,@><f3,a>
.
-( @ , a >
In
and by ( 4 ) w i t h h = h a ( v 2 ) h (v
P
b o t h c a s s , f ( t ( l - v d ) , u ) = l by (6) f o r some d = 1 , 2 , o r 3.
Choose v so v d - I # 0. Then we g e t f z l .
by Lemma 2 0 ( c )
(5b)
Assume
sothat
v2f0
=
j3
and
f ( t, u )
(:K)
1 - v
( t, u ) ,
C
=
=
1.
I f there i s a root
(v)
f (tv,uv).
inthepreceding
Choose v
so t h a t
(o) and (61,
By
i) = f ( t, U / V ) f
whence f[t(l-v+v2) , u )
>
h = h
+ v 2 f 0.
f ( t ( v - v 2 , , u ) = f ( t ,U/ (V-v
= f
and r a n k
< a , r > = 1, s e t
argument and o b t a i n
v -
a
( t , ~ / ( l - - V )=) f ( v t , u ) f ( ( l - Y )t , u )
1 and
f
1.
I f t h e r e i s no
such
d\ ,
t h e n Z . i s of t y p e
a = 8+ 2
t h i s c a s e , however,
and
Cn
f
a
i s a long root.
and
with
cp x a ( v )
Since
Thus,
(ktu2) with
( y x g ( t ) , y x Y ( u ) ) = g r f x b + y ( +t u ) x 8+2
by Lemma 33.
roots.
In
g
E
C,
commutes w i t h a l l f a c t o r s b u t t h e
l a s t by ( 5 - a ) , i t a l s o cOI?-Imutes w i t h t h e l a s t .
Assume a = p and r a n k C = 1. A t l e a s t we have
f ( t , u ) = f ( t v2 , u v 2 ) , t , u ~ k v, ~ k ' ", u s i n g h = h a ( v ) i n t h e
(5c)
argument above.
We may a l s o assume t h a t
[k
1
i s n o t a prime.
\
If it were, t h e n
and ( 5 )
xa(t)
and
xa(u)
would be immediate.
would b e powers of
R e f e r r i n g t o t h e proof of
s e e i t w i l l s u f f i c e t o b e a b l e t o choose
a r e s q u a r e s and
1
-v
1 + w 2
=
-
2,
characteristic
squares.
v
v-v2f
2
0,
this t o o i s possible.
,
where
>_ 25 i n t h e p r e s e n t c a s e ,
]kl
by
a ((t+u)c-'1)
ha(a)
is
x
One proves
The element
i s i n
itself.
x cp xa ( t )cp x,(u)
p r e s e r v e s t h e r e l a t i o n s (B).
E C.
so t h a t
T h i s completes t h e proof o f ( 5 ) .
t h i s transform i s a l s o
f (t,u)
are
k
S i n c e a t most 1 3
-.a
x
w
and we need o n l y choose
and 1 - v + v 2 # 0.
x = y X U (tc-l)gxa(uc-l)(rx
(8)
i s f i n i t e of
.
%
preserves t h e r e l a t i o n s (A).
t h e t r a n s f o r m of
k
, we
v, 1 - v
i s p o s s i b l e s i n c e a l l elements o f
2
s e t v =(2w/(l+w2))
Then
v a l u e s of w a r e t o b e avoided and
(4.1, ( 5 )
If
(5b )
this
((1-ut?)/(l+w2))
(7)
so t h a t
1- v + v2 # o .
v 2 #0,
Otherwise,
#o,
v
xa(l)
f
=
C,
andhence
However, by (31,
6x
31
(t+u)rl
.
We have
1 by i n d u c t i o n on
n,
the
number o f r o o t s of t h e form
i s just
f
(5).
>
n
If
ia+jP i
,
aF.
j
n = 0, t h i s
If
0 , t h e i n d u c t i v e h y p o t h e s i s and ( 7 ) imply
s a t i s f i e s ( 6 ) , and t h e n t h e argument i n ( 5 a ) may be used.
( 9 ) p,
t
p r e s e r v e s t h e r e l a t i o n s (B )
.
t.
T h i s f o l l o w s from ( 4 )!,
T h i s c o m p l e t e s t h e p r o o f o f t h e theorem.
Exercise:
i s t h e o r i g i n a l Lie a l g e b r a w i t h c o e f f i c i e n t s
Assume
t r a n s f e r r e d by means o f a C h e v a l l e y b a s i s t o a f i e l d
c h a r a c t e r i s t i c d o e s n o t d i v i d e any
>
i s indecomposable o f rank
(a)
The r e l a t i o n s
1.
[Xa,Xp]
defining s e t for
1.
N
a,0
Prove:
#
k
whose
0. A l s o assume
C
X
a + P # 0 , form a
a,B a + $ '
H i n t : d e f i n e Ha = [Xu, X-,]
= N
and show t h a t t h e r e l a t i o n s o f Theorem 1 h o l d .
(b)
(c)
& de ,
=
t h e derrved algebra of
Every c e n t r a l e x t e n s i o n of
Hint:
C o r o l l a r y 1:
%
2.
splits.
p a r a l l e l t h e p r o o f o f Theorem 10.
The r e l a t i o n s
c e n t r a l extension of
t
(A) ,(B), ( B ) c a n b e l i f t e d t o any
G.
Corollary 2 :
(a)
G'
i s c e n t r a l l y closed.
splits.
u . c. e .
Each o f i t s c e n t r a l e x t e n s i o n s
I t s Schur m u l t i p l i e r i s t r i v i a l .
It y i e l d s t h e
o f a l l t h e Chevalley groups of t h e given t y p e ,
and c o v e r s l i n e a r l y a l l of t h e p r o j e c t i v e r e p r e s e n t a t i o n s
o f t h e s e groups.
(b)
If
k
i s f i n i t e o r more g e n e r a l l y a n a l g e b r a i c e x t e n s i o n
o f a f i n i t e f i e l d , t h e n ( a ) holds w i t h
G
1
r e p l a c e d by
G.
Proof:
T h i s f o l l o w s from v a r i o u s o f t h e g e n e r a l i t i e s a t t h e
-7
beginning o f t h i s section.
E.g.,
then
if
is finite,
k
pi ( k ) ,
SLn(k) ,
and
>4
lkl
Spi.nn(k)
and
sL2(9)
a l l h a v e t r i v i a l Schur
m u l t i p l i e r s , and t h e n a t u r a l c e n t r a l e x t e n s i o n s
SP, 4 ' ' ~
, S~
pinn4SOn
Corollary
1:
Assume
G, G
?
E'k
with
C = ker n
Proof:
a
E
,
vn = u ) ,
,
n
and
a r e as above.
is
k*
implies t h e r e e x i s t s
G; i . e . ,
is a l s o d i v i s i b l e .
generate
f ( t , u ) = h a ( t ) h a ( u ) hr:. ( t u ) - l
We have
C.
u E k"
we c a n f i n d
.w E k"
in
G ~ '
by Lemma
f (t,vw2) = f (t, v ) f ( t,$)
n
By i n d u c t i o n , we g e t . f ( t , $ n ) = f (t,$)
Since f o r
If
t h e n t h e Schur m u l t i p l i e r of
Elements o f t h e f o r m
39(a).
,
SLn 4 PSL,
a r e a l l universal.
i n f i n i t e and d i v i s i b l e ( u E k * , n E Z
v
i s excluded,
f o r arbitrary
such t h a t u
=
w2n
,
n..
the
p r o o f i s complete.
Corollary 3a:
including 2,
If
then
Corollary 3q:
e x t e n s i o n of
i s i n f i n i t e and d i v i s i b l e by a s e t of p r i m e s
kt
If
i s a l s o d i v i s i b l e by t h e s e p r i m e s .
C
i s i n f i n i t e and d i v i s i b l e , t h e n any c e n t r a l
k"
by a k e r n e l which i s a r e d u c e d g r o u p (no d i v i s i b l e
G
s u b g r o u p s o t h e r t h a n 1) i s t r i v i a l ;
Proof:
reduced.
that
tp C
(
Let
?y ~q
Since
= TC
C
- ker y
E
.
.
i s a homomorphism
, it
splits.
be a c e n t r a l extension of
?
,
i . e.
is a
(n,G )
Since
Hence
C = ker
cp C
0 : G>
-
U.C.
=
e,
TI
1 and
G
with
t
we have FO $ G 4 E
ker
so
IS d i v i s i b l e r s o i s
k e r eo
E so t h a t 8 n
2
.-.k e r n
=
cp
.
.
Thus, t h e r e
Tnerefore,
y e g = n on
Gorolkry
22:
G'
and
If
k"
ye
-
- 1 on
I,.
i s i n f i n z t e and d i v i s i b l e , t h e n any f i n i t e
dimensional p r o j e c t i v e r e p r e s e n t a t i o n of
?ep~~~e~+,~.ti_on.
uniquely t o a
J
!.Ti_~?~;-lr
P r o o f : Assume 0- : G >
o- : G -> PsL(V). Let
projection.
and t h u s
ker
( y ,E )
of
I
p
Example
we have
Y~ ( x , Y ) E E ,
be t h e natural
is finite
k e
Consider t h e c e n t r a l e x t e n s i o n
Now
her ? = I
0 :G
>
-
E
: v ) T -GCX
A -
i s roduced,
X kerp
with
SL(V)
= 1
on
G.
Q' = Y'CI: G + S L ( V ) c GL(V)
then
s a y s , f o r example, t h a t e v e r y f i n i t e
dimensional r e p r e s e n t a t i o n o f
one.
we have
G,
~ = { i x , ylox
) = p y , XEG, yESL
where
Corollary 3c
;
-
G
-+ PSL(V)
f : SL(V)
i s reduced.
G
Y s( x , Y ) =
.
l
Since
S i n c e dim V i s f i n i t e , we have
s o by C o r o l l a r y 3 b ,
1f
PGL(V).
y/ ( x , y ) = x , ( x , y ) E E .
and
can b e l i f t e d
G
sLn(C )
can b e l i f t e d t o a l i n e a r
(The n o v e l t y i s t h a t t h e r e p r e s e n t a t i o n i s n o t assumes t o b e
continuous. )
Theorem 3-.. 1 :.
11 2
1s an lnaclcomposable r o o t s y s t e m , i f
--.---P,
chark= p d O : ,
A x , B2
type
and i f
0.r
( n , ~) ' u n i q u e l y
t h e k e r n e l h a s no
Proof:
--
2
COVbrS
::BG
and /i;1
( i . e . t:e
G
=
3
E
exclude
of type
a l l c c n t r a l e:ctensions o f
lkl =
2 , .Z. o f
A ~ ) , then
G
f o r which
p-torsion,
By Tneorem 1-0;
~ . I P could
154
assume
Jk
or
/ k( =
9.
However, t h e p r o o f does not u s e t h l s assumption o r Theorem 10.
If
,
E
i s a c e n t r a l extension of
G
such t h a t
C = ker
h a s no
p-torsion,
t h e n we w i s h t o show ( A ) , ( B ) ,
and
can
(B')
\
E.
be l i f t e d t o
(1) Assume, C
so t h a t
Y c px , ( t )
Choose cpxa ( t )6 E
P
( y s a ( t ) )= l , a E z , t E k
We
i s divisib!e
by
.
a ( t ! and
=.:
p.
0
cp x i s .
c l a i m r e l a t i o n s ( A ) , ( B ) and ( B ) h o l d on t h e
(la)
then
a,p
If
rp x a ( t !
and
ipx
(u)
B
9 x, ( t ) xp ( u ) y x a ( t 1 - l
.
a + B not a r o o t ,
areroots,
commute,
with
cpxa(u)
= f
t,uEk.
1 = f P which i m p l i e s
powers, we g e t
f = 1
a + ~ f0 ,
We have
E C.
f
and
Taking
since
p-th
h a s no
C
p-torsion.
(lb)
The r e l a t i o n s ( A ) h o l d .
v x a ( t )$ x a ( u )
p
= f
ya(t+u) , f E C,
(2)
General cass.
(2a)
C
p-torsion.
the
CF@ Q/ker
2.-
(2b)
p r o d u c t of
by
of
D-
C ' which i s d i v i s i b l e by
FQ
z
Q
8
=
E
and
C
P
with
i s a d i v i s i b l e group.
= 1 where
D
P
P
projects faithfully into
which i s d i v i s i b l e and has no
Conclusion of p r o o f .
D
C r! D
p-torsion,
C
of a f r e e
Hence
D, s a y , and
Thus,
9
i s a d i v i s i b l e g r o u p , and we
z7 - C Pa2 Q .
has no
C
p-component
D / D ~= C '
3
i
Moreover, s i n c e
as b e f o r e .
We have a homomorp'hism
A b e l i a n g r o u p I: o r t o C. Now
C = ~ / k e r8
f = 1
,
can b e embedded i n a g r o u p
F
powers of
Y
Exercise:
can i d e n t i f y
we g e t
p-th
R e l a t i o i l s ( B ) and ( B ) a l s o hold.
(LC)
and h a s no
Taking
Form
p-torsion.
E
7
=
EC
?
,
the direct
C amalgamated, and d e f i n e
y 7 ( e c ' ) = W ( e ) , e E E, c
P
EC
F
is
.
0
9
:E
-->
G
Now
(y",E')
s a t i s f i e s t h e assumptions o f (1) s o t h e r e l a t i o n s
( A ) , ( B ) , (B' )
can be l i f t e d t o
E
T
.
However, by Lemma 32
f
,
l i f t e d group i s i t s own d e r i v e d group and hence c o n t a i n e d i n
Coroll.ary 1: Every pro j e c t i v n r e p r e s e n t a t i o n of
characteristic
p
i n a f i e l d of
G'
G'
is a
p-group.
These a r e e a s y e x e r c i s e s .
f
n .: G .->G
S i n c e t h e k e r n e l of t h e map
b e t h e Schur m u l t i p l i e r of
o f some i n t e r e s t .
heor or em
12:
above t u r n s o u t t o
G, its structure f o r
arbitrary is
k
The r e s u l t i s :
(Moore, ~ a t s u m o t o ) Assume
r o o t system and
k
i s a n indecomposable
C
Ik
1 >4
.
k,
if
G'
a field with
If
is the universal
G
Chevalley group b a s e d on
E
and
by ( A ( B , ( B
n
i s t h e n a t u r a l m a p from
with
E.
can b e l i f t e d t o a l i n e a r one.
C o r o l l a r y 2 : The Schur m u l t i p l i e r o f
Proofs:
the
C = k e r n;
,
andif
t h e Schur m u l t i p l i e r of
t o t h e a b s t r a c t group
A
i s t h e group d e f i n e d
G,
then
g e n e r a t e d by t h e symbols
G
C
?
to
i s isomorphic
f ( t, u ) ( t, u
subject t o the relations:
(a)
f (t,u)f (tu,v)
and i n t h e c a s e
=
f ( t , u v ) f ( u , v ) , f ( 1 , ~= )f ( u , ~ )= 1
Z i s n o t of t y p e
Cn ( n
2
G
1)(cl = A ~ ) t h e
E
k")
I
(ab )
f
i s bimultiplicative
I n t h i s case r e l a t , i o n s
(cl)
f
(a) -(el
.
may be r e p l a c e d by
(ab')
and
i s skew
The isomorphism i s given by
,
? : f ( t ,u)->ha(t)ha(u)ha(tu)-l
a
a
f i x e d long r o o t .
Remark:
These r e l a t i o n s a r e s a t i s f i e d by t h e norm r e s i d u e symbol
- i n c l a s s f i e l d t h e o r y , which i s a s i g n i f i c a n t a s p e c t of Moorets work.
P a r t i a l Proof:
(1) I f
a
fia
i s t h e group generated
a f i x e d long r o o t , t h e n
ha ( t ) h a ( u ) h a ( t u ) - l
,a
.
%a
C
(in
G' )
by a l l
ha(t),
We know t h a t
E C , t E k9, form a g e n e r a t i n g s e t f o r
C.
Using t h e Weyl group we can narrow t h e s i t u a t i o n t o a t most two
roots
a,p
<B,a>
=
a
with
1 and
long,
(h u
p
s h o r t , and
( t ), h p ( u ) )
Hence,
= hB(t
)-'
=
(a,B) > 0 .
by Lemma 37 ( f )
.
T h i s shows
a
w i l l suffice.
(2)
i s a mapping onto
(3)
i s a homomorphism.
hold i f
f (t, u )
.
.
h a ( t )ha ( u ) h ( t u ) - l
The r e l a t i o n s
a
A s p e c i a l c a s e ( u =I) of
(e)
has been shown
.,--.mp,-,7,.-.
!,.. -.:. .. .
,
We must show t h a t t h e r e l a t i o n s
i s r e p l a c e d by
( a ) a r e obvious.
F?. .
T h i s f o l l o w s from (1).
C.
,
.
.-
,
.
i n Lemma 3 9 ( d ) .
The o t h e r r e l a t i o n s
t h e commutator r e l a t i o n s connecting t h e
( 3 ' ) Assume
Y'
there is a root
X
i s n o t of t y p e
so t h a t
a
, c ,d
b
>
h f s and t h e
Thus
1
w's.
I n t h i s case
Cn.
=
f o l l o w from
f (t , v )
hd' ( ~ )( ft, ~ ) h ( U ) - l = h a ( t u )ha (U)-' ha ( U V ) ha ( t U V ) - l = f (t,u) -lf (t,Uv)
o r f ( t , u v ) = f ( t , u ) f ( t , v ) . By r e l a t i o n ( c ) , f ( u v , t ) = f ( u , t ) f ( v y t ) .
=
(4) rp i s
a n isomorphism.
e x p l i c i t model f o r
Now l e t
of
G
n
.
b e a connected t o p o l o g i c a l group.
G
i s a couple
' group and
G'
This i s done by c o n s t r u c t i n g a n
E
( n , ~ ) such t h a t
i s a homomorphism o f
E
E
i s a connected t o p o l o g i c a l
onto
1 in
1 in
which i s a l o c a l isomorphism.
i.e.,
which maps a
G
neighborhood o f
G;
A coverinG
homeomorphically o n t o a neighborhood o f
A covering i s
u n i v e r s a l i f i t c o v e r s a l l o t h e r c o v e r i n g groups. If
a u n i v e r s a l c o v e r i n g , we s a y t h a t
G
id,^)
is
i s s i m p l y connected.
Remarks .:
(a)
A covering
( n , ~ ) of a connected g r o u p i s n e c e s s a r i l y
c e n t r a l a s was n o t e d a t t h e beginning o f t h i s s e c t i o n .
(b)
If a u n i v e r s a l covering e x i s t s , t h e n i t i s unique and
each o f i t s coverings of o t h e T c o v e r i n g groups i s unique.
This f o l l o w s from t h e f a c t t h a t a connected group i s
g e n e r a t e d by any neighborhood of 1
.
(c)
If
i s a Lie group, t h e n a u n i v e r s a l covering f o r
G
e x i s t s and s i m p l e connectedness i s e q u i v a l e n t t o t h e
G
p r o p e r t y t h a t e v e r y c o n t i n u o u s l o o p can b e shrunk t o a
Theorem 1 3 :
If
-
-
point
( s e e C h e v a l l e y , L i e Groups o r Cohn, L i e ~ r o u p s . )
i s a u n i v e r s a l Chevalley group o v e r
G
a s a L i e group, t h e n
a
viewed
i s simply connected.
G
B e f o r e p r o v i n g Theorem 1 3 , we s h a l l f i r s t s t a t e a lemma
whose proof we l e a v e a s an e x e r c i s e .
Lemma 41:
If
tl,t 2 , . . .
-
.
Iti\ < € , A =
1,2,=..,n
,t
a r e complex numbers such t h a t
n
and X ti = 0 , t h e r e e x i s t ti and
i =l
]ti+t-1 < E .
*such that
3
Proof of Theorem 1 3 :
Let
( n , ~ ) be a covering of
i s i n v e r t i b l e s o we may s e t
in
G.
of
G
We s h a l l show t h a t
rp
A
,
,
E3 i . e . ,
onto
that
t.
3
=
hL
,
id,^)
covers
holdon the
C
Locally
on some neighborhood o f
n
1
can b e extended t o a homomorphism
rp
can b e extended t o a l l o f
B
G.
G
n , .
I t s u f f i c e s t o show
so t h a t t h e r e l a t i o n s
yxts.
Consider t h e r e l a t i o n s
(A
Since
a ( t )rpxa(u) = rpxa
$D
y
i s l o c a l l y an isomorphism,
(A) holds f o r
It1
xa ( t ) =
E, lul
E.
rpxa(ti).
If
rp
see that
cp x u ( t ) i s w e l l d e f i n e d .
t ,u
E
k.
there is
t
E>
E K ,t
Ct,,
=
such that
] t i ]< E
,
Clearly, (A) then holds f o r
A l t e r n a t i v e l y , we c o u l d n o t e t h a t
a l l y equivalent t o
0
Using i n d u c t i o n and Lemma 4.1, we
then set
all
aEC.
(t+d
Xa
and hence simply connected.
is topolcgicThus,
m
Xa
e x t e n d s t o a homomorphism o f
Clearly t h e extension of
into
to
$9
E
and
let
S be t h e s e t of r o o t s of t h e form
let
X
be roots
l a + j p (1,j
be t h e c o r r e s p o n d i n g u n i p o t e n t subgroup o f
*
Topologically,
XS
homomorphism of
cn
i s equivalent t o
hence simply connected.
A s before
into
E,
holds.
'
i s unique.
a
(B), l e t a ,p
To o b t a i n t h e r e l a t i o n s
(A)
f o r some
a f
2
f3
,
EZ'), and
G.
n,
and i s
can b e extended t o a
P,
and t h e r e l a t i o n s
(B) h o l d .
This e x t e n s i o n i s c o n s i s t e n t w i t h t h o s e above, by t h e uniqueness
of t h e l ~ ~ t t e r .
.
We now c o n s i d e r
-1 x a (-1)
wa(-l)
)
xu it-$
x,(t-1)xda(l-t
t
Hence, if
Thus,
i s near
,
1 in
where
-+'
pa( u )
= y ha
Since
a
is central).
( t)?ha (u)rpha( t)
-1
wa(-1)
=
xY = y-1 x y .
y ha ( t ) i s n e a r
then
~ o h , ( t ) i s m u l t i p l i c a t i v e near
where ( r e c a l l t h a t
(c)
h a ( t ) = x a ( t ) x-a (--t-')xa(t)
1 in
1 and hence Abelian every--
We t h e n have
= r ? h a ( t 2 u ) ( o h a ( t 2 ) - l by Lemma 3 7 ( f ) .
'has s q u a r e r o o t s , we have
?ha
i s m u l t i p l i c a t i v e , i .e . ,
holds.
SLn( Q ) , Spn ( a; ) , and
Spinn(
) a r e simply
connected. These c a s e s can a l s o b e proved by i n d u c t i o n on n.
Examples:
-
(See C h e v a l l e y , Lie Groups, Chapter 11.)
Remarks :
(a)
If
then,relations
i s r e p l a c e d by
E.
i n t h e preceding discussion,
( A ) , (B) can be l i f t e d e x a c t l y a s b e f o r e .
c~h, i s s t i l l m u l t i p l i c a t i v e i f one o f t h e two arguments i s
Also
positive.
Further
long r o o t ,
n a (-114
and, i f
= 1
(b)
ker n
t.ype
( - 1 )8
or
i.s g e n e r a t e d by
(n
Cn
=
>
-
y h a (-1)2,a
1) i s excluded,
a fixed
then
Prove a l l o f t h i s .
1.
Moore h a s c9:lstructed a u n i v e r s a l c o v e r i n g of
h a s determined t h e fundamental group i n c a s e
is a
k
u s i n g a p p r o p r i a t e m o d i f i c a t i o n s of t h e d e f i n i t i o n s
and
G
p-adic
(here
field,
is
G
t o t a l l y discnnnectec'.. )
.(c)
Let
b e a Chevalley group o v e r
G
n : G' 4 G
i n g u n i v e r s a l group, and
If
k, G
9
t h e correspond--
t h e n a t u r a l homomorphism.
i s a l g e b r a i c a l l y c l o s e d and i f o n l y a p p r o p r i a t e c o v e r i n g s a r e
k
t
a l l o w e d , t h e n (n,G )
i s a u n i v e r s a l covering o f
G
i n t h e sense
of a l g e b r a i c groups.
We c l o s e t h i s s e c t i o n wit..ll a r e s u l t I n which t h e c o e . f f i c i e n t s
may come from any r i n g ( a s s o c i a t i v e w i t h
based i n p a r t on a l e t t e r from J. Milnor.
let
GL(R)
1). The development i s
Let
R
b e t h e r i n g , and
b e t h e group o f i n f i n i t e m a t r i c e s which a r e e q u a l t o
t h e i d e n t i t y everywhere e x c e p t f o r a f i n i t e i n v e r t i b l e b l o c k i n
t h e u p p e r l e f t hand cc::;i:;.-.
b e t h e subgroup of
l+tE
is
unit.
GL(R)
,. . .
GLn (R! C G L ( R ) , n = 1 , 2 ,
Le-5 E(R)
g e n e r a t e d ?y t h e elementary m a t r i c e s
T:lus,
( t E ~ , i : ! j ~ i , j = I - ~ 2 ,where
~ , * ) E, . - - i s t h e u s u a l m a t r i x
For example, i f
R
13
i s a f i e l d , then E(R) = SL(R),
a
simple group whose double c o s e t decomposition i n v o l v e s t h e i n f i n i t e
symmetric group.
I n d e e d , if
R
i s a E u c l i d e a n domain, t h e n
Lemma 42 :
=eGL(R)
(a
E(R)
(b)
E(R) =
Proof:
The r e l a t i o n
7
E(R)
and hence a l s o
B'E(R)
, 1 + Ek J ) = 1+ t E 1- 3DGL(R).If x , y 5 GL,(R)-,
(1+ t Eik
x<ly-l
E E (R)
We c a l l
K ~ ( R )= G L ( R ) / E ( R )
because i n
GL2,(R)
(b)
then
we have
t h e Whitehead g r o u p o f
R =
H[G]
i s a E u c l i d e a n domain, t h e n
IC1(R)
i s used i n topology.
shows
The c a s e i n which
R.
This concept
i s of particular
interest.
Example:
If
R
group of u n i t s .
By Lemma
K2(R) = ker a
=
,
the
(See IvIilnor, Whitehead Torsion_. )
42
and ( i v ) ,
.
T h i s n o t a t i o n i s p a r t l y motived by t h e f o l l o w i n g
E(R)
has a
u.c.e.
( ~ , u ( R ) ) . Set
e x a c t sequence.
1->
R*
K~ ( R )
->
U ( R ) ->GL(R)
->K~(R)
->
1.
i s a f u n c t o r from r i n g s t o Abelian groups w i t h t h e f o l l o w i n g
K2
t
p r o p e r t y : if R -> R
i s o n t o , t h e n so i s t h e a s s o c i a t e d map
Remark:
i s known t o t h e l e c t u r e r i n t h e f o l l o w i n g c a s e s :
K2
(a)
If
i s a f i n i t e f i e l d (or an algebraic extension
R
4
~f a f i n i t e f i e l d ) ,
(b)
If
R
(c)
If
R =
then
K2 = 1.
i s any f i e l d , s e e Theorem 1 2 .
Zy t h e n
J
K =~ 2. ~
Here ( a ) f o l l o w s from Theorem 9 and t h e n e x t theorem, and a p r o o f
of ( c ) w i l l b e s k e t c h e d a f t a r t h e remarks f o l l o w i n g t h e c o r o l l a r i e s
t o t h e next theorem.
Theorem I-,!+:
Let
U ( R ) b e t h e a b s t r a c t group g e n e r a t e d by t h e
x. . ( t ) ( t
symbols
R, i
1J
$
1,2,.
j , i ,j =
. .)
subject t o t h e
relations
(A) x. . (t,) i s a d d i t i v e i n t .
1J
If
x :
then
u ( R ) + E (R)
( ~ , u ( R ) ) is a
Proof:
so t h a t
(a)
x
i s t h e homomorphism g i v e n by
u. c. e .
x i s central.
i s a product o f
t h e subgroup o f
U(R)
Now by ( A )
(B)
and
,
for
x
If
x. -1s
1J
x. ( t )->l+t E. . ,
1J
1J
E(R).
,
kern
1,j < n.
with
g e n e r a t e d by t h e
any element o f
choose
n
Let
xknTs (K # n ,
Pn
l a r g e enough
Fn
k =
be
1,2,
can be e x p r e s s e d a s
.. )
T[r x k n ( t k ) .
Since i n
isomorphism.
i
n
.
Also by ( A ) and ( B )
Thus,
and s i n c e
IP,
n
x
particular,
,
.
x. . ( t ) Pn x. .(t)-' C
-- P n i f
1J
x
i s i n t h e center of
(b)
i s universal.
n
13
y E P n , then
x P n x -1 E P ,
If
*
i s a n isomorphism we have
commutes w i t h a l l
i s an
nlpn
*
xkn(t).
x n k ( t ) and h e n c e w i t h a l l
with a l l
Thus,
t h i s form i s unique,
E(R)
(x,y)
n ( x , ~ )= 1,
=
In
1.
x
Similarly,
commutes
x. . ( t ) = (xi,(t) x - ( l ) 1.
'
1J
nJ
u(R).
From ( B )
,
it f o l l o w s t h a t
U(R) = ~ u ( R ) .
Hence it s u f f i c e s t o show i t c o v e r s a l l c e n t r a l e x t e n s i o n s .
( Y, A )
of
be a c e n t r a l extension of
A.
and l e t
E(R)
be t h e center
C
We must show t h a t we c a n l i f t t h e r e l a t i o n s ( A ) a n d ( B )
t o A.
Fix
Yi. J ( t ) E
i ,j i
Y-'
and choose
j
p f i ,j.
Choose
($1
x. - ( t ) s o t h a t
(y. ( t )
(1))= y i j ( t ) .
IP
Y~~
w i l l prove t h a t t h e y f s s a t i s f y t h e equations (A) and ( B ) .
and
1~
(bl)
If
q
k
Choose
Since
Lct
yik(t)
y q j (1),
, then
i & j, k f
and w r i t e
,
yik(t)
1,
C
with
y l
.
y~
9
E
C-
(u)
.( u )
Hence
J
f i x e d , i s Abelian.
(b2)
iyij(t)
(b3)
The r e l a t i o n s ( A ) h o l d .
i ,j
y q J- ( u ) commute.
y 4 j ( u ) = 3 ( y X q ( u ), y q J ( 1 ) ) , c
commutes up t o a n element o f
i t commutes w i t h
and
We
The p r o o f i s e x a c t l y t h e
same a s t h a t o f s t a t e m e n t ( 7 ) i n t h e p r o o f o f Theorem 1 0 .
(b4)
yij(t)
If
q f p,i,j, set
by
w
in
($1
i s i n d e p e n d e n t of t h e c h o i c e o f
w = y q p (1)y p q ( - l ) y q p ( 1 ) . T r a n s f o r m n g
and u s i n g ( b l ) we g e t
(*I w i t h
(b5)
The r e l a t i o n s ( B ) h o l d .
(
I f , a , b ,c
q
i n p l a c e of
p.
We w i l l u s e :
a r e elements of a group such t h a t
a
p.
(*)
commutes w i t h
then
(
(
c
and s u c h thAt
(b,c)
( a , ( b , c ) )= ( ( a , b ), c ) . Since
a
,( b c c
Now assume
a
commutes w i t h
commutes w i t h
The o t h e r c o n d i t i o n s i n s u r e
are distinct.
i ,j ,k
Choose
(a$)
and
c,
c, ( a , ( ~ , c j )
( ( a , b ) , ( b , c ) c ) =( ( a , b ) , c ) -
q f i ,j , k ,
so t h a t
This completes t h e proof of t h e theorem.
Let
d e n o t e t h e subgroup o f
un(R)
C o r o l l a r y 1: If
n
C o r o l l a r y 2:
R
If
- 5,
>
then
Un(R)
U(R)
generated by y. . ( t )
1J
i s c e n t r a l l y closed.
i s a f i n i t e f i e l d and
n
2 5
then
SL~(R)
i s c e n t r a l l y closed.
Proof:
En ( R )
T h i s f o l l o w s from C o r o l l a r y 1 and t h e e q u a t i o n s
=
S L ~ ( R )= u,(R).
Remarks:
It f o l l o w s t h a t i f
(a)
R
i s a f i n i t e f i e l d and i f
S L ~ ( R ) i s not c e n t r a l l y closed, t h e n e i t h e r
IR 1 5
4
and
n
5
4.
]R
=
9, n = 2
or
The e x a c t s e t of e x c e p t i o n s i s : b ~ ~( 4 ) ;
S L 2 ( 9 ) , SL3 ( 2 1 , SL3 ( 4 1 , SL4 ( 2 )
* Exercise:
(b)
Prove t h i s .
The argument above can be phrased i n t e r m s o f r o o t s , e t c .
A s s u c h , i t c a r r i e s o v e r v e r y e a s i l y t o t h e c a s e i n which a l l r o o t s
have one l e n g t h .
(c)
The o n l y o t h e r e x c e p t i o n i s
~ ~ ( 2 ) .
By a more c o m p l i c a t e d e x t e n s i o n of t h e argument, it can
a l s o b e shown t h a t t h e u n i v e r s a l Chevalley group of t y p e
B
n
o r Cn
o v e r a f i n i t e f i e l d ( o r an a l g e b r a i c e x t e n s i o n o f a f i n i t e f i e l d )
i s c e n t r a l l y closed i f
n
i s l a r g e enough.
number o f u n i v e r s a l Chevalley g r o u p s w i t h
k
Hence, o n l y a f i n i t e
indecomposable and
Z
f i n i t e f a i l t o be c e n t r a l l y closed.
Now we s k e t c h a proof t h a t
The n o t a t i o n
...
U,Un,
K2 (
a) i s
above w i l l b e used.
a g r o u p o f o r d e r 2.
The p r o o f depends on
t h e following r e s u l t :
*ij
,(I
(1) For
n
> 3,
j = 1
.
i
, . n
SL,(Z)i s
g e n e r a t e d by symbols
subject t o the relations
j
x i j-1 X i j
, h 1. .J
=w-.
1J
'
then
h. 2_ = 1
1J
.
with t h e usual
x - - ( 1 and u s i n g
1J
1J
( t ) = X . - ( l ) t, we s e e t h a t t h e r e l a t i o n s ( B ) h e r e imply
Xij
1J
t h o s e of Theorem 14. S i n c e t h e l a s t r e l a t i o n may b e w r i t t e n
Identifying
x
( - 1 = h ( 1 and + 1
a r e t h e o n l y u n i t s of
1J
1J
S L , ( ~ ) d e f i n e d by t h e u s u a l r e l a t i o n s A
Bc
,
h
Perhaps t h e r e a r e o t h e r r i n g s , e.g.,
which t h i s r e s u l t h o l d s .
of
we have
86.
t h e p-adic i n t e g e r s , f o r
For t h e proof of (1) s e e W. Magnus,
Acta Iqath. 64 ( 1 9 3 4 ) , which g i v e s t h e r e f e r e n c e t o N i e l s e n , who
proved t h e key c a s e
n
=
3 (it t a k e s some work t o c a s t N i e l s e n l s
r e s u l t i n t o t h e above f o r m ) .
-1
= X
-1
2l
,
The c a s e
n
=
2,
with
(B)
r e p l a c e d by
i s s i m p l e r and i s proved i n an appendix
t o Kurosh, Theory o f Grpups.
Now l e t
xi
If
Cn
(2)
I
n
I.
2 3,
then
,
,
w
h
-
1J
r e f e r t o elements of
is the kernelof
i s g e n e r a t e d by
Cn
A s u s u a l , we o n l y r e q u i r e
'
nn : U n
h122 , and
(c)
when
.
Z) .
(a)->S L n ( z )
and
2 2 = 1.
(h12)
1,j = 1,2,
and
h12
E Cn
,
2
= 1 amounts t o d i v i d i n g by <h12>
which t h u s e q u a l s
h12
1
The r e l a t i o n h h h-l = hT2
which may b e deduced from
23 1 2 23
Setting
n
'
Un(
,
( B ) as i n t h e proof of Lemma 3 7 , t h e n y i e l d s
There i s a n a t u r a l map
Assume n o t .
? c4
~ ~x 1 J(
12
=
un(Z)
->
.
h;
Un
(W),
,
o n t o h12(-1)2
which ( s e e
h12
Remark ( a ) a f t e r t h e proof o f Theorem 13) g e n e r a t e s t h e k e r n e l o f
u
>
-
1 .
h
This maps
SLn( @ ) .
s i m p l y connected.
Thus
Since
R) i s c e n t r a l l y c l o s e d , hence
SLn( R ) can b e c o n t r a c t e d t o SOn
S k (
(by t h e p o l a r d e c o m p o s i t i o n , which w i l l b e proved i n t h e n e x t s e c t i o n )
I
I
which i s n o t simply c o n n e c t e d s i n c e
Spin,
-+
SOn
is a nontrivial
c o v e r i n g , we have a c o n t r a d i c t i o n .
I
It now f o l l o w s from ( 2 ) , (3) and Theorem 14 t h a t
J K ( 2)
~ I
By C o r o l l a r y 1 above t h e same c o n c l u s i o n h o l d s w i t h
r e p l a c e d by any SLn( I) w i t h
Exercise:
kn
with
Let
k
SAn
again
be
54, x
a field.
n
= 2.
SL(.~)
- 5.
>
t r a n s l a t i o n s of t h e underlying space,
I..,
SAn
i s t h e group of a l l
(n+l)
SAn
j =
(n+l)
m a t r i c e s o f t h e form
i s generatedby
...1
1
(I)
.
x
iJ
[ ; :]
where
( t ) ,t E k , i f j, i = 1 , 2
x
E a h , YE^"
,..., n ,
Prove:
If t h e r e l a t i o n
( c ) hi (t)
i s multiplicative.
i s added t o t h e r e l a t i o n s ( A ) and ( B ) o f Theorem 1&,a
complete s e t of r e l a t i o n s f o r
(2) If k
is finite,
(c)
SAn
i s obtained.
may b e omitted.
( 3 ) If n i s l a r g e enough, t h e group d e f i n e d by ( A ) and (B)
is a
u. c. e .
for
SAn
.
( 4 ) Other analogues o f r e s u l t s f o r
We remark t h a t
SLn.
S A Z ( c ) i s t h e u n i v e r s a l c o v e r i n g group o f t h e
inhomogeneous b r e n t z group, hence i s o f i n t e r e s t i n quantum
mechanics.
5
8.
V a r i a n t s of t h e Bruhat lemma.
k, B
...
(a)
G =
Let
be a Chevalley group,
G
We r e c a l l (Theorems 4 and
a s usual.
U BwB ,
4'):
a d i s j o i n t union.
wew
w
For each
(b)
on t h e r i g h t .
W
E.
,
,
BwB = BwUW
w i t h u n i q u e n e s s of e x p r e s s i o n
Our purpose i s t o p r e s e n t some a n a l o g u e s of ( b ) w i t h
applications.
a
For each simple r o o t
of
w,
,
N/H
in
Theorem 1 5 :
For each simple r o o t
w e W
, is
chosen i n
a l s o denoted w a
representatives f o r
For each
G =
a
B~\G,
-
a YX-a
a
and assume t h a t t h e r e p r e s e n t a t i v e
,
group of r a n k 1, Ba = B n G a
<x
we s e t
B,)
a
,
let
Ya
o r more g e n e r a l l y f o r
choose a minimal e x p r e s s i o n
... Y..
0
.
be a system of
= waw~
a product of r e f l e c t i o n s r e l a t i v e t o simple r o o t s
B w B = BYaYD
GGL
a, $
\
B BwaB
wg
. .
BwB
Since
G
C,
-
=
BvlGBwGwE
=
BwCL BY
=
BY Y
G
P
as
Then
w i t h u n i q u e n e s s of e x p r e s s i o n on t h e r i g h t .
BG = Ba waB a ' t h e second c a s e above r e a l l y
i s more g e n e r a l t h a n t h e f i r s t . We have
Proof:
... Y8
... Y8
( b y Lemma 2 5 )
(by induction)
( b y t h e c h o i c e of
YCL )
.
.
NOW
assume
P F
- b 9 Yayp
YyY8 -
bycyB
or
Then
by a
.
Bw~B
9
-1
... y y = b v yo ... y ymym
9
etc.
9
YyYz
0
with b, b
9
,
e B
-1
ymy,
e B
0 0
0
We have
The second c a s e can n o t o c c u r s i n c e t h e n t h e l e f t s i d e
would b e i n
Bww-B
0
BwB
and t h e r i g h t s i d e i n
From t h e d e f i n i t i o n of
by i n d u c t i o n t h a t
Y~
Y,
0
v
- yy
it f o l l o w s t h a t
,
.
->
G
( b y Lemma 2 5 ) ,
9
y j = yC,
,
and t h e n
whence t h e u n i q u e n e s s i n
Theorem 1 5 ,
Lemma 43:
Let
( s e e Theorem 4
c :
0
,
SL2
Cor. 6 ) .
b e t h e c a n o n i c a l homomorphism
CI
Then
s a t i s f i e s t h e c o n d i t i o n s of
Ya
Theorem 1 5 i n each of t h e f o l l o w i n g c a s e s ,
k = Q(resp.R )
(b
of t h e e l e m e n t s of
SU2
with
-
i s t h e image under
qa
b
>
0
of
,
a-
I
--
(c)
Ya
( r e s p . SO2) ( s t a n d a r d compact f o r m s )
-6
a
t h e form
and
If
e
i s a p r i n c i p a l i d e a l domain
(commutative
-0J,
with l ) , e
i s t h e g r o u p of u n i t s ,
k
i s t h e quotient f i e l d ,
i s t h e image under ma
of t h e e l e m e n t s of S L 2 ( a ) of
Ta b l
t h e form
w i t h c r u n n i n g t h r o u g h a s e t of r e p r e s e n t a t i v e s
and
YCI
for
- o ) / ,
.I,
and f o r each
,
c
a r u n n i n g o v e r a s e t of
r e p r e s e n t a t i v e s f o r t h e r e s i d u e c l a s s e s of
Proof:
We have ( a ) by Theorem 4
( b ) and ( c ) we may assume t h a t
0
Ga
mod
o
applied t o
is
SL2
and
Gc
c
.
.
To v e r i f y
BCI t h e
B2
s u p e r d i a g o n a l subgroup
SL*(Q)
k e r cpu C B2
since
can be c o n v e r t e d t o one of
SU2
.
Any element of
by a d d i n g a m u l t i p l e of
t h e second row t o t h e f i r s t and n o r m a l i z i n g t h e l e n g t h s of t h e
rows.
N
,
(B2(C)
f) SU, )\su,
with
k
as in (c).
such t h a t
b, d
in
pa
+ qc
We choose
= 0
so t h a t
o
whence ( b )
ad
-
(a)
wUw = wa% a
1
.
B2(Q) \ s L ~ ( Q )
[:
Now assume
a, c
bc = 1
11
in
o
SL2(k)
r e l a t i v e l y prime and
.
Multiplying t h e preceding
,
and
follows.
(c)
The c a s e ( a ) above i s e s s e n t i a l l y Theorem 4
. w p x P ... w6X
.
B~ ( k )
we g e t a n element of
S L ~ ( =~ B) 2 ( k ) S L 2 ( a )
Remarks:
Then
( u s i n g unique f a c t o r i z a t i o n ) , and t h e n
m a t r i x on t h e r i g h t by
Thus
.
SL2 (Q) = B2 ( C ) .SU2
Thus
P
since
i n t h e n o t a t i o n of Theorem 1 5 , by
Appendix I1 25, o r e l s e by i n d u c t i o n on t h e l e n g t h of t h e e x p r e s s i o n .
(b)
I n ( c ) above t h e c h o i c e can be made p r e c i s e i n t h e f o l l o w i n g
cases:
(1) a
and
and
= a ;choose a ,
(2)
o = F[X]
dg a
<
dg c
(3)
o
=.z
(p-adic
P
a
(F
c
so t h a t
a field);
.
O<_a< c
choose s o t h a t
c
i s monic
.
i n t e g e r s ) ; choose
a n i n t e g e r such t h a t
0<
- a
<
c
c
a power of
p
.
7~~~K+i+;~p~+~p?:~.%:3
,
:
..-
I n what f o l l o w s we w i l l g i v e s e p a r a t e b u t p a r a l l e l
developments of t h e consequences of ( b ) and ( c )
above.
In
k = @ f o r definiteness, t h e case
we w i l l t r e a t t h e c a s e
(b)
k =
being s i m i l a r .
L e m a 44:
Let
(a)
and
[Xa,
H ~ ] be a s i n Theorem 1.
There e x i s t s a n i n v o l u t o r y seniantomorphism
( r e l a t i v e t o complex c o n j u g a t i o n of
%X,
=
-X-a
qj Ha
and
(b)
-
- Ha
{ X , Y]
z'
of
such t h a t
)
a
f o r every r o o t
2 t h e form
On
@
5
.
(X, 6 , Y )
d e f i n e d by
in
t e r m s of t h e K i l l i n g form i s n e g a t i v e d e f i n i t e .
Proof:
T h i s b a s i c . r e s u l t i s proved, e.g.,
i n Jacobson, L i e
a l g e b r a s , p. 147.
Theorem 16:
Let
L i e group over
(a)
such t h a t
G
Proof:
viewed a s a
There e x i s t s a n a n a l y t i c automorphism
= x
-Ci
The group
(-t) and
K = G
maximal compact subgroup of
holds
@
d.
cx,(t)
(h)
be a C h e v a l l e y group o v e r
G
oh ( t ) = h
G
Q
( z -l)
r of
for a l l
of f i x e d p o i n t s of
c is a
and t h e decomposition
G = BK
ff
G
o:
and
(Iwasawa d e c o m p o s i t i o n ) .
Let
con j u g a t i o n , and
be
p
i n Lemma 44 composed w i t h complex
t h e r e p r e s e n t a t i o n of
used t o d e f i n e
G
.
Applying Theorem 4
P
,
5 t o t h e Chevalley groups (both equal t o
COP.
G) c o n s t r u c t e d from t h e r e p r e s e n t a t i o n s
we g e t a n automorphism o f
yo71
,
of
which a s i d e f r o m complex c o n j u g a t i o n
G
,
and
JD
s a t i s f i e s t h e e q u a t i o n s o f ( a ) , hence composed w i t h c o n j u g a t i o n
From Theorem 7 a d a p t e d t o t h e p r e s e n t
s a t i s f i e s these equations,
s i t u a t i o n ( s e e t h e remark a t t h e end o f
i s a n a l y t i c , whence ( a ) .
x,
by t h e serniautomorphism
o;;
45:
K = Gr
Let
(a) Kc="E
(b)
5 ) it f o l l o w s t h a t
We o b s e r v e t h a t i f
a j o i n t r e p r e s e n t a t i o n of
LMuna
3
G
i s d e f i n e d by t h e
o- i s e f f e c t e d by c o n j u g a t i o n
then
of Lemma 44
Kn G,
.
, Ka
=
SU2
( s e e Lemma 4 3 ( b ) ) , hence
B ~ H= ~ (=h
e
r
f o r each simple r o o t
HI 1
)1
YG
.
c
C
Kc
.
1)
,
A
= 1 for a l l
L
6
(global weights ) 3
hi(ti)
=
( s e e Lemma 2 8 )
= maximal t o r u s i n
Proof:
and
on
The k e r n e l o f
q,
: SL2
c:.
->
K
1
( t i /=
11
.
i s contained i n
Gc.
15
cr p u l l s back t o t h e i n v e r s e t r a n s p o s e c o n j u g a t e , s a y
.
SL2
S i ~ l c et h e e q u a t i o n
we g e t ( a )
=
-
x
h a s no s o l u t i o n s
.
Since c h a ( t ) =
hc(T -')
(here
.
y
r2x
5
and
A
,
p ( hc ( t ) )= t P ( H c )
a r e c o r r e s p o n d i n g w e i g h t s on
qt!
and
H)
,
,
h,(t)
and the
h
H
E
, so
that
A
weights
P
there are
1
fi
l i n e a r l y independent w e i g h t s
for a l l
, for
p r o d u c t of t h e
fi
, then
p
i
all
.
If
c(h) =
::
'G
,
AHi)
n
is universal, then
Bb
ihi(-) j
, hence
.
Since
i
we s e e t h a t i f
[tiln= 1 f o r some
G
circles
4
, then
hi(tij
-'f o r a l l
( j ~ ( h )=( 1 f o r a l l
i f a n d o n l y if
4
h =
-
c(o-h) = p(h)
If
fi
=
o-h = h
have
-
IP(~)=
[ 1
Iti/
, we
H
generate
104
>
0
, whence
is t h e
is a t o r u s ; if n o t ,
we have t o t a k e t h e q u o t i e n t by a f i n i t e group, t h u s s t i l l have a
torus.
Now if
fi
a ( h ) (a
E
h
E
Hg
i s g e n e r a l enough, s o t h a t t h e numbers
X) a r e d i s t i n c t and d i f f e r e n t from
1
, then
, the
Gh
,
is H , by t h e uniqueness i n Theorem
c e n t r a l i z e r of h i n G
:
4 , s o t h a t Hc i s i n f a c t a maximal Abelian subgroup of Gr ,
which p r o v e s t h e lemma.
Exercise:
Check o u t t h e e x i s t e n c e of
h
and t h e property
Gh = H
above.
By Theorem 1 5 and
Now we c o n s i d e r p a r t ( b ) of Theorem 16.
Lemmas 4 3 ( b ) and 45 ( a ) we have
(BwB) C T -C BrK u...Kg
,a
SU2
H
n = 0
.
By t h e same r e s u l t s
i s being used).
G , K1
be a compact subgroup of
E
BK
K
Thus
with
.
b
Since
1
2,
E
K1
B, y
E
K
,
2
K
.
and t h e n
Assume
x
b
with
uh
=
i s compact, a l l e i g e n v a l u e s
a r e bounded, whence
Then a l l c o e f f i c i e n t s of a l l
un
h
E
K
E
G,
is
Let
K1
=
( T h i s a l s o f o l l o w s e a s i l y from Lemma 4 4 ( b ) 1
compact.
h
=
compact s e t s i n c e each f a c t o r i s ( t h e
compactness of t o r i and
x = by
G
.
K1
u
E
Write
U,
bc ( ,n n )
by Lemma 4 5 ( b ) .
a r e bounded s o t h a t
u = 1
.
-
K i s s e i - ~ i s i m p l ea n d t h a t a
It can b e shown a l s o t h a t
Remark:
complete s e t of s e m i s i m p l e compact L i e g r o u p s i s g o t f r o m t h e
above c o n s t r u c t i o n .
C o .r o .l l a. r y 1: L e t
b e of t h e same t y p e a s
G'
G, K
l a t t i c e c o n t a i n i n g t h a t of
natural projection.
Proof:
f
Then
7tK
=
t
f
and
= Gr,
K
w i t h a weight
G
n:
G
1
->
.
T h i s f o l l o w s f r o m t h e f a c t p r o v e d i n Lemina 45 t h a t
i s g e n e r a t e d by t h e g r o u p s
Examples:
--
If
G = SLn(C)
(b)
If
G = SOn(fJ
, then
, then
K = Sun
.
f i x e s siinultaneously
K
C x ixn + l - i a n d C x 1. 7i , h e n c e e q u a l s SOn ( K )
p a c t f o r m ) a f t e r a change o f c o o r d i n a t e s . Prove t h i s .
t h e forms
(c)
'
n
--
(~iy2ncl-i
1
SUn(IL))
If
,
G = S P ~ ~ ( ~ ' t)h e n
- X2n+1-i y . )
and
1
(compact f o r m ,
=
K
.
q,SU2
(a)
the
G
L' x i q
K
(com-
f i x e s t h e forms
, and
quaternions).
i s isoinorphic t o
For t h i s s e e C h e v a l l e y ,
L i e g r o u p s , p. 22.
I
(d)
Tile have isomorphisms and c e n t r a l e x t e n s i o n s
,
I
I
5 (p) ,
4 (R
S U 2 ( l ~ )--->
SO
S U ~ ( ~ ) '
snL(E) ->
so6(R)
(compact f o r m s ) .
>
a
-
SO
,
This follows from ( a ) , ( b ) , ( c ) , Corollary 1 and t h e equivalences
-2:
-
Tlie group . K
.
.
Since
C o r o l l a r y 3 : If
T
The limp
~ \ f i under
->
6
B\,
G = BK
.
B\G
T
Corol_lary:
.
\t
G
(b)
If
T\K
t h e n a t u r a l map.
->
Bk
, hence
,
i s c o n t i n u o u s and con-
l e a d s t o a c o n t i n u o u s map of
T = B
nK
and
is compact, t h e map i s a homeomorphism.
(
(a)
, then
H6
1 - 1 and o n t o s i n c e
which i s
Since
k
T\
s t a n t on t h e f i b r e s of
into
K
denotes t h e maximal t o r u s
i s homeomorphic t o
.-Proof:
i s g e n e r a t e d by t h e groups
is connected, s o is
SU2
P
T\r
K
A s a l r e a d y remarked,
Proof:
C__n
v,SU2
i s connected.
is contractible t o
I(
is universal, then
G
.
K
i s s i m p l y con-
nected.
A
Proof:
Let
H = AT
, so
__r_
i1
that
G = BK
n e s s of e x p r e s s i o n .
for a l l
UAK
.
On t h e r i g h t t h e r e is unique-
->
is contractible t o a point,
G
is universal, then
hence s o i s
Ii
$=
..,6
TY,.
Proof:
p E L]
Then we have
is compact it e a s i l y f o l l o w s t h a t
G
i s a homeomorphism.
G
is contractible t o
Since
.
K
UA
If a l s o
i s s i m p l y connected by Theorem 1 3 ;
.
C o r o l l a r y 5 : For
a,B,.
G
.
fi
0
K
Since
UA x K
t h e n a t u r a l map
=
A
>
= ih E Hl,u(h)
w
E
Y set
b e a s i n Theorem 1 5 .
..Y6 , w i t h
,
( B W B ) =~ BwB A K =
Then
K
=
V lcw
and l e t
and
W
uniqueness of e x p r e s s i o n on t h e r i g h t .
This f o l l o w s from Theorem 1 5 a n d Lerrma 4.3 ( b )
.
Remark:
Observe t h a t
i s e s s e n t i a l l y a c e l l s i n c e each
K
& ( c o n s i d e r t h e v a l u e s of
i s homeomorphic t o
i n Lemma 4 3 ( b ) )
a
A t r u e c e l l u l a r decomposition i s o b t a i n e d by w r i t i n g
u n i o n of c e l l s .
6:
T
.
as a
p e r h a p s t h i s decomposition can b e u s e d t o g i v e
K
a n e l e m e n t a r y t r e a t m e n t of t h e cohornology of
Corollary
ya
B\F
T\K
and
.
have a s t h e i r ~ o i n c a r gpolynomials
.
t 2 N ( w ) They have no t o r s i o n .
C
w &I?
Proof:
We have
.
2N(w)
dimension
,a
wUw
homeomorphic t o
B\BWB
c e l l of r e a l
S i n c e each dimension i s even, it f o l l o w s t h a t
t h e c e l l s r e p r e s e n t independent elements of t h e homology group
and t h a t t h e r e i s no t o r s i o n ( e s s e n t i a l l y because t h e boundary
o p e r a t o r l o w e r s dimensions by e x a c t l y 1), whence Cor
Y,
n a t e l y one may u s e t h e f a c t t h a t each
Remark:
. 6.
dlter-
&
i s homeomorphic t o
The above s e r i e s w i l l be summed i n t h e n e x t s e c t i o n ,
where it a r i s e s i n c o n n e c t i o n w i t h t h e o r d e r s of t h e f i n i t e
Chevalley g r o u p s .
C o r o l l a r y 7:
For
w
E
s i o n a s b e f o r e and l e t
W
let
S
w
=
..w 6
w,.
b e a minimal expres-
d e n o t e t h e s e t of elements of
W
each
of which is a produ.ct of some subsequence of t h e e;cpression f o r
w
.
Proof:
and
%
Then
If
TaYa =
TG = T 17 K,
K,
, we
have
u
.
II
w t ES w f
K, = TaYa U TCL
by t h e c o r r e s p o n d i n g r e s u l t i n
.T6Y6 by Lemma 4 3 ( b ) .
-Kw > TKu ...K , and we have
6
BwB = B*T,Y,..
so that
( t o ~ o l o g i c a lc l o s u r e ) =
Hence
by Lemma
SU2
%=
.
4.5 ( a )
Now
T-TaY a . 0 . T 6 Y 6
e q u a l i t y s i n c e each f a c t o r
,
.
on t h e r i g h t i s compact, s o t h a t t h e r i g h t s i d e i s compact, hence
closed.
C
K ,,K a.
Since
K
K
e
W
w
if
i,
81
E
a
\i and
is sim-
w w,?,
u
W
p l e , by Lemma 2 5 , Cor. 7 f o l l o w s .
Corollary 8:
C o r o l l a r y 9:
(a)
T = K1
(b)
$
The s e t
i s i n t h e c l o s u r e of e v e r y
.
i s c l o s e d i f and only i f
w
=
of Cor. 7 depends o n l y on
S
t h e minimal e x p r e s s i o n c h o s e n , hence may be w r i t t e n
Proof:
-
-Kw
Because
Lemma 46:
r o o t s negative.
Proof:
-
w
W
b e t h e element of
Then
1
.
w
, not
on
S(w) ,
S(wo)
, and
W
=
0
= wmcl..
s i n c e if
n = N
.wil
W
E
.
.
w
wl...w
wo
=
=
.
w ~ ..wm.. .wn
- N ( w ) , and
w0
Corollary 10:
m
n
+
=
N
wo
N(wo)
.
w &S(wO)
we see that
If
=
i s a s above and
w0
i s one f o r
m = N(w)
i s t h e number of p o s i t i v e r o o t s t h e n
N
segment of
Then
let
which makes a l l p o s i t i v e
rn b e a minimal express i o n as a p r o d u c t of s i m p l e r e f l e c t i o n s and s i m i l a r l y f o r
wmlw
Assume
.
d o e s n ' t depend on t h e e x p r e s s i o n .
wo
Let
Kw
,
Looking a t t h e i n i t i a l
.
wo
=
w wP
.. .w 6
is a
minimal e x p r e s s i o n , t h e n
(a)
.
K = Kw
0
(b)
Proof :
(y
K = KLK $ . . . K6
(a)
By Cor
(b)
By ( a )
'
. 7 and
Lemma 46.
K = TK,Kr.Kg
s i m p l e ) , then absorb t h e
get ( b ) .
TXls
.
We may w r i t e
i n appropriate
T =
K
1s
'd
Ty
to
E x e f c i s e : If
G
above, show t h a t
(a)
Remarks:
...
wo ,a ,P ,
i s any Chovalley group and
G = BGaG
If
.
B . . .G 6
and
a r e r e p l a c e d by
SU2
are as
and
in
SO2
accordance w i t h Lemma 4 3 ( b ) , t h e n e v e r y t h i n g above goes through
except f o r Cor. 4 , Cor. 6 and t h e f a c t t h a t
torus.
'
I n t h i s case each
corresponding a n g l e s i n Cor
K,
T
i s no l o n g e r a
is a c i r c l e since
. 1 0 ( b ) , which we
SO2
is.
The
have t o r e s t r i c t
s u i t a b l y t o g e t u n i q u e n e s s , may be c a l l e d t h e E u l e r a n g l e s i n
analogy w i t h t h e c l a s s i c a l case:
G = S L ~ ( N ,K =
sOj(R)
,
Ka ,Kg = { r o t a ti o n s around t h e
(b)
, x-axis 1 ,
If K, i s r e p l a c e d by
BwB
formula f o r
z-axis
is obtained.
BwB = B K ,
(Prove t h i s . ]
i n Cor. 7, t h e
If
(or
R)
is
r e p l a c e d by any a l g e b r a i c a l l y c l o s e d f i e l d and t h e Z a r i s k i t o p o l ogy i s u s e d , t h e same formula h o l d s ,
So a s not t o i n t e r r u p t t h e
p r e s e n t development, we g i v e t h e proof l a t e r , a t t h e end of t h i s
section.
Theorem 1 7 :
(C o r
R,, K
(Cartan).
=
G6
Again l e t
a s above, and
(a)
G = KAK
(b)
In ( a ) the
G
be a Chevalley group over
A = {h
/r
E
H I I ~ ( h>
) 0
for a l l
( C a r t a n decomposition).
A-component
conjugacy under t h e Weyl group.
is determined u n i q u e l y up t o
110
Proof:
(a)
G = BK
(Theorem 1 6 ) , t h e r e e x i s t elements i n
assume
x
Given such an element
%*.
y
1a[
.
u = 1
up (up
B>O
choose
a:
then if
a
>
n
( )
,
fl
choose
y
c
so
exp cHu
=
We w r i t e
H
that
E
r
commutes w i t h
i
G,
We now choose
We must show
[a1
r
)
~
7
=
a
~
7
,
uU
$
u
H
-
1 f o r some simple
, such
n
Write
that
ua
( u , @ )> 0
+ 1,
and
~ ~ ( l ) ~ w ~ ( l )and
- ' proceed by
by
-
7
u ua
is t h e s e t of p o s i t i v e r o o t s ) .
choose
Killing
+ 1 , then
u
simple s o t h a t
replace
.
n
.
W
w6 ,
E
1 HI , t h e
=
if
We may assume
of minimum h e i g h t , s a y
i n d u c t i o n on
P
EB) .
E
1
exp H ( H
=
and
.
U4
is con2act).
K
(recall that
.KxK
We w i l l r e d u c e ( * ) t o t h e r a n k 1 c a s e .
, then
B <n
h t vi a
a
1a(
set
This f o l l o w s from;
can be i n c r e a s e d .
u =
write
This norm i s i n v a r i a n t under
t o maximize
that
, we
, then
a)
H = AT
By t h e decompositions
y = ua
u n i q u e l y determined by
norm i n
.
G
E
u
1
# p-Ial
E
Then we w r i t e
cH,
= exp H
with
a
=
E
A, a
=
a,a
1
.
,
exp H
Ha
, set
Then
a
i s orthogonal t o
1
(here
elementwise and i s orthogonal t o
7
aU r e l a t i v e
t o t h e b i l i n e a r form corresponding t o t h e norm i n t r o d u c e d above.
By
f o r groups of r a n k 1, t h e r e e x i s t
( )
1
Yu,auZ
aa
=
7 - 1 7
= yu y
y-a
a,a
E -1
7
.
n GO.
and
Since
Ga
1
do),
r a n k 1 case.
an e x e r c i s e .
) ; u , ~ - ' E u .
1a
7
>
.
a
y, z
Then
E
K,
such t h a t
1
normalizes XP- { a ] ( s i n c e
1 1 2
S i n c e /a,a I = 1a:12 + l a ' [ '
This case, e s s e n t i a l l y
G
=
SL2
7
yuaz = yu uuaaa z
,will
and
be l e f t a s
(b)
-1k2
6 X = kla
x
.issume
,s o
x
G,
E
that
=
x6x-l
-
klakZ
as in ( a ) .
kla 2 kl-1
=
--------I ,./
since
fi
Lemma
47:
group)
, they
,u(o-a) = $ ( a )
=
a(a-l)
If elements of
H
r a = a -1
Here
.
A
fi
;A E L
for a l l
Then
a r e conjugate i n
.
(any Chevalley
G
a r e c o n j u g a t e under t h e Weyl group.
This e a s i l y f o l l o w s from t h e uniqueness in Theorem
,
x
By t h e lenma
above u n i q u e l y determines
jugacy under t h e Weyl group., hence a l s o
in
a
a2
t
4
.
up t o con-
s i n c e square-roots
a r e unique.
A
Remark:
A + = ja
We can g e t uniqueness i n ( b ) by r e p l a c i n g
A
E
Ala(a) >
-=
1 for a l l
G >0)
A
by
This f o l l o w s fromAppen-
d i x I11 33.
Corollary:
crx
Let
P
c o n s i s t of t h e elements o f
which s a t i s f y
G
x-I and have a l l e i g e n v a l u e s p o s i t i v e .
=
.
CP
(a)
11.
(b)
Zvery
p
E
P
i s c o n j u g a t e under
u n i q u e l y determined up t o conjugacy under
(c)
G
KP
=
, with
K
t o some
a
E
A
,
( s p e c t r a l theorem).
11
uniqueness on t h e r i g h t ( p o l a r decom-
posit ion) ,
Proof:
u--1: p
a
.
(a)
T h i s h a s been n o t e d i n ( b ) above.
(b)
\ie can assume
=
ak-I
(Since
a
.
Thus
k
p
=
ka
E
KA
commutes w i t h
, by
a2
t h e theorem.
, hence
Apply
a l s o with
i s d i a g o n a l ( r e l a t i v e t o a b a s i s of weight vec-
t o r s ) and p o s i t i v e , t h e m a t r i c e s commuting w i t h
a
have a c e r t a i n
b l o c k s t r u c t u r e which does n o t change when it i s r e p l a c e d by
'2
k
=
i and
lc = a
-'-'
p = a .
.
P
t h e d e f i n i t i o n of
2
1
1
1
.
Then
a
.)
A.
pa
E
P
K
Since
,
so that
i s compact,
i s u n i p o t e n t by
k
k
=
1
.
Thus
The uniqueness i n ( b ) fol.l.ows a s b e f o r e .
(c)
If
that
x
with
ki
Then
p1 = k;1k2p2
=
E
x
E
, then
G
kl 1c2 l<ila k2
E
KP
and
E
P
K
pi
.
x
.
.
-
Thus
k1 a k 2
a s i n t h e theorem, s o
G = KP
.
Xssuriie
By ( b ) we can assume t h a t
4s i n ( b ) we conclude t h a t
whence t h e uniqueness i n ( c ) .
Example :
If
G = SLn
(a) , s o
that
A = { p o s i t i v e diagonal matrices
K = Sun (@)
3,
P = !positive-def i n i t e Hermitean m a t r i c e s
1,
t h e n ( b ) and ( c ) r e d u c e t o c l a s s i c a l r e s u l t s .
,
-1
klpl
p2
=
k2p2
E
A
kl k2 = 1
,
.
We now c o n s i d e r t h e c a s e ( c ) of Lemma 43.
The development
i s s t r i k i n g l y p a r a l l e l t o t h a t f o r c a s e ( b ) j u s t completed a l though t h e r e s u l t s a r e b a s i c a l l y a r i t h m e t i c i n one c a s e , g e o m e t r i c
i n t h e other.
4.
-4.
a ,e ,k,Y,
Throughout we assume t h a t
Lemma 4 3 ( c ) and t h a t t h e Chevalley group
based on
.
k
We w r i t e
are as i n
under d i s c u s s i o n i s
G
f o r t h e subgroug of elements of
Ge
M
whose c o o r d i n a t e s , r e l a t i v e t o t h e o r i g i n a l l a t t i c e
in
e
G
,a l l
lie
.
48:
is a s i n Theorem 4 '
If
Proof : If
,
Cor. 6, t h e n
i s a E u c l i d e a n domain t h e n
e
9L2 ( e )
is generated
by i t s u n i p o t e n t s u p e r d i a g o n a l and s u b d i a g o n a l e l e m e n t s , s o t h a t
t h e lemma f o l l o w s from t h e f a c t t h a t
t
i n t e g r a l polynomial i n
if
p
p
i s a prime i n
(all
a/b
E
k
s
.
and
such t h a t
x,(t)
e
P
a ,b
E 8
,
with
Remark:
A v e r s i o n of Lemma 48 i s t r u e i f
Since n o = e
2
P P
we have our r e s u l t .
in
a , b ,c , d
b
n eP
(p
esting cases.
b
prime t o
p)
then
e . g . by u n i q u e f a c t o r i z a t i o n ,
e
i s any commutative
with i n t e g r a l c o e f f i c i e n t s (proof omitted)
=
at
e
i s g e n e r i c a l l y e x p r e s s i b l e a s a polynomial
proof j u s t g i v e n works i f
e =
a s an
i s t h e l o c a l i z a t i o n of
CG,
-
vc;[,a
M
I n t h e g e n e r a l c a s e it f o l l o w s t h a t
v,SL2(e)
ring since
a c t s on
e
.
The
i s any i n t e g r a l domain f o r which
maximal i d e a l ) , which i n c l u d e s roost of t h e i n t e r -
..
.. .
.. ..
Lemma 49:
Write
0K
K = G e , KU
= Gan K
.
= ( u ~ K ) ( H ~ K )
(a)
B
(b)
UnK= [ n x q ( t a ) [ t a & @ I
(c)
H
nK
a>O
=
(h
~ l z ( h E) eii
E
.
(a)
Proof:
_
1
b
If
t o a b a s i s of
(b)
, hence
K
rr
u =
If
End ( M )
in
(c)
A
p(h)
must be i n
defining
0
h
If
G
u
, relative
h
must a l s o .
x,(t,)
E
U
n K , then
...
by i n d u c t i o n on
and t h e p r i m i t i v i t y
(Theorem 2 , Cor. 2 ) we g e t a l l
E
HA K
,.
a..b
fact i n
.
ta E e
i n d i a g o n a l form as above, t h e n
A
f o r each weight
e
, in
i t s diagonal
xa(t) = 1 + tXu +
heights, t h e equation
Xa
L]
E
.
C K,
, then
uh e B f l K
fi
p
made up of weight v e c t o r s ( s e e Lemma 18, Cor.
M
3 ) , must be i n
=
A
for a l l
Ya
( d ) ~ P ~ S L =~ Ka( ~ ) Hence
of
.
e
of t h e r e p r e s e n t a t i o n
p
s i n c e t h e sum of t h e s e w e i g h t s is
( t h e sum i s i n v a r i a n t under
W)
.
If we w r i t e
h =
rr
hi(ti)
.I-
t:
and u s e what has j u s t been proved, we g e t
n
>
0
, whence
(d)
GL'
=
B,V
f o r some
e'p
.I.
ti
Set
Bay,
E
by unique f a c t o r i z a t i o n .
e'
E
.
= y,SL2(e)
S,
By Lemma 4 9 ,
Y,C
by Lemma 4 3 ( c ) and
s i o n f o l l o w s from:
Ban K
C S,
.
S,
, the
S,
C K, .
r e v e r s e inclu.
I
-
NOW if
x = ~ ~ ( t ) h , ( t ' s~ B,T)
)
K r
.
I
-
then
t
E
@
that
x
E
6 c u ( e ) ~ - ~ ( e )=> Sa
Theorem 18:
and
Let
t"'
E
e*.
e ,k,G
(Iwasawa decomposition)
by
and
.
Since
(a), (b)
, whence
K = G8
)
( c ) applied t o
(d.)
Ga
,
SO
.
be a s above.
Then
G = BK
115
Proof:
w
every
BwB
By Lemmas 4 3 ( c ) and 4 . 9 ( d ) ,
, so
W
s
that
Kw
C o r o l l a r y 1: W r i t e
(a)
U
K =
Kw
=
.
K
BwB
for
.
. .. Y 6 , w i t h
(B n K)Y,.
=
. ..Y6 C- BK
BY,.
.
K,
w&ilJ
(b)
BK
G =
=
B fi K
g i v e n by Lemma 4 9 ,
and on t h e r i g h t t h e r e i s u n i q u e n e s s of e x p r e s s i o n .
Remark:
K = G
T h i s normal form i n
8
h a s a l l components i n
whereas t h e u s u a l one o b t a i n e d by imbedding
Go
i s g e n e r a t e d by t h e g r o u p s
%
Corollary 2:
Proof:
1;
By Lemiila
C o r o l l a r y 3:
by
(x,(t)lci
Proof:
If
E
G
in
G
8
doesn't.
.
49 and Cor. 1.
Z, t
K
i s a E u c l i d e a n domain, t h e n
e
e 41
i s generated
.
Since t h e corresponding r e s u l t holds f o r
SL2(e)
, this
f o l l o w s f r o m Lemma 4 9 ( d ) and Cor. 2.
Example:
by
Assume
[x,(l)]
.
Z, k
e =
has
rank
>
P-7
.
We g e t t h a t
Gz
is generated
The normal form i n Cor. 1 can b e u s e d t o e x t e n d
N i e l s e n ' s theorem ( s e e
C
$
=
2
, is
(1-)
on p. 9 6 ) from
Gz
whenever
indecomposable, and h a s a l l r o o t s of e q u a l
l e n g t h ( W . Wardlaw, T h e s i s , U .
C. L. A. 1966).
i f t h e form c o u l d be u s e d t o h a n d l e
proof is q u i t e i n v o l v e d .
p r e s e n t i n poor s h a p e .
e a r l i e r development
SL~(Z) t o
K
3L
3
It would b e n i c e
(z) i t s e l r " s i n c e N i e l s e n T s
The c a s e of unequal r o o t l e n g t h s i s a t
In analogy with t h e f a c t t h a t i n t h e
i s a s i m p l e compact g r o u p if
decomposable, we have h e r e :
Every normal subgroup of
is i n -
C
G
z
is
116
X
f i n i t e o r of f i n i t e i n d e x if
rank
>
-
.
2
Exercise:
X
i s indecomposable and has
The proof i s n ' t e a s y .
Prove t h a t
i s f i n i t e , and i s t r i v i a l if
Gz/.@Gz
.il ,B2
i s indecomi,osable and n o t of t y p e
Returning t o t h e general s e t up, if
I
we w r i t e
[-
- 2-r
IxlP
for 'the
i'
-
p
r
with
a
and
b
i s a prime i n
101- = 0
.
and
e
.
p
k
be a s
above, a p r i n c i p a l i d e a l domain and i t s q u o t i e n t f i e l d ,
e
f i n i t e s e t of i n e q u i v a l e n t primes i n
t
P
It
E
k
.
Then f o r any
- t p l p< tz
for a l l
, and
E
>
0
there exists
p
E
S
and
ltlc
S
f o r each
t
k
E
,
Q
and
P
prime t o
(Approximation theorem) : L e t
Theorem 1 9 :
.
G2
p-adic norm d e f i n e d by
;r = p a / b
if
or
a
p e S,
such t h a t
<_ 1 f o r a l l primes
(31s
Proof:
t
P
=
We may assume e v e r y
pra/b
a s above.
a = cpS + db
t
E
P
By choosing
and r e p l a c i n g
a/b
.
e
s
by
To s e e t h i s w r i t e
> -r
d
,
and c and d s o t h a t
we may assume
b = 1
.
If we t h e n m u l t i p l y by a s u f f i c i e n t l y h i g h power of t h e p r o d u c t
S
of t h e elements of
n
we now choose
, we
achieve
2--n
<
so that
E ,
r
2
e
=
0
, for
:,i? ,
all
ep
p
=
E
e/'i3n
P &S
f
P' g~
so that
f . pn + g e
P P
2
=
1
,
and f i n a l l y
t
= '2
.
S
,
g e t
P P P
If
then
,
we a c h i e v e t h e r e q u i r e m e n t s of t h e theorem.
Now g i v e n a m a t r i x
Ixlp
=
maxla
1
ij P
.
x
=
( ai j )
over
k
,
we d e f i n e
The f o l l o w i n g p r o p e r t i e s a r e e a s i l y v e r i f i e d .
(2)
< l:qpl~lp
lxylp
lxilp
( 3 ) If
-
xi
I x - xplp
a ~ ~P o l" *~* ~~~ il [
<
p
for a l l
E
.
S
E
t
= x a ( t p ) with
p
E
x
pi
E
~
q
)
S
.
Xu,
then
is an i n t e g r a l polynomial i n
by (1)and ( 2 1 above.
al ,a2,.
for a l l
p
E
S
.
..
Thus our r e s u l t
I n t h e g e n e r a l c a s e we
SO
xp
that
--
xplxp2*"
By t h e f i r s t c a s e t h e r e e x i s t s
so that
xi'%a:
if
Xui
so that
G
x = xa(t)) t c k
If
, and
k
- Ilp <-- It - tPIP , s o t h a t
choose a sequence of r o o t s
x
E
a r e contained i n some
P
f o l l o w s from Theorem 1 9 i n t h i s case.
with
x
IxIq 5 1 f o r a l l
and
xa(t)
- tplp
5 [xp(plt
a Chevalley group over
G
S
.
t
ixxil
and s i m i l a r l y
I x - xplp
-- yiIp
IynIPIxi
p * * ~ o
Then t h e r e e x i s t s
E k
P
lxlq i m a x ltls, 1 because
P
,...,n , t h e n
fi
Assume f i r s t t h a t a l l
Proof:
x
i = 1,2
(Approximation theorem f o r s p l i t groups ) : Let
f o r each
G
E
for
be a s i n Theorem 1 9 ,
e ,k,S , E
P
<m
yilp
Theorem 20:
x
= lyilp
E and
S
lxiIq<1
if
q f S .
Weset
x3x1x
*... .
Then t h e conclusion of t h e theorem holds by ( 3 ) above.
With Theorem 2 0 a v a i l a b l e we can now prove:
Theorem 21:
(Elementary d i v i s o r theorem) : Assume
a r e a s before.
4
a(h)
E
e
Let
A+
be t h e s u b s e t of
f o r a l l positive roots
A
a
.
H
e ,k,G, K = G
d e f i n e d by:
8
I
+
(a)
G = K4 K
(b)
The
mod H
,
K
A
+
(Cartan decomposition).
component i n ( a ) i s u n i q u e l y determined
i . e . mod u n i t s ( s e e Lemma 49) ; i n o t h e r words, t h e
A
s e t of numbers
)
G]
I
( h )1 p
weight of t h e r e p r e s e n t a t i o n d e f i n i n g
is.
Example:
and
c o n s i s t s of t h e d i a g o n a l elements
A'
such t h a t
ai
i s a m u l t i p l e of
-
Proof of theorem:
-=
i n which
ai+l
d i a g (al , a 2 ,
for
i = 1,2,.
h a s a s i n g l e prime, modulo u n i t s .
e
,
. . ., a n )
.. .
F i r s t we r e d u c e t h e theorem t o t h e l o c a l c a s e ,
t u r e i n t h i s case.
Assume
of primes a t which
x
write
G = SLn(k) , K = SLn(e)
The c l a s s i c a l c a s e o c c u r s when
x
E
G
.
Let
Assume t h e r e s u l t
be t h e f i n i t e s e t
S
f a i l s t o be i n t e g r a l .
For
p
E
, we
S
,
for the local ring a t p i n e
and d e f i n e
and
P
K~
+ i n t e r m s of e a s K and A+ a r e d e f i n e d f o r e
A
By t h e
P
P
?
l o c a l c a s e of t h e theorem we may w r i t e x = c a c ' w i t h
P P P
P
c
c E K
and a E A+ , f o r a l l p E S
S i n c e we may choose
P' P
P
P
a
s o t h a t $(a ) i s always a power of p and t h e n r e p l a c e a l l
P
P
a
by t h e i r p r o d u c t , a d j u s t i n g t h e c f s a c c o r d i n g l y , we may
P
+ , and i s i n t e assume t h a t a
i s independent of p , i s i n
P
1
g r a l o u t s i d e of S
We have c a c x - l = 1 w i t h a = a
for
P
P P
1
By Theorem 20 t h e r e e x i s t c , c E G s o t h a t
p E S
e
.
.
.
.
I c - cPIP <
[cplp for
p
all
p
E
S
.
I
1
I
S
and
Iclq<l
t
c , and
P
By p r o p e r t i e s ( l ) ,( 2 ) , ( 3 ) of
same e q u a t i o n s h o l d f o r
I
E
t
c
and
1
easily verified that
I c 1 5 1, 1 cpl <- 1
q # S , the
'
-1
[ c a c x - 1lp5 1
for
I Ii3 , it i s
r -1
and / c a c ,x -
for
now
11, <_ 1 ,
119
whether
cat
i
p
-1
x
is i n
S
, so
that
K
E
or not.
c
Thus
+
x c KA K
E
K, c
t
K
E
a s required.
and
The uniqueness i n
Theorem 21 c l e a r l y a l s o f o l l o w s from t h a t i n t h e l o c a l case.
We now consider t h e l o c a l c a s e ,
in
.
e
17.
p
being t h e unique prime
The proof t o f o l l o w i s q u i t e c l o s e t o t h a t of Theorem
Let
h
h
be t h e subgroup of a l l
a r e powers of
, and
p
i n addition a l l %(h)
5
For each
A+
redefine
( 2>
a
Proof:
-
;;(a) = pp ( H )
ha(cap na )
a =
,!:( Ha
.
.
t
If
for a l l
If
H
and
x
H = AT
u
E
Since
for
a:
+
Kii K
.
E
.i
if
.
fi
, so
that
la1 = [ H I
p
% ( a ) = Pp ( H )
H
t
Then
, the
the
,
c,
with
, then
H
= H ,
H = l o g a , a = pH
P
K i l l i n g norm.
Now assume
From t h e d e f i n i t i o n s i f
x
T = 1-1
y = ua
E
This norm
.
We want
K
then
KxK
with
G
E
,
There i s only a f i n i t e number of p o s s i b i l i t i e s
a =
, then
[p(H)I ,u
r e s e n t a t i o n ) i s bounded below (by
t h e m a t r i x of
Z.
E
i s a power of
p(a)
Thus by Theorem 18 t h e r e e x i s t s
,
U, a
E
?kX ,
A
is i n v a r i a n t under t h e Weyl group.
t o show
e S r , na
E
a r e a s above, we w r i t e
and i n t r o d u c e a norm:
.
.
p
i s a second p o s s i b i l i t y f o r
H'
p(H ) = p(H)
a
c,
E
p
of t h e L i e a l g e b r a
-8-
with
being u n i t s , nay be o m i t t e d , s o t h a t
H = I: n,H,
Ha
f o r a l l weights
7
Write
$(h)
out u n i t s , s o t h a t
t h e r e e x i s t s a unique' H
A
that
such t h a t a l l
, casting
t h e 22-module g e n e r a t e d by t h e elements
1 , such
H
a r e nonnegative powers of
0)
A
E
E
*"x
-n
a weight i n t h e given repif
i s i n t e g r a l , because
n
i s chosen s o t h a t
(pi'(ti)]
are the
weights is
, so
0
la[
.
that
.
the lattice
a l s o above s i n c e t h e sum of t h e
is c o n f i n e d t o a bounded r e g i o n of
H
We choose
u = n'u,
If
, and
y)
d i a g o n a l e n t r i e s of
(u,
E
supp u
and t h e n minimize
y = ua
, we
)(-a )
above s o a s t o maximize
set
supp u = { a J u ,
+ 11
subject t o a lexicograpliic ordering
of t h e s u p p o r t s based on a n o r d e r i n g of t h e r o o t s c o n s i s t e n t
with a d d i t i o n (tlius
supp u
<
t
supp u
i n one b u t n o t i n t h e o t h e r l i e s i n t h e s e c o n d ) .
.
u = 1
a
E
Suppose n o t .
.
supp u
, we
K
were n o t i n
extreme l e f t i n t h e e x p r e s s i o n f o r
We c l a i m
K and aulu,a
We c l a i m ( * ) u,
u,
If
a
means t h a t t h e f i r s t
k K
for
could move it t o t h e
and t h e n remove it.
y
The
new terms i n t r o d u c e d by t h i s s h i f t would, by t h e r e l a t i o n s ( B ) ,
u
correspond t o r o o t s h i g h e r t h a n
u,
a = pH
set
$ 1
2c = <H,Hu>
P'I2
f o r some s i m p l e
, choose
a,=
p
by a product of
y
cHu
,a
r
c
t
u.
, as
T
so that H = H
H
= p
, a = a,a T
'
2 , so
supp u
would be
Now a s i n t h e proof of Theorem
y i e l d s t h e second p a r t of ( * ) .
get
that
Similarly a s h i f t t o the r i g h t
diminished, a contradiction.
17 we may con j u g a t e
, so
wg (1)' s
well a s (
.
cHa
(all in
)
K)
to
We w r i t e
is orthogonal t o
,H
We o n l y k n o w t h a t
t h a t t h i s may i n v o l v e a n a d j u n c t i o n of
which must e v e n t u a l l y b e removed.
t h e n a f t e r r e d u c i n g (,:)
If we b e a r t h i s i n mind,
t o t h e rank 1 case, exactly a s i n t h e
proof of Theorem 1 7 , what remains t o b e proved i s t h i s :
[
Lemma 5 1 : Assume
,
y = ua = '
rPC
:
t-I
t
I
,
1
~ o ~ P-c]
J I
with
2c E Z ,t E k ,
121
t
tp-"
and
e
.
e
Then
c
can be i n c r e a s e d by a n i n t e g e r
Proof:
>
n + 2c
and
t
Let
=
ey
-n
.
J.
with
e
e"'
E
.
K
by m u l t i p l i c a t i o n s by elements of
c + n
by t h e assumptions, s o t h a t
0
>
( c + n[
(c(
on t h e r i g h t by
.
If we m u l t i p l y
x
that
.
&iK
E
Thus
A+
c o n j u g a t e t o an element of
f u l l y represented i n
.
G = KAK
K
u = 1
.
(
t
m o r p h i c a l l y onto
Then
=
K
+
11
F i n a l l y every element of
A
is
w,(l)
E
.
K)
I
+
X
ith fundamenCa1 w e i g h t ,
G->G
c, c
Vi
is t h e
n
and
49(d) and Theorem 1 8 ,
n
maps
A
1
+
i s o-
is u n i v e r s a l .
G
Gi
is indecomposable.
G
Vi
an
Let
L-module w i t h
hi
hi
E
ii
and
as
the
i
t
be
t h e corresponding Chevalley g r o u p ,
t h e corresponding homomorphism, and pi
T
corresponding lowest weight. Assume. now t h a t x = cac
with
If
component.
G
.
G = KA'K
i s a d i r e c t product of i t s indecomposable f a c t o r s s o
G
highest weight,
n i:
y = a
Thus
Thus we may assume t h a t
t h a t we may a l s o assume t h a t
the
p
so
and from Lemma 49 t h a t
.
Lo
A,
n a t u r a l homomorj~hism, it follows from Lemna
nK
get
Thus
i s t h e u n i v e r s a l group of t h e same t y p e a s
Cor. 2 t h a t
> -c
E
It remains t o prove t h e uniqueness of t h e
G'
c, c + n
and
under t h e Weyl g r o u p , which i s
(every
K
>
, we
in
which proves t h e lemma, hence t h a t
I
z,
n > 0
on t h e l e f t by
y
, both
[1-e'1pn+2c
E
n
Then
a
E
'1
+
Set
A
,ui(a) = p
-n
.
E
G
,
Each weight
i n c r e a s e d by a sum of- p o s i t i v e r o o t s , Thus ni
i
11
i
i s t h e s m a l l e s t i n t e g e r such t h a t p nia is i n t e g r a l , i .e. such
on
is
,ti
I
n
p in.x
that
n. c
is since
1
1
.
x
u n i q u e l y determined by
t i c e of weights
and
(pi = w0h 1
.)
n-c
1
a r e i n t e g r a l , t h u s is
1
Since
{pi]
i s a b a s i s of t h e l a t -
, this
y i e l d s t h e uniqueness i n t h e
l o c a l c a s e and completes t h e proof of Theorem 21.
If
Corollary 1
e
K
is n o t a f i e l d , t h e group
i s maximal i n
i t s commensurability c l a s s .
Proof;
K
Assume
t
i s a subgroup of
By t h e theorem t h e r e e x i s t s
I
t h e diagonal matrix
(
rization
I
Remark:
The c a s e
If
numbers) and t h e
e =
Z
Zp
and
Some e n t r y of
k = $
P
We may assume t h a t
Then
hence a l s o i s
K
equations s i n c e
is a subgroup of
K',
a
,
-G
G
K
n o t bounded s o t h a t
G =
.
r ) SL(V,e)
K =
fIfn
.
.
G
be t h e a l g e b r a i c c l o s u r e of
E
K
i s of some importance h e r e .
We w i l l u s e t h e f a c t t h a t
Chevalley group.
a
,a
(p-adic i n t e g e r s and
P
p-adic topology is u s e d , t h e n K i s a maximal
e =
i s a good e x e r c i s e . )
that
t
i s n o n i n t e g r a l s o t h a t by unique f a c t o -
a
compact subgroup of
-k
f
A /I K
E
properly.
IK~/KI isinfinite.
C o r o l l a r y 2:
Proof:
-
a
K
containing
G
G
and
k
nSL(V,k)
Since
i s compact.
e
(The proof
is u n i v e r s a l .
Let
t h e corresponding
(Theorem 7 , Cor. 3 ) , s o
i s compact, s o i s
,
End(V,e)
t h e s e t of s o l u t i o n s of a system of polynomial
i s a n a l g e b r a i c g r o u p , by Theorem 6.
containing
, by
K
1
K
t h e theorem.
If
K'
properly, there exists
Then
i s n o t compact.
[ [ a n l p in
E
z]i s
Remark:
and
We observe t h a t i n t h i s c a s e t h e decompositions
4-
G = KA K
G = BK
a r e r e l a t i v e t o a maximal compact subgroup j u s t
a s i n Theorems 1 4 and 1 7 .
Also i n t h i s c a s e t h e c l o s u r e formula
of Theorem 1 6 , Cor. 7 h o l d s .
Exercise
( o p t i o n a l ) : Assume t h a t
&,
$
and t h a t K
P
subgroup d i s c u s s e d above.
or
i s a Chevalley group over
G
i s t h e c o r r e s p o n d i n g maximal compact
Prove t h e commutativity under convolu-
t i o n of t h e a l g e b r a of f u n c t i o n s on
which a r e complex-valued,
G
c o n t i n u o u s , w i t h compact s u p p o r t , and i n v a r i a n t under l e f t and
.
K
r i g h t m u l t i p l i c a t i o n s by elements of
(Such f u n c t i o n s a r e
sometimes c a l l e d z o n a l f u n c t i o n s and a r e of importance i n t h e
harmonic a n a l y s i s of
antiautomorphism P
a
and
K
, and
t
,
that
that
v
9
G
.)
of
G
Hint:
prove t h a t t h e r e e x i s t s an
vxa(t)
such t h a t
--a( t ) f o r a l l
p r e s e r v e s e v e r y double c o s e t r e l a t i v e t o
p r e s e r v e s Haar measure.
= I
.1. much h a r d e r e x e r c i s e
i s t o determine t h e e x a c t s t r u c t u r e of t h e a l g e b r a .
Next we c o n s i d e r a double c o s e t decomposition of
i t s e l f i n t h e l o c a l case.
K = G8
We w i l l u s e t h e f o l l o w i n g r e s u l t , t h e
f i r s t s t e p i n t h e proof of Theorem 7.
Lernma 52:
Let
a l g e b r a of
81 w i t h
b e t h e L i e a l g e b r a of
its coefficients transferred t o
number of p o s i t i v e r o o t s , and
made up of p r o d u c t s of
x
E
G
write
( t h e o r i g i n a l Lie
G
XaTs
xY1 = Z c . ( x ) Y
J
j
Y
Y
and
.
,
. .. Y
Hi's
]
with
,N
k)
a b a s i s of
Y1 =
/\
Xa
a>O
Then
x
E
U-HU
the
.
nN
For
if and o n l y i f
Theorem 22:
that
i s a l o c a l p r i n c i p a l i d e a l domain,
e
i s i-ts unique prime, and t h a t
p
and
G
a r e a s before.
.
(b)
.
G
BIwBI
(c)
w,e w i t h t h e l a s t component of t h e r i g h t
mod Uw
2P
= B WU
I
.
u n i q u e l y determined
Proof:
k
BI = U-H
U
is a subgroup of Ge
p e e
Ge = U BIwBI
( d i s j o i n t ), if t h e representatives f o r
wew
a r e chosen i n Ge
(a)
W in
ilssurne t h a t
Let
denote t h e r e s i d u e c l a s s f i e l d
Chevalley group of t h e same t y p e a s
By Theorem 18, Cor. 3 r e d u c t i o n
t h e u s u a l subgroups.
y i e l d s a hoinomorphisrn
of
n
.
C
- U-HU
(1) nal(u=H-u-)
8 8 8
-e , and
over
G
e/pe, G;;
G
8
A s is easily seen
t h e b a s i s of
Yls
.
Now assume
by t h e lemma a p p l i e d t o
G
a c t i n g on
nx
B--,Hz
,...
mod p
nNl
as
acts integrally relative t o
Ge
E
, whence
Ge
.
8
We c o n s i d e r
i n Lemma 52.
G--
onto
the
.
U=H-U0
0 8
cl(x)
+
Then
and
0
cl(rur) $; 0
x e U-HU
a g a i n by t h e lemrI?a.
(2)
-1la,rx:
(3)
BI = n-1B-
ker n
From t h i s and
that
(4)
.
S
x
E
x
UiHeUe
.\ssurne
E
, and
E
.
T C - ~ B ~ Then
then t h a t
-
n 1
x
c:
E
U-BU
by (1).
it f o l l o w s a s i n t h e proof of Theorem 7 ( b )
Ge
Completion
_ ----_of p r o o f :
simply a p p l y
.
C
- U-HU
x
E
BI
.
By ( 3 ) we have ( a ) .
t o t h e decomposition i n
Ge
To g e t ( b ) we
relative t o
Bg
.
We need only rei-rlark t h a t a c h o i c e a s i n d i c a t e d i s always p o s s i b l e
s i n c e each
w,(l)
E
Ge
.
From ( b ) t h e e q u a t i o n i n ( c ) e a s i l y
follows.
(Check t h i s . )
Assume
blwul
.
E U
Then bl-1b2 = wulu2-1,-1
Ui
W,@
-1
U1"2
U ~ , Pand ( c ) f o l l o w s .
Remark:
The subgroup
BI
= b
E
2
wu
0 U-8
BI
with
2
=
U"
P
bi
E
BI,
, whence
above i s c a l l e d a n Iwahori s u b g r o u p .
It was i n t r o d u c e d i n a n i n t e r e s t i n g paper by I w a h o r i and Matsumoto
No. 25 ( 1 9 6 5 ) ) . There a decomposition
( P a b l . Math. I.H.E.S.
which combines t h o s e of Theorems 2 1 ( a ) a n d 2 2 ( b ) can b e f o u n d .
The p r e s e n t development i s c o m p l e t e l y d i f f e r e n t from t h e i r s .
There i s a n i n t e r e s t i n g c o n n e c t i o n between t h e decomposition
G
8
=
UB
,
Go = ~ ( B w B ) ~t h a t
~ W Babove
~
and t h e o n e ,
as a subgroup of
Corollary:
9, and t h a t
w
n : Ge
inherits
, namely:
G
Assume
Ge
->
E
W
,
that
S(w)
i s a s i n Theorem 1 6 , Cor.
i s , a s above, t h e n a t u r a l p r o j e c t i o n .
r
Then n(BIwBI) = BgwBG , and ~ ( B w B =) ~
BGw Be
Hence
w' ES ( w )
if e is a topological f i e l d , e.g.
o r b9 , t h e n n(BwB)@
8
G-
u
-
a,R
i s t h e t o p o l o g i c a l c l o s u r e of
n(BIwBI)
Proof:
The f i r s t e q u a t i o n f o l l o w s from
above.
Write
w
=
waw P " '
( + ) (BwB), = ( B W & B ) ~ ( B W
B)@.
>-
P
( B W ~ B ) ~xCa(p)
Thus
.
n(BwaB)B
and
From t h i s , (::),
..
= BI
~ ~ ( and
1 )is a u n i o n of
(BwUB),
, proved
Then
by Theorem 1 8 , Cor. 1.
=
Be G a j S
C
- B0Ga , e
t h e d e f i n i t i o n of
quired expression f o r
-
n B
';
a s i n Lemma 2 5 , Cor.
>- B~ !J BZ; wuBG
i t y a l s o holds s i n c e
.
S(w)
.
B
6
Now
double c o s e t s .
The r e v e r s e i n e q c a l -
by Theorem 1 8 , Cor. 1.
, and
n(BwB)@ now f o l l o w s .
Lemma 2 5 , t h e r e -
Our purpose i s t o prove Theorem 23 below which g i v e s
Appendix.
t h e c l o s u r e of
T
w
write
<_ w
9 , i . e . if w
under v e r y g e n e r a l c o n d i t i o n s .
BwB
w
if
t
r
with
S(wJ
E
We w i l l
a s i n Theorem 1 6 , Cor,
S(w)
i s a s u b e x p r e s s i o n ( i , e . t h e product of a subse-
w
quence) of some minimal e x p r e s s i o n of
a s a product of s i m p l e
reflections.
Lemma 53 : The f o l l o w i n g a r e t r u e .
(a)
for
w
If
,i t
(b]
w
1
i s a s u b e x p r e s s i o n of some m i n i m a l . e x p r e s s i o n
i s a s u b e x p r e s s i o n of a l l of thein.
I n ( a ) t h e subexpressions f o r
w
T
can a l l be t a k e n t o
be minimal.
(c)
The r e l a t i o n
(d)
If
(resp.
w-la
wo
(e)
. -
Proof :
(a)
w
>
E
is transitive.
W and u i s a s i m p l e r o o t s u c h t h a t
, then
wwa
for a l l
w
0)
>w
<
-
>w
E
W
( r e s p a waw
w
two minimal e x p r e s s i o n s f o r
n
=
d i s t i n c t simple r e f l e c t i o n s ,
order
w1w2)
(b)
If
.7
and 9 i n a
t h e e q u a l i t y of
( a s a product of s i m p l e r e f l e c -
t i o n s ) i s a consequence of t h e r e l a t i o n s
I
.
It i s a d i r e c t consequence of t h e f o l l o w i n g
f a c t , which w i l l be proved i n a l a t e r s e c t i o n :
(wl ,w2
>0
.
T h i s was proved i n Theorem 1 6 , Cor
r a t h e r roundabout way.
> w)
wa
n
w1w2..
.
=
w2w1..
.
terms on each s i d e ,
.
w
T
=
w1w2.
..Wr
i s a n e x p r e s s i o n a s i n ( b ) and
it i s n o t minimal, t h e n two of t h e terms on t h e r i g h t can b e
c a n c e l l e d by Appendix I1 21.
- - , .
-
e
-
,
w
for
ww,
that
(c)
BY ( a ) and ( b ) .
(d)
If
>w ,
>
.ws
w1w2..
0 and
. .w swa,
wl.
then
(e)
*I
wu
i s a minimal e x p r e s s i o n
wwU,
i s one f o r
by Appendix I1 1 9 , s o
and' s i m i l a r l y f o r t h e other case.
T h i s i s proved i n Lemma 4 6 .
Now we cone t o our main r e s u l t .
Theorem 23 : L e t
be a Chevalley group.
G
Assume t h a t
k
is. a
n o n d i s c r e t e t o p o l o g i c a l f i e l d and t h a t t h e t o p o l o g y i n h e r i t e d by
G
I
a s a m a t r i c group over
Proof:
r
(b)
w
Let
Y1
Yw = /'\ Xwa
is used.
k
Then t h e f o l l o w i n g condi-
<- w .
be a s i n Lemma 52 and more g e n e r a l l y
w
for
E
W
a>O
c o e f f i c i e n t of
x
For
.
xY1
in
Yw
.
E
let
G
c w ( x ) denote t h e
We w i l l show t h a t ( a ) and ( b j a r e
equivalent t o :
(c)
cwf
(a)
=>
(0
or a r o o t )
sents
(c).
w
in
is not i d e n t i c a l l y
We have
xg(t)X,
u + jB
W
in
,
N/H
X,
+
I
BwB
t J xj
.
with
X
Thus
( * ) BwBY1
C
-
of weight
j
nwXa = cXhram( c f 0) i f
and
.
=
0 on
nw r e p r e -
k'"~, +
higher
terms i n t h e o r d e r i n g g i v e n by sums of p o s i t i v e r o o t s .
is not i d e n t i c a l l y
(c)
=>
(b).
maximal t h e n
0 on
f
Bw B
, hence
a l s o n o t on
We u s e downward i n d u c t i o n on
w
1
=
w0
, the
element of
~ ( w ' ).
BwB
Thus
, by
cwT
(a).
If t h i s is
W making a l l p o s i t i v e
w
r o o t s n e g a t i v e , and t h e n
1,
+ wo .
wT
I
>
N(wuw )
Choose
Since
,
c
W
B W B ~ B W ~ W B, we s'ee t h a t
so that
wan' .
<
-. w
<w
'(BwB) ) 0
c
. wuW
.
and
0
above.
or
0
(b)
Wc;W
.
In t h e first case
wc.'a
I n t h e second c a s e i f
<w
w'
<0
0 ,
then
by Lefiuna 5 3 ( d ) which p u t s u s back i n t h e f i r s t c a s e ,
w,
and conclude t h a t
=>
(a).
w'
u BwaB .
<
-w
contains
starting
.
a
if
is simple, then
=
The l e f t s i d e i s c o n t a i n e d i n t h e r i g h t , a n a l g e b r a i c
g r o u p , hence a c l o s e d s u b s e t of
-1
w
By t h e d e f i n i t i o n s and t h e u s u a l c a l c u l u s of double
cosets, t h i s is equivalent t o :
B
$
r(Bw,wB)
c
while i f n o t we nay choose a minimal e x p r e s s i o n f o r
with
Assume
, hence
BwabwB C
-
>
wT-la
+
(BwB)
1
wawl <_ waw
or
by Lemma 5 3 ( c ) and ( d )
WW
,
by ( c ) and (::)
0
simple s o t h a t
r.
1
N(w )
w
=
and t h e t o p o l o g y on
1
, hence
also
.
Since
Bw,B
contains
is not d i s c r e t e , i t s c l o s u r e
k
,
B
G
proving t h e r e v e r s e i n e q u a l i t y and
completing t h e proof of t h e theorem.
Remark:
I n case
above is
k
t o r e s u l t s obtained e a r l i e r .
Z a r i s k i t o p o l o g y on
and
k
Chevalley ( u - n p u b l i s h e d ) .
&,a o r aP ,
In case
k
t h e theorem r e d u c e s
i s i n f i n i t e and t h e
a r e u s e d it becomes a r e s u l t of
G
Our proof i s q u i t e d i f f e r e n t from h i s .
I
Exercise:
wa
>
0
,
(a)
w
If
prove t h a t
and c o n v e r s e l y i f
positive roots
,
E
wwu
w
r
<_
a
and
W
>w
w
(compare t h i s w i t h Lemma 5 3 ( d ) ) ,
then
, ..a r
i s a p o s i t i v e r o o t such t h a t
( )
t h e r e e x i s t s a sequence of
such t h a t if
wi
=
w,
then
!
1
w wl..
and
.wielui
( )
>
for a l l
0
i
and
1
w wl...w
r = w
.
Thus
w
f
<w
a r e equivalent.
(b) It seems t o us l i k e l y t h a t
alent to:
t h e r e e x i s t s a permutation
such t h a t
w nii
1
-=
wu
a > 0 ; o r even t o :
n
1
w <
-.. w
i s a l s o equiv-
of t h e p o s i t i v e r o o t s
i s a sum of p o s i t i v e r o o t s f o r every
t
C (w a
u>o
- wu)
i s a s u n of p o s i t i v e r o o t s .
I
1
,
Theorem 2 4 :
V
Let
W b e a f i n i t e r e f l e c t i o n group on a r e a l space
of f i n i t e dimension
I(S)
G
,
t h e a l g e b r a of polynomials on
S
( aJ
I(S)
i s g e n e r a t e d by
dependent elements
(b)
The degrees of t h e
Then:
homogeneous a l g e b r a i c a l l y in-
6
,... ,I& .
I . f s , say
J
z(dj
-
dl
,...,d4 , a r e
1) = N
, the
uniquely
number of
j
positive roots.
(c)
.
For t h e i r r e d u c i b l e Weyl groups t h e
dits
a r e a s follows:
Our main g o a l is :
Theorem 25 :
field
k
of
(a)
q
Let
G
,
Il
determined and s a t i s f y
I
W
t h e s u b a l g e b r a of i n v a r i a n t s under
V
b e a u n i v e r s a l Chevalley group over a
elements and t h e
diTs
a s i n Theorem 24.
Then
di
(q
[ G I = qN
- 1)
with
N = Z(di
1
- 1) = t h e number
of p o s i t i v e
roots.
(b)
c =
If
G
i s simple i n s t e a d , t h e n we have t o d i v i d e by
I H O ~ ( L ~ / L * ,k") 1 , given
Remark:
a s follows :
B&
We s e e t h a t t h e groups of t y p e
If
same o r d e r .
4 = 2
groups a r e isomorphic.
and
C
have t h e
t h e r o o t systems a r e isomorphic s o t h e
We w i l l show l a t e r t h a t i f
groups a r e isomorphic if and only i f
q
4
-3
>
the
i s even.
The proof of Theorem 25 depends on t h e f o l l o w i n g i d e n t i t y .
Theorem 26:
Let
W
and t h e
di's
be a s i n Theorem 24 and
-
t
.
-
Z tN'W'
=
(1
t di)/(l t )
WEW
i
We show f i r s t t h a t Theorem 25 is a consequence of Theorems
an indeterminate.
Then
24 and 26.
Lemma 54:
Proof:
and
If
G
i s a s i n Theorem 2 5 ( a ) t h e n
R e c a l l t h a t , by Theorems 4 and 4
BwB = UHwU,
1
,
BwB
WEJ
with uniqueness of expression. Hence
.
I G I = [UIIHIo Z I U w [
w&W
3 3 , I U [ = qN and [U,I
G =
(disjoint)
Now by C o r o l l a r y 1 t o t h e p r o p o s i t i o n of
=
qN ( w )
.
BY Lemma 28,
I H I= ( q
- 1)& .
132
U
Corollary:
i s a p-Sylow subgroup of
c h a r a c t e r i s t i c of
Proof:
, if
G
p
denotes t h e
.
k
N(w) = 0
p [ q N ( W ) unless
.
N(w) = 0
Since
if and o d y
w = l , p ) j ~N (qw )
if
( a ) f o l l o w s from Lemma 54 and Theorem 26.
Proof of Theorem 25:
( b ) f o l l o w s from t h e f a c t t h a t t h e c e n t e r of t h e u n i v e r s a l group
i s isomorphic t o
H O ~ ( L ~ /k*)
L ~ , and t h e v a l u e s of
L ~ / Lfound
~
S3.
in
Before g i v i n g g e n e r a l proofs of Theorems 24 and 26 we g i v e
independent ( c a s e by c a s e ) v e r i f i c a t i o n s of Theorems 24 and 26
f o r t h e c l a s s i c a l groups.
Theorem 24:
functions
Type
[
A&:
Here
,i,.... .,LJ4 +1
W
SL+l
such t h a t
q
=
t h e elementary symmetric polynomials
4, + 1 l i n e a r
permuting
E Wi = 0
C2,.
.
.. ,46+ 1
I n t h i s case
a r e invariant
and g e n e r a t e a l l o t h e r polynomials i n v a r i a n t under
Types B4, C&:
1
,
.. . ,
Here
W
W
.
acts relative t o a suitable basis
by a l l permutations and s i g n changes.
2
mentary symmetric polynomials i n W 1,
. ..,W42
Here t h e e l e -
a r e i n v a r i a n t and
g e n e r a t e a l l o t h e r polynomials i n v a r i a n t under
W
.
Here only a n even number of s i g n changes can occur.
2
Thus we can r e p l a c e t h e l a s t of t h e i n v a r i a n t s f o r B4, W1..
Type Dk:
.4
Theorem 26:
Type
A&
: Here
&
W = SL+l
and
N(w)
i s t h e number
1
133
of i n v e r s i o n s i n t h e sequence
-
.
(w(l,..w(
+ 1))
If we w r i t e
tN(~
t h e)n PLcl(t) = P C ( t ) ( l + t + t 2 +.. .+ t &+l)
W€W="S~*~
a s we s e e by c o n s i d e r i n g s e p a r a t e l y t h e 4 i-2 v a l u e s t h a t
PJt)
1
=
w(d+2)
cantakeon.
L+1
PL(t) = n ( 1 - t 5 ) / ( 1 - t )
Hencetheformula
5=2
f o l l o w s by i n d u c t i o n .
I
Exercise:
D4
)
.
Prove t h e corresponding formulas f o r t y p e s
Bd, Cd
and
Here t h e proof i s s i m i l a r , t h e i n d u c t i o n s t e p b e i n g a b i t
more complicated.
P a r t ( a ) of Theorem 24 f o l l o w s from:
Theorem 27:
Let
be a f i n i t e group of automorphisms of a r e a l
G
I
V
vector space
I
*
polynomials on
I
(a)
II
by
If
of f i n i t e dimension
V
i n v a r i a n t under
Conversely, i f
I
by r e f l e c t i o n s .
Example:
-
Let
G = {+ i d . ]
4 = 2
, then
G
x2 , xy, and y2
Notation:
and
V
I
is g e n e r a t e d
V , So
neous elements of
I
1)
.
algebraically
1) t h e n
have c o o r d i n a t e s
i s generated
G
.
x,y
is not a r e f l e c t i o n group.
I
If
is generated
and no s m a l l e r number of elements s u f f i c e s .
Throughout t h e proof we l e t
polynomials on
I
t h e algebra of
is g e n e r a t e d by
I
I,
Then:
I
i s g e n e r a t e d by r e f l e c t i o n s , t h e n
G
independent homogeneous elements (and
I-
.
G
I
by
and
a l g e b r a i c a l l y independent homogeneous elements (and
4
(b)
,
4
the ideal i n
S
S
be t h e a l g e b r a of a l l
g e n e r a t e d by t h e homoge-
of p o s i t i v e d e g r e e , and
Av
stand f o r
average over
(i.e.
G
-
Proof of ( a ) :
..
11, 12,.
i s not i n t h e i d e a l i n
Then
P1
Proof:
E
I
I1
Suppose
i22
a contradiction.
S
R2,
E
So
.
C((Pi
P1
Assume
I1
d
.
- gPi)/~)Ii=
0
- gP1
E
E
AvPl
= 0
So
,
.. .
S
I1 = AvIl
so that
I
i.e.
>
.
(mod S o )
and l e t
0
.
g
E
G
P1
gP1 (mod S o )
AvPl
E
g
So
We choose a minimal f i n i t e b a s i s
of homogeneous elements of
I
.
E
G
so
Il
d = 0,
.
gPi)
by t h e i n d u c t i o n assumption
.
S
be a r e f l e c t i o n i n
, L ( (Pi -
i
If
, so
But
...,
12,
g e n e r a t e d by
d = deg P1
Then f o r each
5
Then
does not belong t o t h e i d e a l i n
by r e f l e c t i o n s t h i s holds f o r a l l
P1
I1 '
C PiIi
12,.
... .
L = 0
a hyperplane
such t h a t
g e n e r a t e d by
t o the ideal i n
Hence
I*,
...
S
We now prove (1)by i n d u c t i o n on
P1 = 0
such t h a t
g e n e r a t e d by t h e o t h e r s and t h a t
ideal i n
E
f o r some
g e n e r a t e d by
I
.
So
T A ~ R ~ belongs
)I~
Z
a r e elements of
a r e homogeneous elements of
Z RiIi
i*
=
1
'
=
..
P2,.
--
(Chevalley, Am. J,
P
of Math. 1955. )
(1) Assume
P1,
C gp)
g EG
AvP = [GI-'
Since
G
Hence
is generated
and hence
P1
E
So
,...,In
.
for
So
formed
Such a b a s i s e x i s t s by H i l b e r t t s
Theorem.
(2)
The
IiTs
Proof : I f t h e
H(I1,.
.. , I n ) =
Ii
0
a r e a l g e b r a i c a l l y independent.
a r e not a l g e b r a i c a l l y independent, l e t
be a n o n t r i v i a l r e l a t i o n with a l l monomials i n
.
the
Iirs
xl,.
of t h e same minimal d e g r e e i n t h e u n d e r l y i n g c o o r d i n a t e s
- .>x& .
not a l l
Hi
...,Hm]
Let
Hi = H ( I 1 , . .
are
0
.
,In)Ii
.
H
By t h e c h o i c e of
Choose t h e n o t a t i o n s o t h a t
< n)
b u t no s u b s e t of it g e n e r a t e s t h e i d e a l i n
m
I g e n e r a t e d by a l l t h e Hi
Let H. =
V
for
J
i=l j , i Hi
j = m + I,
n where V
e I and a l l terms i n t h e e q u a t i o n
[ H ~ ,
(m
.
...,
j ,i
..
a r e homogeneous of t h e same degree. Then f o r k = 1, 2 , .
,4
n
we have 0 = bH/bxk = X Hi ?di/bxk
i=i
rn
n
= c H ~ ( ~ I + ~ /z ~ vxj ,i
~ ?dj/bxk)
BY ( 1 ) q / b x k
i=l
j =m+l
n
+
z V
31 -/bxk E so
N u l t i p l y i n g by xk
summing over
J
j=m+l j , l
.
.
.
,
k
u s i n g E u l e r t s f o r m u l a , and w r i t i n g
= deg I
we g e t
dj
j
n .
n
d . 1 = C AiIi
where Ai belongs t o t h e i d e a l
dl=l j = m"~1
J j
i=l
+l
i n S g e n e r a t e d by t h e xk
By homogeneity iil = 0 ..
Thus Il
,
+
.
..., In , a
i s i n t h e i d e a l g e n e r a t e d by
( 3 ) The
Proof:
P = C PiIi,
Pi
E
I
.
Pi
generate
IiTs
P e I
Assume
Each
.
S
E
Pi
Proof:
By ( 2 )
index i n
over
n
R (xl,
R ,whence
I
contradiction.
a s an a l g e b r a .
i s homogeneous of p o s i t i v e degree.
i s of degree l e s s t h a n t h e degree of
5
4
x2,.
n
Then
By a v e r a g i n g we can assume t h a t each
by i n d u c t i o n on i t s degree
-
12,
.
P
i s a polynomial i n t h e I i ' s
By G a l o i s t h e o r y
. . ,xn) , hence
>
- 4,
.
P
m(1)
, SO
.
is of f i n i t e
h a s transcendence degree
4,
,
By ( 2 1 , ( 3 ) and ( 4 ) ( a ) h o l d s .
i
-- -
(Todd, Shephard Can. J. Math. 1954. )
Proof of ( b ) :
... ,IZ
Let
I
of d e g r e e s
d
(5)
n(l - t
di)-l =
..
cl,.
Av d s t ( 1
F, EG
i=l
Let
(1 +
-,
I
C
Eit
*
+
p1 p2
el>
.. .
+
E.t
1
E,
-
gt)-l , as a f o r m a l i d e n t i t y
be t h e e i g e n v a l u e s of
,E&
t h e corresponding eigenfunctions
=
'
, respectively.
l d
G
Proof:
be a l g e b r a i c a l l y independent g e n e r a t o r s of
11,
...) .
j
.
Then
det ( 1
t h e t r a c e of
...,x4
- gt)-l
The c o e f f i c i e n t of
i.e.
xl,
and
g
g
tn i s
a c t i n g on t h e
s p a c e of homogeneous polynomials i n xl,. ..,x4 of d e g r e e n
- YlXP2
s i n c e t h e monomials xl
f0rm.a b a s i s f o r t h i s space.
,
...
By
a v e r a g i n g we g e t t h e dimension of t h e s p a c e of i n v a r i a n t homogeneous
.
T h i s dimension i s t h e number of monopolynomials of d e g r e e n
p2
mials Ilp1 I2
of degree n , i e , t h e number of s o l u t i o n s of
.. .
..
t h e c o e f f i c i e n t of
(6)
Proof:
di = [ G I
Wehave
and
-
Z(di
det(1-gt) =
1) = N = number of r e f l e c t i o n s
4
fi1-t)
)
if
g = 1
,
reflection,
a polynomial n o t d i v i s i b l e by
44 -1
otherwise
(1 t )
-
.
,
S u b s t i t u t i n g t h i s i n ( 5 ) and m u l t i p l y i n g by (1 - ' t )'
we have
di-1
(1 + t +
+ t
)
= 1~[-~
+ ~
(1
(1-t)/(l+
+ t()1 - t 1 2 p ( t ) )
...
where
P(t)
d
- .
= G
( 7 ) Let
II
t = 1
is regular a t
tions.
Then
Proof:
Let
be t h e subgroup of
t
and hence
= G
1
1
, and
Ii, di
N
T
G
corresponding J a c o b i a n n o t
Ii, b1~/b1;
C(di
- 1) =
i
I
, so,
N
=
#
N
for a l l
0
1
C(di
G
T
.
.
1
by ( 6 ) .
Hence
J G I = n d i = T d i = IG
dl,
t
Ii
The
di >_ di
Hence
t
a g a i n by ( 6 ) ,
C o r o l l a r y : -The degrees
I
i
- 1)
1
=
i s a r e f l e c t i o n group.
can be
Ii w i t h t h e determinant of t h e
Hence a f t e r a rearrangement of
.
0
t = 1 we g e t
g e n e r a t e d by i t s r e f l e c -
G
refer t o
expressed a s polynomials i n t h e
the
t = 1 we g e t
Setting
D i f f e r e n t i a t i n g and s e t t i n g
G~
G
.
d2,
...
di
1
.
But
t
-
di
I , so
for a l l
G = G
1
.
above a r e u n i q u e l y determined
and s a t i s f y t h e e q u a t i o n s ( 6 ) .
!
Thus Theorem 2 4 ( b ) h o l d s .
.
I
E x e r c i s e : For each r e f l e c t i o n i n G choose a r o o t a
Then
b(I1, 1 2 > * * * )
= T r a up t o m u l t i p l i c a t i o n by a nonzero number.
det
q,
X2,*.*)
I1
. . (I ,,
Remark:
The theorem remains t r u e i f
of c h a r a c t e r i s t i c
of
V
0
i s r e p l a c e d by any f i e l d
and i k e f l e c t i o n h ' i s r e p l a c e d by ~automorphism
with f i x e d p o i n t s e t a h y p e r p l a n e i f ,
For t h e proof of Theorem 2 4 ( c ) ( d e t e r m i n a t i o n of t h e
_I
we u s e :
-
--
-
-
.^-__C-.
.
,
.-.
- .,
*
-
di)
138
Proposition:
w = w1.
Let
. .wL , t h e
a n o r d e r i n g of
h
V
be t h e o r d e r of
and t h e
G
di
product of t h e s i m p l e r e f l e c t i o n s ( r e l a t i v e t o
( s e e Appendix 1.8) ) i n any f i x e d o r d e r .
w
.
.
N = Ch/2
(b)
w contains
(c)
If t h e e i g e n v a l u e s of
Proof:
Let
Then:
(a)
then
be a s i n Theorem 27 and
{mi +
u
11
=
=
exp 2ni/h
idi]
w
a s an eigenvalue, but not
m.
a r e iIJ 1
' 1 <_mi < h
11
- -
.
T h i s was f i r s t proved by Coxeter (Duke
.
m.-J .
u s i n g t h e c l a s s i f i c a t i o n t h e o r y s e e S t e i n b e r g , T.A.M.S.
Can. J_.Math.
1951),
For a proof n o t
c a s e by c a s e , u s i n g t h e c l a s s i f i c a t i o n t h e o r y .
( a ) and ( b ) and Coleman,
1
1959, f o r
1958, f o r ( c ) u s i n g ( a )
and ( b ) .
di
T h i s c a n b e used t o determine t h e
groups.
As a n example we determine t h e
f o r a l l t h e Chevalley
di
for
, s o by ( a ) h = 30 . S i n c e w
1 1 a r e a l l eigenvalues. Since
4 = 8, N = 120
bn(( n ,
30) =
t h e s e a r e a l l t h e eigenvalues.
1, 7 , 11, 1 3 , 17, 1 9 , 23, 29
previously.
E6
and
E7
Exercise :
Remark:
The p r o o f s f o r
Hence t h e
di
a l l i n c r e a s e d by
G2
and
Fq
P8
.
Here
acts rationally
' ~ ( 3 0 )= 8 =
are
1
, as
listed
a r e e x a c t l y t h e same.
r e q u i r e f u r t h e r argument.
Argue f u r t h e r .
The
t o Theorem 2b:
dils
a l s o e n t e r i n t o t h e following r e s u l t s , r e l a t e d
.
(a)
,f.
Let
teristic
0
be t h e o r i g i n a l Lie algebra,
,G
t h e c o r r e s p o n d i n g a d j o i n t Chevalley g r o u p .
a l g e b r a of polynomials on
by
the
a f i e l d of charac-
k
i n v a r i a n t under
G
is generated
a l g e b r a i c a l l y independent elements of d e g r e e
4,
dits
dl,
a
. . ,d4
the
G - i n v a r i a n t polynomials on
onto t h e
(b)
R
to
a r e mapped i s o m o r p h i c a l l y
p. The
W-invariant polynomials on
corresponding
r e s u l t f o r t h e u n i v e r s a l e n v e l o p i n g a l g e b r a of
easily
,
a s above.
T h i s i s proved by showing t h a t under r e s t r i c t i o n from
%
The
then follows
.
If
G
f , the
a c t s on t h e e x t e r i o r a l g e b r a on
i n v a r i a n t s i s a n e x t e r i o r a l g e b r a g e n e r a t e d by
homogeneous e l e m e n t s of d e g r e e s
- 11
(2di
4,
a l g e b r a of
independent
.
/
It i m p l i e s t h a t t h e P o i n c a r e polyno-
T h i s i s more d i f f i c u l t .
m i a l (whose c o e f f i c i e n t s a r e t h e B e t t i numbers) of t h e correspondi n g compact semisimple L i e g r o u p ( t h e group
2di-1
(C i n 5t3) is
(1 + t
1
Proof of Theorem 26:
Let
-
(Solomon, J o u r n a l of A l g e b r a , 1966.)
b e t h e s e t of s i m p l e r o o t s .
b e t h e subgroup g e n e r a t e d by a l l
w
(1) If
support not i n
Proof :
f3 =
If
Z
e,a
aeTT
p
E
n
c o n s t r u c t e d from
K
Wn t h e n
w
wa, a
E
If
n
.
n
C
-
let
W71
permutes t h e p o s i t i v e r o o t s w i t h
.
is a p o s i t i v e r o o t and
w i t h some
e,
>
0, a
p
n
supp p
.
Now
n
wp
then
is
$
plus a
140
vector with support i n
(2)
, hence
i t s c o e f f i c i e n t of
w e Wn
then
C o r o l l .
If
7
it i s t h e same whether we c o n s i d e r
(3)
For
n -c n
(a)
Every
(b)
(a)
w
Hence
w
with
'It
it
it-1
wu
1
Wn
E
, ut
E
.
w
t
E
>
0
=
t
w
and
.
if
w
E
,u
W&
t
it
E
E
t
t
= u , w
'It
W'IGw
E
.
.
Then:
t
w = w w
I1
.
.
t
N(w )
be such t h a t
t
w
Suppose now
.
Wn
<
(IT)
Then
0
so
t
;I
a?
i7
w w u
w u
t
=
8-1
-'n
w w
i?
1
= u
t
= u u
= u
i?
tt
.
has s u p p o r t
by Appendix 11.23 ( a p p l i e d t o
It
.
a e n by Appendix I I . l 9 ( a )
t
C
- '~t S O
W
N(w )
.
I1
w
Now
w ui7-ln
1
It
Wlwrr> 0)
it
for a l l
w
w
w
let
W
I: t N ( " )W,, ( t )
W(t) =
( - l ) n ~ ( t ) /7C~( ti ) = tN , where
r z o t s and
E
w e Wn)
or
Wn)
.
f o l l o w s from ( a ) and (1).
Let
a
Iw
f
wEW
C
W
can be w r i t t e n u n i q u e l y
>0
and
= l . Hence
(4)
Wn
1
W
i s unambiguous ( i . a .
N(w)
E
t
w a
:W
Hence
(b)
E
so that
w w "u i'-ln
Hence
in
t
1
W
w
N(w) = N(w )
In (a)
Then
E
t
w
For any
i s minimal.
define
w
with
Proof:
is p o s i t i v e ,
a
.
wp > 0
so
n
(-11%
= (-1)l n l
N
=
X
.
t N ( w ) Then
w EW,
i s t h e number of p o s i t i v e
.
,
.
We have, by ( 3 ) , W ( t ) / ; " J n ( t ) = C t N ( w ) T h e r e f o r e
weWn
t o t h e sum i n ( 4 ) i s c w t N ( W )
t h e c o n t r i b u t i o n of t h e t e r m f o r
Proof:
4
where
cw =
C
- 1 )
C7-r
wn>o
.
If
w
keeps p o s i t i v e e x a c t l y
k
elements
of
rr
then
r(l
I - ilk
cw=
if
= 0
k
f
0
w
element of
which makes a l l p o s i t i v e r o o t s n e g a t i v e , s o t h e sum i n
Corollary:
Exercise:
(
-
= 1
W
a
Deduce from ( 4 ) t h a t i f
s u b s e t s of
then
Set
D = (v
c- TT s e t
E
D~ = iv
VI
.
$ ~
n - n ]
-
o
Tt
a
for a l l
, and
E
a
E
is a n o p e n f a c e o f
( 5 ) The f o l l o w i n g subgroups of W
.
E (-l)"-p/~n(t-l)
for a l l
(v, a) =
D
a r e complementary
70P
VI(v, a) > 0
E
p
and
Z (-l)n-OL/~n(t) =
n2
forall
(4) is
tN a s r e q u i r e d .
equal t o
n
, the
wo
T h e r e f o r e t h e o n l y c o n t r i b u t i o n i s made by
n, (v,
f o r each
8) > o
D .
a r e equal:
( $ 1 Wn
Proof:
The s t a b i l i z e r of
(c)
The p o i n t s t a b i l i z e r of
(d)
The s t a b i l i z e r of any p o i n t of
(a)
C
- ( b ) because n
Dn
.
(b)
Dn
is orthogonal t o
because
D . i s a fundamental domain f o r
Clearly
( c ) C (d)
=-
.
( d )C (a)
k
~ ( - 1 ) ~
= n(-1)
~
Proof:
ni
W
.
D,
Dn
.
(b)
C
-
(c)
by Appendix 111.33.
by Appendix 111.32.
( 6 ) I n t h e complex c u t on r e a l
of h y p e r p l a n e s l e t
.
k-space by a f i n i t e number
b e t h e number of
i-cells.
Then
.
T h i s f o l l o w s from E u l e r ' s f o r m u l a , b u t may be proved
d i r e c t l y by i n d u c t i o n .
I n f a c t , i f a n e x t r a hyperplane
added t o t h e c o n f i g u r a t i o n , each o r i g i n a l
H
has corresponding t o it i n
an
H
- n,w
n,(w) (n C
planes l e t
D,
congruent t o
Proof:
and
Each c e l l of
c u t from
W)
E
by t h e r e f l e c t i n g hyper-
V
Then
(-l)"nn(w)
Z
a--r
( 5 ) every c e l l f i x e d by w l i e s i n Vw(Vw
=
Applying ( 6 ) t o
- In1
we g e t
.
w
Z
Vw
dim D, = 4
and u s i n g
, where
(-l)nnn(w) = (-1)
L-k
a
k = dim V,
oiyal, s o t h a t i t s p o s s i b l e eigenvalues i n
X-is
a c h a r a c t e r on
W1
,a
X
xew
IW1l-l
.
X(xwxol)
Dn
But
+ 1,
W
By
.
is orthog-
-1
(-l)C-k = d e t w
( 3 )
.
Vlwv = v ] )
subgroup of
denotes t h e induced c h a r a c t e r defined by
=
E
are
V
Hence
p a i r s of conjugate complex numbers.
If
{v
.
= det w
W-congruent t o e x a c t l y one
is
W-
denote t h e number of c e l l s
w-fixed.
K
separating t h e
~ ( - 1 ) ~ remains
n ~
unchanged.
two p a r t s from each o t h e r , s o t h a t
( 7 ) I n .the complex K
is
i - c e l l cut i n two by
- 1)-cell
(i
H
and
.
, t h e n X'
XW( w )
-
(See, e . g . , W. F e i t , Characters of
f i n i t e -.)
3
-
( 8 ) Let
Then
--
X
be a c h a r a c t e r on W
(-l)*Kn(w)
Z
m
-
Proof:
if --l
=
X ( w ) det w
. .
Assume f i r s t t h a t
flxe.9 X-'D
P.?
., . .
'7l
~es
4
X
5
1
.
and
Xn
( X ( W , ) ~ ( ~ , T.T )
w E M .
for a l l
Now
=
xwxnl
E
Ifn
if and only
(by ( 5 ) ) which happens i f and only i f
. Therefore
1,(w)
gives t h e r e s u l t f o r X z 1
.
=
If
n,(w)
X
by
( )
.
w
By ( 7 ) t h i s
i s any c h a r a c t e r t h e n
143
1
=
x
7
T
(8) holds.
SO
( 9 ) Let
M
b e a f i n i t e dimensional r e a l
be t h e subspace of
Wn-invariants,
A
Then
I:
CT
-
A
I(M)
and
( - l ) n d i m I n ( M ) = dim I ( M )
W-module,
I,(M)
b e t h e s p a c e of
,
.
. .
w
E
, and
W
Ta , t h e
p =
i s skew and
Proof:
We have
p
so
If
.
a(f
f
and
Then
I:
wp
,
= -p = ( d e t wa)p
=
Pn
i
V
i s skew and
P(t) =
a
We must show
.
tdi)^l
(-1)"dim
a
tTI
(*)
Let
as usual.
I,(Sk)
.
is a s i m p l e r o o t by
=
( d e t w,)f
and f o r
n C T
a root then
plf
Yn
as
w,f
I:
p
= -f
.
T(l - t d i ) / ( l- t )
be d e f i n e d f o r
t h e c o e f f i c i e n t of
Z
a
if
[di]
and
-
P
are for
let
W
.
.
d
, graded
V
is g e n e r a t e d by s i m p l e r e f l e c t i o n s
( - l ) % p ( t ) / ~ , ( t ) = tN
tNn(l-
on
W
C-TT
Proof:
over
p r o d u c t of t h e p o s i t i v e r o o t s , t h e n
By unique f a c t o r i z a t i o n
( 1 ) Let
idni]
, average
M
d i v i d e s e v e r y skew polynomial on
Appendix 1.11. S i n c e
is skew.
t o be t h e c h a r a c t e r of
use
If
(10)
p
X
In (8) take
Proof:
(-~)*T(I
-t
C n aI: Sk
-1
i
b e t h e a l g e b r a of polynomials
k= 0
A s i n ( 5 ) of t h e proof of Theorem 27
S =
t k on t h e l e f t hand s i d e of ( * ) i s
.
Similarly, using
on t h e r i g h t hand s i d e of ( a ) i s
(lo), the
A
dim I ( S k )
.
c o e f f i c i e n t of
These a r e e q u a l
(12)
Proof of Theorem 26.
We w r i t e (11)a s
and ( 4 ) a s
(t
-
(tN
(-l)n-)/~(t)
C ( - l ) / ~ ( .t )
alP
Then, by i n d u c t i o n on
=
'
Remark:
S t e p ( 7 ) , t h e geometric s t e p , r e p r e s e n t s t h e only simpli-
f i c a t i o n of Solomon~so r i g i n a l proof.
0
Isomorphisms and automorphisms
slC-
.
I n t h i s s e c t i o n we d i s c u s s
t h e isomorphisms and automorphisms ,of Chevalley groups o v e r perfect fields.
T h i s assumption of p e r f e c t n e s s i s n o t s t r i c t l y
n e c e s s a r y b u t it s i m p l i f i e s t h e d i s c u s s i o n i n one o r two p l a c e s .
We b e g i n by p r o v i n g t h e e x i s t e n c e of c e r t a i n autoinorphisms r e l a t e d
t o t h e e x i s t e n c e of symmetries of t h e u n d e r l y i n g r o o t systems.
55:
Lemma
Let
b e a n a b s t r a c t indecomposable r o o t s y s t e m w i t h
C
Let
n o t a l l r o o t s of one l e n g t h .
-
C"
.I.
[a'
t h e a b s t r a c t s y s t e m o b t a i n e d by i n v e r s i o n .
1
= 2a/ (a ,a) a
E
C]
be
Then :
.
b
(a)
c - ~ is
(b)
Under t h e map
a r o o t system.
*
l o n g r o o t s a r e mapped o n t o s h o r t r o o t s
and v i c e v e r s a .
F u r t h e r , a n g l e s and s i m p l e systems of
r o o t s a r e preserved.
( 4 If
P = ( a o + o ) / ( $ o , ~ o ) with
t h e map
a
->
a.
long,
if
a
is long,
if
a.
is s h o r t ,
Do
short t h ~ n
e x t e n d s t o a homothety.
( a ) holds since
The r o o t s y s t e m
Exercise:
Let
s i m p l e ones.
ever
ai
Examples :
a
<a",$'>
Prove t h a t
<@,a>
( b ) and ( c ) a r e c l e a r .
C
is c a l l e d
.
C
n1
.ai
= C
=.
o b t a i n e d i n t h i s way from
C*
t h e r o o t system dual t o
.
J,
.a.
Proof:
a
b e a r o o t e x p r e s s e d i n terms of t h e
i s l o n g i f and o n l y i f
p[ni
when-
is short.
( a ) For
n
>3,
Bn
nld
Cn
a r e d u a l t o each o t h e r ;
146
B~
G2
and
F4
a r e i n d u a l i t y w i t h themselves ( w i t h
(b)
X
for
<c
f3
and
.
B2
4:
, and
$"
with
=2a + $
a, B, a
Let
of t y p e
j3, a
+
+
Then t h o s e f o r
.
*
a
= 'a
+
p*
we g e t a map of
given by r e f l e c t i n g i n t h e l i n e
Theorem 26:
<(a,$) )
E , E"
Let
acteristic
(p
p
a s is
2)
be t h e p o s i t i v e r o o t s
4-
.I-
( a + 28)*
G
28
X q a r e a+., B", (a + p ) *
.
If we i d e n t i f y
d
with
B2
onto i t s e l f .
a ->
$,
.J-
.
p->a,u+P->a+ZP;a+ig->a+g
t h e b i s e c t o r of
=
.
p = 3)
(with
p
L
T h i s i s t h e map
(L
i n t h e diagram below
is
and a d j u s t i n g l e n g t h s .
and
p
b e as above, k
is e i t h e r 2 or 3 ) ,
groups c o n s t m c t e d from
(C ,k)
G , G*
a f i e l d of char-
u n i v e r s a l Chevalley
(x", k ) r e s p e c t i v e l y .
and
pl(cati
Then
JI
there exists a hoinomorphism 9
of
G
into
G'
and s i g n s
c
a
for a l l
E
Z
such t h a t
Y!xa(t))
=
if
xa*( cat )
If
k
is p e r f e c t then
Examples :
'pin2n+lt
(a)
''2n+l
If
k
Y
if
a
a
is long,
is short.
i s a n isomorphism of a b s t r a c t groups.
is p e r f e c t of c h a r a c t e r i s t i c
( s p l i t f o r m s ) , and
Sp2n
2
then
a r e isomorphic.
147
(b)
1
U
t h a t on
and
C*)
Consider
C2, p
=
2,
E
CI,
1
=
.
The theorem a s s e r t s
we have a n endomorphism ( a s b e f o r e we i d e n t i f y . C
2
such t h a t
(1) V ( x a ( t ) ) = x p ( t ) , V ( x p ( t ) ) = xa(t ) ,
2
9 ( x a + p ( t ) ) ' xa+z8( t )
q
(B)
t r i v i a l r e l a t i o n of t y p e
on
U
by Lemma 33.
(tu2)
=
.
~( t ~) )= +~ ~~ +
~ ~ The
( t o n) l y non-
(
is
(2)
Applying
CP
(xa(t), xg(u))
t o (2) gives
This i s v a l i d , s i n c e it can b e o b t a i n e d from ( 2 ) by t a k i n g i n v e r s e s
2
and r e p l a c i n g t by u , u by t
.
I
1
tI
1
I
(c)
G
as
xg(t)
Sp
phism of
S6
andif
4
-1
The map
t+O,*,(t)-lhas
rank 2
(d)
If i n ( b )
( k l = 2 , CP
-
1
while
l e a d s t o an o u t e r automor-
I:-;li!]
since, i n f a c t ,
a s t h e Weyl group of t y p e
with matrix
rank1
.
has
S6
i n (b) is o u t e r , f o r i f we r e p r e s e n t
CP
0
0
S p 4 ( 2 ) 'X S6
A5
.
.
To s e e t h i s r e p r e s e n t
T h i s f i x e s a b i l i n e a r form
0
-1
-
T h i s is s o b e c a u s e , up t o mul2
t i p l i c a t i o n by a s c a l a r , t h e form is j u s t C x . x . ( a . , a . ) = I X xiail
1 3 1 J
Reduce mod 2
The l i n e t h r o u g h al + a + a
becomes i n v a r i a n t
r e l a t i v e t o a b a s i s of s i m p l e r o o t s .
lI
i
.
3
5
and t h e form becomes skew and nondegenerate on t h e q u o t i e n t s p a c e .
Hence we have a homomorphism
y
: S6
->
sp4(2)
.
1t i s e a s i l y
.
148
seen t h a t
= 24(22
f/
ksr
so
- l ) ( z 4 - 1) =
-'
ker
a point
p
.
p3
Since
sp;(2)
{p] U S2
which c o n t a i n s 1 5 p o i n t s .
S2
p
.
such t h a t each of
= 720
Given
These s p l i t
{ p ] y S1
c o n s i s t s of mutually nonorthogonal p o i n t s and t h e s e
a r e t h e only f i v e element s e t s c o n t a i n i n g
There a r e
6!
a c t s on t h e
t h e r e a r e 8 points not orthogonal t o
S1,
IS61 =
i s a n isomorphism.
may be d e s c r i b e d a s f o l l o w s .
i n t o two f o u r p o i n t s e t s
and
= 1
[ s p 4 ( 2 )1 ,
underlying projective space
1
Y/
15=2/5 = 6
such
5
p
element s e t s .
f u l l y by p e r m u t a t i o n on t h e s e 6 s e t s , s o
I
with t h i s property.
Spq(2)
Sp4(2)
->
acts faith-
S6 i s de-
1
I
fined.
Under t h e o u t e r automorphism t h e s t a b i l i z e r s of p o i n t s
and l i n e s a r e i n t e r c h a n g e d .
Each of t h e above f i v e p o i n t s e t s
corresponds t o a s e t of f i v e m u t u a l l y skew i s o t r o p i c l i n e s .
Proof of Theorem 26:
that
(A),
p = 2
If
a s d e f i n e d on t h e
(B),
and
(C)
.
x,(t)
Here
nontrivial relations i n
each
(B)
(A)
1
=
E,
.
We must show
by t h e g i v e n e q u a t i o n s p r e s e r v e s
and
(C)
f o l l o w a t once.
The
are:
i f u , B a r e s h o r t , o r t h o g o n a l , and a+p rZ
if
I
(The l a s t e q u a t i o n f o l l o w s from Lemma 3 3 .
hand s i d e i s of t h e form
~ , + ~ ( N , , ~ t u. ))
[cLI>IBI a n d
<(a,B) = 135'
In the others the r i g h t
If
p = 2
t h e second
e q u a t i o n can b e o m i t t e d and t h e r e a r e no a m b i g u i t i e s i n s i g n .
Becuase of t h e c a l c u l a t i o n s i n Example ( b ) above
CP
preserves
,
.
I
these relations.
Thus
extends t o a homomorphismm
cp
There remains only t h e case
G2, p = 3
.
The proof i n t h a t
case depends on a sequence of lemmas.
Lemma 5 6 :
Let
simple r o o t s ,
be a Chevalley group.
G
n
0 . .
Then:
( a ) wa(l)wB(l)wa(l)
= W W W
p a p * "
on each s i d e )
Proof:
We may assume
n = 3
we assume
Let
of
Ga =
x e G
x
E
8
H
w,(l)w
Similarly
.
[X, ,X,]
=
Ha
-1
Lemma 57:
If
a, b
exp
Proof:
Then
factors
-
x
(where
w = w w
a B
...)
.
For s i m p l i c i t y of n o t a t i o n
wa(l)wg(l)wa(l)wg(-l)wa(-l)wg(-1)
=
G, x = 1
yX,
w,(l)
=
Let
C X - ~ where
y , yX-,
=
.
= cXP
- .
c = +1
(same
x a ( l ) x-a ( - l ) x a ( l )
c
Since
a s above).
we o b t a i n
.
a r e elements of a n a s s o c i a t i v e a l g e b r a over
makes s e n s e t h e n
Consider
(n
.
.
(b)
.
W
( 1 ) G w (-l)w,(-1)
= G w w a = GB and hence
a B
a 0
By t h e uniquenkss i n Theorem 4' ,
x E G
,
a f i e l d of c h a r a c t e r i s t i c
if
to
i s p r e s e r v e d by
, proving
\
Then t h e product of t h e f i r s t f i v e f a c t o r s
Exponentiating and u s i n g
=
X,
Consider
8
that
G ,
By t h e u n i v e r s a l i t y of
y = w,(l)wg(l)wa(l)
c
in
is universal.
G
<yayX-a> .
is i n
x
.
, so
be d i s t i n c t
... = w B ( l ) w a ( l ) w p ( l ) ...
Both s i d e s map
1
W
in
B
a , f3
( n f a c t o r s on each s i d e ) i n
W W W
(b)
w w
t h e order of
Let
0
, if
both commute with
[a,b]
and
exp(a + b ) = exp a exp b exp(-[a,b]/2)
f ( t ) = e x p ( - ( a + b ) t ) e x p a t exp b t e ~ ( - [ a , b l 2t /2)
.
1
.
a formal power s e r i e s i n
-
= (-(a+b) + a
= f(0) = 1
.
t
[b,a]t + b
-
D i f f e r e n t i a t i n g we g e t
[a,b]t)f(t) = 0
.
Hence
F
f (t)
f(t)
.
Now assume
i s a Chevalley group of t y p e
G
of c h a r a c t e r i s t i c
, and
0
over a f i e l d
G2
t h a t t h e corresponding r o o t system is
a s shown.
(1) Let
w
to
=
y = wa(l)w (1) be an element of
( r o t a t i o n through 60'
w w
a F3
Chevalley b a s i s of
-x WY
yxx =
Proof:
yX
(*u) c c c 2
bwkw
and
Xw2,
and
XeW2,
=
Y
so that
I
(2)
.
Then t h e
.
Then
( * ) c Y = c-
8 ' and
(by Lemma 5 6 ( h ) ' . Adjust t h e s i g n s of
-
cL -.
- C~~~
i n the
= -1 f o r a l l
(a)
.
.
-1
,
and a d j u s t t h e s i g n s of
Xm
X
->
1
;
:
a
It is c l e a r from ( " ) and ( a * ) t h a t
i n t h e same way.
C4
(clockwise) )
corresponding
can be a d j u s t e d by s i g n changes s o t h a t
= cbXwb, c X = +1
= -1
-1 f o r a l l
may make
%
for a l l
Let
G
$
w-orbit through
a
.
S i m i l a r l y we
b/ i n t h e w-orbit through
I n (1)we have
Nwzywg =
-Nk,6
$
.
for a l l I f , 6
.
(b)
We may a r r a n g e s o t h a t
It t h e n f o l l o w s t h a t
( a ) f o l l o w s from a p p l y i n g
y
+
N ~ 6,
is(
=
- r < i < q]
-+ ( r+
1)
is t h e
, and
in a
and a r r a n g e t h a t
-
N
Q ,a+2@--
.
to
'
= 2 .
and
and u s i n g (1).
Y, t h e n
6-string through
and 6+
N '& ,-6 have t h e same s i g n
6
q ( r + 1 ) ) By changing t h e s i g n s of a l l
XJ
.
w-orbit we can p r e s e r v e t h e c o n c l u s i o n of (1)
N ~ , =8
=
'3
Na+B,B
2
.
By ( a ) and ( * ) we have
=
C x ~ C X ~ +X2@ 11
B y=
,CxaJx811 , s o
that
3
.
Now
N a+2P , ~ ~ a , a + 3 ~
= N
= l .
N
a,a+3P
a9P
1f (1)and ( 2 ) h o l d t h e n :
'alBN u+2p,a+g
(3)
a+B, B
= 3
,a+2@
[Xg,X61
N~,a+2 ~N a + p , ~ + 2 =
p jN-a-8, 2a+3p = - 3 ~ 8 , ~
-
N
N
( f o r t h e i r product is
Xr f o r
= 1 and
y, 6 a r e r o o t s and
I n t h e proof of ( b ) we u s e (:K) i f
{
a ,B
= l .
N","+3~
Proof:
N
Hence
.
( a ] (x,(t) , x $ ( u ) ) = xa+B(tu)xa+3p ( - t u 3 ) x ~ + ~ $ ( - ' 2u )x2a+3s( t 2 U 3 )
2
( (tu ))) = xa+2B(2tu)xa+3B(-3tu lX2a+38( 3 t 2 u )
( b ) ( ~ ~ + ,~x B
Proof:
(a)
2
'a+2p +
+
I
get
By ( 2 )
3
x (u)X,
8
.
=
( e x p ad uX )Xu = Xa
8
P I u l t i p l y i n g by
- uXa+P
-t
and e x p o n e n t i a t i n g we
2
x ( u ) x , ( - t ) x (-u) = exp(-tXa-tu 3 Xa+3B
B
B
) e x ~ ( t u X ~ + ~ XM2$
-tu 1
3
2
) x ~ + ~ ~ )x2a+3p(-t2u3/2)xa+B(tu)xa+2B(-tu
( " ~ u
)x2a+38( 3 t 2 u 3 / 2 )
1
=
1
by Lemma 57, which y i e l d s ( a )
.
The proof of ( b ) i s s i m i l a r .
In
2
-
(c)
(e)t h e tern on Che wight hand
i y + j6
r o o t of t h e form
.
a*
wh
ecz;E?.~pon&to #e
N
The c o e f f i c i e n t i s
.
Y,
We have
t a k e n t h e o p p o r t u n i t y of working o u t a l l of t h e n o n t r i v i a l r e l a -
U
t i o n s of
However, we w i l l only u s e them i n
exp1,icitly.
c h a r a c t e r i s t i c 3 when t h e y s i m p l i f y c o n s i d e r a b l y .
(4)
i s r o t a t i o n through 60'
w
if
( a ) There e x i s t s an automorphism
such t h a t
G
sT
(-t)
6xy(t) =
then
of
8
for a l l
Y & C , t E k .
If c h a r a c t e r i s t i c
(b)
endomorphism
CP
of
G
, then
k = 3
r
such t h a t i f
i s t h e p e r m u t a t i o n of
t h e r o o t s g i v e n by r o t a t i o n through 30'
-
yxa(t) = I s y ( - t )
lxr,(t3)
fa']
Proof :
Y
if
is long,
if
is s h o r t .
then
Take 8 ta be the imer aut;omorphism by the element
of (1).
(b)
The r e l a t i o n s ( A )
Now on t h e g e n e r a t o r s CP 2 = f3 0
hence
CP2
and
a r e c l e a r l y preserved.
(C)
y/ , where
extends t o an endomorphism of
i n verifying t h a t the relations
\t/i
.
G
<(x
1
( x $ t ) , x 6 ( u )) =
1
,6 ) = <()f,6)
serves
1
R($ , 6 )
preserves
, and
.
R(ry,r6)
if
CP
->
~
~
preserves
~
x,(t
cp
with
If
CP
R(X,6)
does n o t p r e s e r v e
,
)
it
<(X,6) =
For i f R(Y,6)
i j
+u )~ , if
~
(
c
then
CP
To show t h i s it is enough t o show t h a t
.
3
This i m p l i e s t h a t
(F,6)
each of t h e a n g l e s 30°, 60°, 90°, 120°, 150'.
is t h e r e l a t i o n
xa(t)
a r e p r e s e r v e d by
(B)
s u f f i c e s t o show t h i s f o r one p a i r of r o o t s
I
t h e r e exists an
R(rx,r6)
preCP
then
~
~
t
153
does n o t p r e s e r v e
p2
I
extends t o a n endomorphism.
p a i r s of r o o t s
For
a
respectively.
a f t e r applying
k = 3)
9
(y,6) w i t h
< ( ~ , 6 )= 30°, 60°, 90'
+ 2p
.
For
I
I
For
x= a, 6 = a
we t a k e
take
=
(~,6=
) (a, a
+
.
3B)
Then
150'
we t a k e
(x,6) = (a,$)
(a,p) a n d t h e p o s i t ~ i v e r o o t s y s t e m r e l a t i v e t o
.
= 1 we have N
-a ,u+f3 = 1
Na,$
= -2
By
= 2 we have
N~ ,a+@
B
f o r a l l short roots
we r e t u r n t o
Corresponding t o
changing t h e s i g n of
X~
the original situation.
.
wa
Since
maps t h e f i r s t system o n t o
is s h o r t ,
if
t h e second,
R(r,6)
war
G
, and
s o t o prove t h a t
it i s s u f f i c i e n t t o prove t h a t
t h a t ( a ) i s p r e s e r v e d by
I.e.,
8 , 2 a + 3f3,
+
f o r t h e p o s i t i v e r o o t system
Nx,6
e x t e n d s t o a n automorphism of
I
< ( y , 6 ) = 30°, 60°, 90°, 120°, 1 5 0 ~ .
<()(,6)
Na+s,
(Note t h a t
for
( x a + @ ( - t ) , x g ( - u ) ) = ~ ~ + ~ ~ ( which
- t u )i s ( b ) ,
a n d corresponding t o
serves
ROf,6)
Here we have commutativity b o t h b e f o r e and
We compare t h e c o n s t a n t s
(-a,a + $ )
It remains t o v e r i f y
< ( x , 6 ) = 120'
converts ( c ) t o
relativeto
c o n t r a d i c t i o n s i n c e CP2 = 0 o
( s i n c e ( e ) becomes t r i v i a l s i n c e c h a r a c t e r i s t i c
9
a valid relation.
I1
,a
R(y,6)
xx(t)
->
7
o 9
CP
does, i . e .
k11:(t3)
r
(t)
if
is long,
if
is short.
is t h e r e f l e c t i o n i n t h e l i n e bisecting
t h a t t h e following equations a r e consistent:
pre-
<(a,p))
,
154
t
t
( a ) f o l l o w s from ( a ) by r e p l a c i n g
taking inverses.
(4)
T h i s proves
by
u3, u
.
We now complete t h e proof of Theorem 28.
c a s e of t h e f i r s t s t a t e m e n t i s
, and
t
by
of t y p e
G
The o n l y remaining
G2, p
=
3
.
If
\b
G
=
G q t h i s f o l l o w s from ( 4 ) above.
( )
o r not is immaterial because
i s d e t e r m i n e d by
.
b a s i s of
and
C
( )
I n f a c t , whether
a u n i v e r s a l Chevalley group
is perfect.
k
Then
.
Since
.I<
Hence
Remark:
e x i s t s on
k
If
,
G'
i.e.
i s n o t p e r f e c t , and
t h e subgroup of
G
i s l o n g , by
if
kP
and
k
and
(C)
s o does
i s a n isomorphism.
V
rp : G
e x i s t s on t h e
->
G
, then
i s p a r a m a t e r i z e d by
is s h o r t .
e
kp
any f i e l d between
a
i n which
CPel
( A ) , (B) ,
preserves
CP
maps one s e t of
CP
g e n e r a t o r s one t o one o n t o t h e o t h e r s o t h a t
rp-I
o r t h e Chevalley
f o l l o w s from Theorem 2 9 below.
Assume now t h a t
generators.
%
i n d e p e n d e n t l y of
k
G = G*
Here
kP
TG
k
is
if
can b e r e p l a c e d by
t o y i e l d a r a t h e r weird simple
group.
Theorem 29:
(L,
( L ~ ,t =
from
Let
=
[X,'
and
G
G
{Xa,Ha[a
'Ha' la
t
be Chevalley g r o u p s c o n s t r u c t e d
L, k)
E X ] ,
7
1
X ], L
E
t
and
, k) ,
t h a t t h e r e e x i s t s an isomorphism of
a
->
a
t
such t h a t
isomorphism
CP : G
rpxa(t) = x a t ( c a t )
take
ea
=
+1 i f
->
L
G
maps o n t o
t
for a l l
a
or
L
1
-a
E
Z, t
E
X
t
Assume
taking
Then t h e r e e x i s t s an
&,(a
and s i g n s
a
respectively.
onto
X
k
is s i m p l e .
.
a
E
X)
such t h a t
Furthermore we may
155
By t h e u n i q u e n e s s theorem f o r L i e a l g e b r a s w i t h a g i v e n
Proof:
:
r o o t s y s t e m t h e r e e x i s t s an isomorphism
that
Y/xa
YH,
=
Ha'
=
0)
(of c h a r a c t e r i s t i c
and
with
ea = 1 i f
a
-+ ( r +
-
€a
=
construct
1) = Nut , p r
Let
G'
.r
i s simple.
By Theorem 1,
R'
be a f a i t h f u l r e p r e s e n t a t i o n of
g
Then
%
By i n d u c t i o n on h e i g h t s e v e r y
\1/
o
can be used t o c o n s t r u c t
'? = i d .
-a
or
-
=
S U C ~
~ ~ s base f i e l d f o r
( F o r t h i s s e e , e .g. J a c o b s o n , L i e Algebras . )
Na,~
f ----> J'
L which
x C . l ( & & ( t ),) S O t h a t
i s a r e p r e s e n t a t i o n of
.
G
used t o
Then
xa(t) =
meets our r e q u i r e m e n t s .
Remarks:
(a)
28 w i t h
t?
Suppose
r e p l a c e d by
t r a n s p o s e of
.
v / ii ,
i s i n f i n i t e and we t r y t o prove T'heorem
k
.
t
Then we must f a i l .
For t h e n t h e
.!.
H". t o t h o s e on
m a p p i n g c h a r a c t e r s on
H
,
.*I
'maps
'
t: i n t h e i n v e r s i o n a l manner of Lemma 55, hence
onto
This explains t h e r e l a t i v e treatment
can n o t be a homomorphism.
of l o n g and s h o r t r o o t s .
(b)
If
k
i s a l g e b r a i c a l l y c l o s e d and we view
G
and
-81
G-'.
a s a l g e b r a i c groups t h e n
'P
i s a homomorphism of a l g e b r a i c
groups and a n isomorphism of a b s t r a c t g r o u p s , b u t n o t a n isomorphism of a l g e b r a i c groups ( f o r t a k i n g
pth
r o o t s (which i s nec-
e s s a r y f o r t h e i n v e r s e map) i s n o t a r a t i o n a l o p e r a t i o n ) .
(c)
For t y p e
r e s u l t holds f o r
C2
G2
and
t h e r e i s a n endomorphism
,
Fq
characteristic
,
characteristic
such t h a t
k = 3
k
(a similar
= 2)
,
in
zk
.P : Xa
{-xra
->
7-dimensional
if
a
is long
X
i d e a l spanned by a l l
3-
H
and
Cor
This l e a d s t o a n a l t e r n a t e proof of t h e e x i s t e n c e of
Corollary:
(a)
Let
.
If n o t , and i f
p
->
1
.
cr
s e x t e n d s t o a per-
cr of a l l r o o t s which a l s o i n t e r c h a n g e s l o n g and s h o r t
r o o t s and is s u c h t h a t t h e map
a
crf
CT
i s d e f i n e d a s above, t h e n
must i n t e r c h a n g e l o n g and s h o r t r o o t s and
mutation
.
r extends t o a n automor-
If a l l r o o t s a r e e q u a l i n l e n g t h t h e n
C
V
be a n indecomposable r o o t s y s t e m ,
C
a n a n g l e p r e s e r v i n g permutation of t h e s i m p l e r o o t s ,
phism o f
short.
p r a
if
a
a
->
ra
if
a
is long,
i s s h o r t i s a n isomorphism of r o o t systems.
The p o s s i b i l i t i e s f o r
cr a r e :
( i ) . 1root length:
An(n>2):
0-0
...
0-0-
b
L
-'
(ii) 2 r o o t l e n g t h s ,
6"= 1
L
0-0
o
o
i n a l l cases.
(b)
Let
s t r u c t e d from
occur assume
be a f i e l d and
.
(C ,k)
k
a be a s i n ( a ) . If two r o o t l e n g t h s
Let
or
a
bc(I(~at)
(
PP
t
->
G
t
type
C
-
ker n
p
( b ) If
i s u n i v e r s a l t h e e x i s t e n c e of
G
If
.
Now
k
ker n C
-
+2.
center
k
9
Remark:
k f 2
,'s
L1
To show t h a t . rp
and u n l e s s
G'
can
If
C
1
2
t h e c e n t e r of
YC
C
is
D2,
, then C
~k*)
/ L =~ (, L ~ / L ~ ) ', g i v i n g
of
C
G
i s of t y p e
G
c e n t e r of
=
H ~ ~ ( L
sucn t h a t
is of
G
?
Now suppose
a correspondence between subgroups
and
is not universal l e t
G
it i s n e c e s s a r y t o show t h a t
G
i s c a n o n i c a l l y isomorphic t o
Since
1
is short.
to
GI
and c h a r a c t e r i s t i c
Lo
=
E,
a
cyclic, s o t h e r e s u l t follows.
between
Then
) if
with c h a r a c t e r i s t i c
D2n
.
i s l o n g o r a l l r o o t s a r e of one l e n g t h ,
be t h e u n i v e r s a l covering.
be dropped from
PP ker n
-is of
a
f o l l o w s from Theorems 28 and 29.
n: G
L
G
if
( a ) is c l e a r .
Proof:
6L=
and s i g n s
G
If
i s s i m p l e ) such t h a t
4
yxa(t) =
of
.
p
, assume
cha,racteristic k f 2
t h e r e e x i s t s an e;?.5cmorphism 9
if
a Chevalley group con-
G
is p e r f e c t of c h a r a c t e r i s t i c
, and
DZn
type
k
C
?
G
t
and l a t t i c e s
i f and only i f
C
?
,
L
r LC
-L
.
r L = L
t h e r e s u l t follows.
that
The preceeding a - ~ ~ r n e nsth o w d f o r DZn i n c h a r a c t e r i s t i c
ker n
corresponds t o
a n autcmo~phi.smof
according t o
L
and
fixing
G
H
and permuting t h e
cr can e x i s t only if r L
=
L
.
*
Remark:
Automorphisms of
G
of t h i s t y p e a s well a s t h e i d e n t i t y
-
automorphisms.
a r e c a l l e d graph -
Exercise:
Prove
(b)
By imbedding
X , such
by a l l
above i s o u t e r .
(a)
cp
G2
a s t h e subgroup g e n e r a t e d
i s l o n g , show t h a t i t s graph automor-
a
that
in
A2
phisms can be r e a l i z e d by i n n e r automrphisrns of
D4 i n F4, Dn
for
a
simple
.
, and
E6
E7
in
.
Let
r~x,(t) = x,(fat)
for all
a
E
V
Similarly
k"
E
be extended t o a homomorphism of
f
Then t h e r e e x i s t s a unique automorphism
that
.
58: Let G b e a Chevalley group over k , fa
Lemma
k*
Bn
in
.
G2
for a l l
Lo
of
G
into
such
.
Z
Consider t h e r e l a t i o n s ( B ) , ( x a ( t } , xg(u))
i j
Applying P we g e t t h e same t h i n g w i t h
xia+ja( c i j t u )
r e p l a c e d by f a t , u r e p l a c e d by f u ( f o r fia+jp = fi fj ) .
Proof:
=
t
.
IT
$
The r e l a t i o n s
and
(A)
a $
a r e c l e a r l y preserved.
(C)
The unique-
ness is clear.
Remark:
phisms
Automorphisms of t h i s t y p e a r e c a l l e d dia,gonal automor-
.
Exercise:
Prove t h a t every diagonal automorphism of
r e a l i z e d by c o n j u g a t i o n of
Example : Conjugate
If
G
SLn
G
in
G
G(F) by an element i n H(E) .
by a diagonal element of
GLn
is r e a l i z e d a s a group of m a t r i c e s and
automorphism of
k
t h e n t h e map
can be
g : x,(t) ---> x,(t 2'
e r a t o r s extends t o a n automorphism of
G
.
.
i s an
)
on gen-
Such an automorphism
i s c a l l e d a f i e l d automorphism.
Theorem 30:
Let
G
b e a Chevalley group such t h a t
C
is
.
..<
,...
I..
indecomposable and
k
is perfect.
G
Then any automorphism of
can be expressed a s t h e product of an i n n e r , a d i a g o n a l , a graph
and a f i e l d automorphism.
Proof:
(1) The automorphism
cr H
is done t h e n
= H
crU
a l l simple
u
If
k
acteristic
r
.
U
is f i n i t e ,
k)
=
.
U
xya
and
cr
U-
=
z-,
=
)(
is a p-Sylow subgroup
for
( p = char-
by t h e c o r o l l a r y of Theorem 25, s o by Sylowfs
Theorem we can normalize
crU
a=
. If this
permutation g of t h e
= U , CT U-
and t h e r e e x i s t s a
simple r o o t s such t h a t
.
can be normalized by m u l t i p l i c a t i o n
T
by an i n n e r automorphism s o t h a t
Proof:
G
6 be any automorphism of
Let
k
If
c r ' b y an i n n e r automorphism s o t h a t
i s i n f i n i t e t h e proof of t h e corresponding
statement i s more d i f f i c u l t and w i l l be given a t t h e end of t h e
U-
w
E
-
(steps (5)
proof
E
U
.
U = 1
Since
U-
Thus
w
.
o U = U
For now we assume
, s o cr Urn = UWUWU-l
f o r s ome
0 u = 1, U W U W U - ' ~ ~ U = 1 and hence
=
w0
-- --1.
6 ~=-UU u
SO
r by t h e i n n e r automorphism corresponding t o
<U
=
U , o-U- = U-
normalizer of
permutes t h e
U-
. Now
. Hence
(B ,B)
group i f and only i f
=
u
=
UH
=
< f i x e s B f\ Bwa, a
"
u
-1
Now
simple.
=
H
Normalizing
we g e t
U , B- = U-H
normalizer of
Now ( B u BwaB) T\ U- =
Bw,Bwa -1 = BW' :
B# -a y and
t h e s e groups.
= BW';
B
double c o s e t s .
w
.
U
i s conjugate t o
W, u
wuw-ln
(12)).
.
Also
=
r
B U BwB ( 0f 1) i s a
Therefore
.
6 permutes
Since f o r
B
LJ
BwaB
160
X-a
1, B X - , ~ U - =
BAU-=
Thus it s u f f i c e s t o show
T h i s h o l d s by Theorem
permuted by
4.
, wa must show Bwa-1 n
BWZ'W~
TI u-wo = Bw,-1wo 0 woU
Therefore t h e
s i m p l e ) i f and o n l y i f
(2)
a
.
The automorphism
-
?a
It i s t h e n t r u e t h a t
(1)
.
Further
J
,a
i s empty.
simple, a r e
The p e r m u t a t i o n
X-B
% a and
i s empty.
B
(a,
commute
+B.
cr can b e f u r t h e r n o r m a l i z e d by a
d i a g o n a l automorphism s o t h a t
u
.
,Ts
o- a n d s i m i l a r l y f o r t h e
i n b o t h c a s e s i s t h e same s i n c e
I
,Ts
,--
o-xa(l) = x
(1) f o r a l l s i m p l e
Pa
o - x - ~ ( ~=) x
(1)
7'"
o-wa(l)
and
preserves angles.
I n t h e proof of ( 2 ) we u s e :
Lemma 59:
Let
xa(t)x-,(u)xa(t)
a
be a r o o t ,
t
w,(t)
Proof of ( 2 ) :
a
I
and
$
orderof
We can a c h i e v e
cxa(l)
by a d i a g o n a l automorphism.
0-x -a (-1) = x
(-1) and hence
7"
simple
are
w w
a B
o-(w,(l)wg(l))
E
.
k
Then
in
roots.
Then
W =
orderof
mod H = o r d e r of
u = -t
-1
SL2
=
, where
x
it i s immediate.
(1) f o r a l l s i m p l e
f"
By Lemma 59 w i t h
o-wa(l)
<(a,p) =
=
n
w
,Pa,
(1)
- /
.
t = 1
Suppose
a
whe~e n =
wa(l)w(l)modH= orderof
B
in
W
.
Hence
<(a,P)
Pa('3v)B ) o-. can b e f u r t h e r normalized by a g r a p h automorphism
= <(
so that
' f = l *
,
.
It s u f f i c e s t o v e r i f y t h i s i n
Proof:
roots
kZc, u
= Xma ( u ) x , ( t ) ~ _ ~ ( u i) f a n d o n l y i f
i n which c a s e b o t h s i d e s e q u a l
I
E
By t h e C o r o l l a r y t o Theorem 2g a graph au-tomorphism e x i s t s
Proof:
corresponding t o
or
, or
F4
p = 3
i s of t y p e
C
a
a + 28
C2
ga =
are roots,
we get
O-
F4
xa
, in
gB
=
a
.
.
type
.
G2
F i n a l l y , if
(Y.
+ B
Xa+ZB
=
Suppose
and
=
a+B
Since
'
Xacg .
Hence
s o t h e required
k = 2
S i m i l a r l y it e x i s t s if
C
is of
, characteristic
D2n
is of t y p e
I:
z
if
w a ( l ) Xpwa(-l)
7
xB
C2
Then t h e r e e x i s t simple
, and
c X a + 2 ~=
graph automorphism e x i s t s .
or g L = L
t h e notation there.
a n d v a + 2 8 commute s o do
and
= +2
Hence c h a r a c t e r i s t i c
o=N,+B,B
-
,
Z is of t y p e
s h o r t , such t h a t
j3
8,
if
G2
f 1
and
p, a l o n g ,
and
Applying
char k f 2
or
p = 2
i s of t y p e
I:
if
and
D2n
i s of t y p e
roots
t' provided t h a t
i s extended i n t h e obvious way, t h e n g L
L by
P
t h e remark a f t e r C o r o l l a r y ( b ) t o Theorem 28, s o t h a t t h e graph
k f 2 ,and
=
automorphism e x i s t s by t h e c o r o l l a r y i t s e l f .
(4)
6
cr
= 1
that
bxa(l)
=
1
I
(i.e.
if
of
k
=
.
xa(f(t))
We have
then
a
and d e f i n e
f : k
a d d i t i v e , onto,
f
.
k e r n e l o f t h e map
t
h,(t)
U, CU-=
a
We w i l l show t h a t
= wa(f(t))
rw,(t)
=
U-,
6 is a f i e l d
.
Proof : F i x a s i m p l e r o o t
o-xa(t)
o-U
o- s a t i s f i e s
x a ( l ) f o r a l l simple r o o t s
automorphism.)
'
can now b e normalized by a f i e l d automorphism s o
Therefore
->
is m u l t i p l i c a t i v e ,
h,(t)
f
f
->
k
by
i s an autornorphism
f (1)= 1 and by Lemma 59
6h,(t)
=
ha(f(t))
is c o n t a i n e d i n
.
Since t h e
[;tl) and
i s m u l t i p l i c a t i v e up t o s i g n .
.
Assume
f(tu) = af(t)f(u)
We must show
.
a = 1
(where
= f ( ( t + 1)u) = bf(t + l)f(u)
(where b
= b ( f ( t ) + 1 ) f( u ) = b f ( t ) f ( u ) + b f ( u )
= 1
b
-b
.
Thus if
$ 1 so that a
Let
diagram
a = b, t h e n
= 1 again.
(8
.
=
f (u)
b ( t + 1,u)
=
-+1)
Hence
a = b = 1
Hence
.
if one e x i s t s )
corresponding t o
( x a ( t ) , xe(u) ) =
t
+
.
T,
.
$
Let
r fixes
Then
..
X ~ + ~ ( L ~ U ) .
f i r s t with
u
g
=
(+
or
- a)f ( t )
(b
, then
a f- b
If
is an automorphism.
f
a
be a n o t h e r simple r o o t connected t o
$
Applying
= f (tu)
af ( t ) f ( u ) + f ( u )
Then
t, u + 0).
a = a ( t , u ) = +l and
i n t h e Dynkin
be t h e automorphism of
xa+p.
-
Consider
i s independent of
1 t h e n with
t
=
k
t ,u)
.
r e p l a c e d by
1, u
we g e t
6 ( x a ( t ) , ~ ~ ( =1 ()x u)( f ( t ) ) ,x p ( l ) ) = x a + $ ( + f ( t ) ) " -
.
6 ( x a ( 1 ) , x B ( t ) ) = ( ~ ~ (x 1
B ()g ()t ) ) = x
( + ~ ( t ) ) * I-n*e i t h e r
a+B
case t h e
term on t h e r i g h t i s ~ x , +
(tt
~) . Hence
Ya+$
f ( t )= g ( t )
Since
f
of
k
C
i s indecomposable t h e r e is a s i n g l e automorphism
so that
6xa(t)
=
xa(f(t))
Applying t h e f i e l d automorphism
ization
f = 1,
simple.
These elements g e n e r a t e
i.e.
We now assume
r U
=
U
of
(1)
k
f-l
f i x e s every
G
f o r a l l simple
to
G
.
w e g e t t h e normal-
xa(t), wG(t)
so
a
6= 1
for
a
.
i s i n f i n i t e and consider t h e normalization
.
( 5 ) Assume t h a t
k
is i n f i n i t e , t h a t
a d d i t i v e and m u l t i p l i c a t i v e groups, t h a t
A.
A
and
and
Mo
M
a r e its
are infinite
,
I
subgroups such t h a t
MoAo m
C H0
group.
If
Proof:
Let
F
M/Mo
i s f i n i t e and
A/Ao
,
then
A.
is a torsion
.
= A
be t h e a d d i t i v e group g e n e r a t e d by
Mo
.
Then
,
F
i s a f i e l d f o r it i s c l o s e d under m u l t i p l i c a t i o n and a d d i t i o n and
f
if
-
fr-lf-r
since
f
then
0
.
F
E
f-'
Now f o r
a
is i n f i n i t e and
Fa
some
$
F, f
E
.
$ 0
( 6 ) If
Hence
B, U
a
E
F
E
A, a
Fix
of
and
k)
x,fl
= {t21h,(t)
so
M 0A0
E
C A,
ha
.
"
=
B B 2~U .
( 7 ) If
A
Bo]
.
with
Now
B,/U
ya
c
A
i n (5).
A.
Bo
is nontrivial
Thus
fa
E
for
A.
.
( t h e a d d i t i v e subgroup
Set
Mo
= x,(t
2
U)
M / M ~ i s t o r s i o n s o by ( 5 )
BY
is abelian
is a
Bo
.
U
h,(t)x,(u)h,(t)-l
(4)
is i n f i n i t e and
i.e.
Since
A
with
Mo
x a ~ ~itcr,o
,
Bo
f-l
is i n f i n i t e and
k
x,
and i d e n t i f y
u
0, Fa r ) A.
, t h e n DBo =
B
so
.
C
- Uo
are as usual,
subgroup of f i n i t e i n d e x i n
Proof:
+
>0
r
is f i n i t e .
A/Ao
FAo
E
f o r some
(*)
>- x a .
D B ~
B B C~U
~hus
.
is a connected s o l v a b l e a l g e b r a i c group t h e n
is a connected u n i p o t e n t group.
T h i s f o l l o w s from:
Theorem ( L i e - - K o l c h i n L Every connected s o l v a b l e a l g e b r a i c group
A
i s r e d u c i b l e t o s u p e r d i a g o n a l form.
P r o o f : 'We u s e i n d u c t i o n on t h e dimension of t h e u n d e r l y i n g space
V and t h u s need only t o f i n d a common e i g e n v e c t o r and may assume
V
is i r r e d u c i b l e .
Let
A1
=
PA
.
By i n d u c t i o n on t h e l e n g t h
of t h e d e r i v e d s e r i e s of
xlv = X ( % ) v
that
.
Let
for a l l
A1
A1
and hence permutes t h e
V
A
rl
E
A1
VX
A1
trivially.
E
V~
, which
$
V, v
0
such
a r a t i o n a l character
v
.
A
normalizes
a r e f i n i t e i n number.
i s connected t h i s i s t h e i d e n t i ~ ypermutation.
a c t s by s c a l a r s .
has determinant
Since
X
A1,
is i r r e d u c i b l e t h e r e i s only one
i.e.,
v
be t h e space of a l l such
on
Since
there exists
A
Since
vx
A1 =
and it i s a l l of
DA
V
each element of
,
A1
1 s o t h e r e a r e only f i n i t e l y many s c a l a r s .
i s connected a l l s c a l a r s a r e 1
Thus
Since
, that
A / / A ~ i s a b e l i a n and a c t s on
is
V1
A1
acts
and hence has
a common e i g e n v e c t o r .
An a l g e b r a i c v a r i e t y i s complete i f whenever it is imbedded
densely i n a n o t h e r v a r i e t y it i s t h e e n t i r e v a r i e t y .
exact d e f i n i t i o n s e e Mumford, Algebraic Geometry)
Examples:
The a f f i n e l i n e i s not complete.
i n the projective l i n e .
( F o r a more
.
It can be imbedded
The f o l l o w i n g a r e complete :
(a)
A l l projective spaces.
(b)
A l l f l a g spaces.
(c)
\
where
and
B
-G
i s a connected l i n e a r a l g e b r a i c group
i s a maximal connected s o l v a b l e subgroup.
/
(See Seminaire Chevalley, Exp 5
- 10.)
We now s t a t e , without p r o o f , two r e s u l t s about connected
a l g e b r a i c groups a c t i n g on complete v a r i e t i e s .
( 8 ) B o r e l l s Theorem:
A connected s o l v a b l e a l g e b r a i c group
a c t i n g on a complete v a r i e t y f i x e s some p o i n t .
This i s a n e x t e n s i o n
165
of t h e Lie-Kolchin theorem, which may be r e s t a t e d :
every connected
s o l v a b l e a l g e b r a i c group f i x e s some f l a g on t h e u n d e r l y i n g space.
We need a refinement of a s p e c i a l c a s e of it.
If
( R o s s n l i c h t , A n n a l i , 1957. )
Theorem:
P
4
i s a connected
u n i p o t e n t group acking on a complete v a r i e t y
is d e f i n e d over a p e r f e c t f i e l d
over
k
, then
Notation:
k,
it c o n t a i n s one f i x e d by
Let
Ck
-
-Bx
Assume
w
We can t a k e
fore
u
wa(l)
.
u- = Bu-
k
i s d e f i n e d over
x
Bk
(10)
Since
d e f i n e d over
7s
where
k
k
u- = wuw-'
E
-
I\u , u-
k
u-'
.
.
k
Proof:
, it
choose
for a l l
x
i s d e f i n e d over
also.
a
-Bx
E
A,
B
a c t on
-B\E
d e f i n e d over
, and
i.e.,
f i x e d by
t h e n by (10) x
xax-'
E
By ( 8 )
by r i g h t m u l t i p l i c a t i o n .
k
B
E
G
.
for a l l
.
A
By ( 9 ) we can
We have
a
defined
.
G-conjugate t o a subgroup of
is
-G
i s a connected u n i p o t e n t subgroup of
4
Ek
E
U-
-
We make
there exists
.
= HkG
If
(11)
t
There-
Now
See t h e proof of Theorem 7, C o r o l l a r y 3.
Proof:
.
k
a s i n Theorem 4
k , x = wu
i s d e f i n e d over
a pr8duct of
IT ,
c o n s t r u c t e d over
i s onto.
i s d e f i n e d over k
a n d hence
.
whose c o o r d i n a t e s l i e i n
(8\€)
is d e f i n e d over
over
g, B
k,
t h e s e t of elements i n
( 9 ) The map G k ->
Proof;
A
contains a point
V
if
if e v e r y t h i n g
b e a Chevalley group over an i n f i n i t e f i e l d
G
t h e a l g e b r a i c c l o s u r e of
and
, and
k
,
V
E
d
,
-Bxa
SO
=
that
Bx
166
.
C
-
xtlxX1
If
(12)
crU = U
Since
k
C
- ti
xiil
is u n i p o t e n t
A
.
i s i n f i n i t e and p e r f e c t , t h e n o r m a l i z a t i o n
of (1) can be a t t a i n e d .
wB
Proof:
is s o l v a b l e s o
-G
group of
containing
o-B
( t h e s m a l l e s t a l g e b r a i c sub-
is s o l v a b l e .
TB)
(s)o
, the
Hence
connected component of t h e i d e n t i t y , i s s o l v a b l e and of f i n i t e
B
. (x)o
=
. Let A = Do - B ~ . By
index i n
in
B
.
and d e f i n e d over
k
x
xlx-l
E
such t h a t
G
normalization
But
6 U
C
-U
By ( 6 )
C
C
-A
Hence
.
By (11)t h e r e e x i s t s
xcr~x-l
h a s been a t t a i n e d .
U
Then
, i.e. , the
U
C
-
rX1lJ
.
is maximal w i t h r e s p e c t t o b e i n g n i l p o t e n t and c o n t a i n i n g
U
no elements of t h e c e n t e r of
rw1u
(7) A
o-U
.
Bo of f i n i t e i n d e x
i s connected, u n i p o t e n t
f o r some
0
= U
so
Therefore
.
is f i n i t e
Let
D
be t h e group of d i a g o n a l automorphisms modulo
D 2
..e
k:::/ kip
of
L1/LO
If
k
Aut G/Int G
is solvable.
Prove :
horn(^^, k*)/
=(b)
(Check t h i s . )
k
t h o s e which a r e i n n e r .
(a)
.
If
Corollary:
Exercise:
6 U = U
G
[Homomorphisms e x t e n d a b l e t o
where t h e
ei
.
Ll j
a r e t h e elementary d i v i s o r s
.
is f i n i t e ,
D
C
, the
c e n t e r of t h e correspond-
i n g u n i v e r s a l group.
(c)
D = 1 if
k
is a l g e b r a i c a l l y closed or if a l l
ei = 1
.
s e m i l i n e a r mapping of t h e u n d e r l y i n g s p a c e composed w i t h e i t h e r
I.e.,
t h e identify o r t h e inverse transpose.
every automorphism
is induced by a c o l l i n e a t i o n o r a c o r r e l a t i o n of t h e u n d e r l y i n g
p r o j e c t i v e space.
Over
(b)
G2
or
.
or
0
e v e r y automorphism of
F4
2
is inner.
(c)
The t r i a l i t y automorphism
exists
( ( s o )
if characteristic
.
Spin*
for
and
(d)
or
9
PSOn
, but
PSO8
So8
not f o r
Aside from t r i a l i t y e v e r y automorphism o f
,
SOn
( s p l i t form) is induced by a c o l l i n e a t i o n of t h e under-
lying projective space
Q:
Z ~
(n
- 2)/2
if
n = 4
~
P
which f i x e s t h e b a s i c q u a d r i c
.+ If ~ n -
x = ~0
is~ even, t h e r e e x i s t two f a m i l i e s o f
dimensional subspaces of
P
Q
entirely within
(e.g.,
t h e two f a m i l i e s of l i n e s i n t h e q u a d r i c s u r f a c e
x1X4 + x2x 3
= 0)
.
The g r a p h automorphism o c c u r s because t h e s e
two f a m i l i e s can b e i n t e r c h a n g e d .
Theorem 31:
1
1
(Z ,k )
G
and
with
G'
acteristic
kt
, and
type
Let
G , G'
E , Z,
1
b e Chevalley groups r e l a t i v e t o
indecomposable,
a r e isomorphic.
k =
either
Bn, C n ( n
If
k
characteristic
k
X
>
.- 3 )
k , kT p e r f e c t .
,
( X ,k)
Assume
is f i n i t e , assume a l s o chart
i s isomorphic t o
.
Then
X
and c h a r a c t e r i s t i c
t
k
i s isomorphic t o
or else
k =
X, X
t
a r e of
characteristic
k t = 2 .
Proof:
A s i n ( 1 ) and ( 2 ) of t h e proof of Theorem 30 we can
cr s o t h a t
normalize
o-U
= U
is an a n g l e p r e s e r v i n g map of
else
C, C
t
are
characteristic
Corollary:
Bn, Cn(n
kt = 2
, o-x,(l)
1
> 3) .
B n > Cn ( n
>
-
3)
>
-2
acteristic
k ='
characteristic
kt
characteristic
p =
C
C
f 2
k
k =
*k .
t h e Chevalley
t h e n t h e assumption charcan be dropped i n Theorem
,
k
and
r a n k C >_ 2
[GI
.
- --
get s t u c k s e e A r t i n , Comm. Pure and A P P ~ . Math., 1955)
SL2(b)
P
or
t
A s i n (4.)
makes t h e l a r g e s t prime power c o n t r i b u t i o n t o
a r e exceptions i n case
t
a r e not isomorphic.
r a n k Z , rank 1'
if
.Pa~ e n c e
i n t h e second case.
If
p
C
(1) where now
As in (3) characteristic
* Exercise:
(Hint:
7
onto
C
,
x
Over a f i e l d of c h a r a c t e r i s t i c
groups of t y p e
31.
=
rank C , r a n k C
PSL2(5) SLg(2) N ~ S L 2 ( 7 ) * )
t
>
-2
f a i l s , e.g.,
then
If you
(There
5
Some t w i s t e d Groups.
11.
I n t h i s s e c t i o n we s t u d y t h e group
of f i x e d p o i n t s of a C h e v a l l e y group
G
under an automorphism
We c o n s i d e r o n l y t h e s i m p l e s t c a s e , i n which
hence a c t s on
W
= N/H
and permutes t h e
Go_
r .
Xcvs
.
N,
U , H , U-,
rfixes
Before launching
I"
i n t o t h e g e n e r a l t h e o r y , we c o n s i d e r some examples:
( a ) G = SLn
it h a s t h e form
x
a
'02 n
If
o-x=ax
cr i s a n o n t r i v i a l g r a p h automorphism,
'-'a -'(where
x
and
0
is t h e transpose
r fixes x
We s e e t h a t
if and o n l y if
If
.
xax
0
= a
.
i s symmetric, we g e t
i n characteristic 2
a
If
G,
i s skew, we g e t
= SOn (split form)
Gb=
.
.
Spn
The g r o u p
d o e s n o t a r i s e h e r e , b u t it c a n b e
r e c o v e r e d as a subgroup of
S02n+1,
namely t h e one i F s u p p o r t e d nby
t h e long roots.
Let
t
-
as fixed f i e l d .
then
be a n i n v o l u t o r y automorphism of
If
r i s now modified s o t h a t
Sun ( s p l i t f o r m ) .
Gg=
a d i v i s i o n r i n g provided
If
W
V
t
point subspaces
Vc
and .
,W
c r x = ax
-> t
ko
9-la-l
9
k
is
i s a n anti-automorphism.
c a c t s on
g e n e r a t e d by t h e r o o t s and
V
and
W
and h a s f i x e d
wri s a r e f l e c t i o n group on
w i t h t h e c o r r e s p o n d i n g i v r o o t s t f b e i n g t h e p r o j e c t i o n on
t h e original roots.
having
-
T h i s l a s t r e s u l t h o l d s even i f
i s t h e v e c t o r space over
i s t h e Weyl g r o u p , t h e n
k
To s e e t h e s e f a c t s , we w r i t e
Vb
n = 2m
-t-
Vr
of
1 or
n = 2rn
0
index
omitted i n case
ui:
d e f i n e d by
- ~ (# j i) and
T
bi -- WWi - U i l i > 0 ] .
C
,
. ..
r
w(lJi
- LJj) = Ck - W-& .
i.e.,
group a c t i n g on
[wi]
if
i f and o n l y i f
-9
(b)
w
i f and o n l y if
- Wj )
w(Di
= Wk
Wr
We s e e t h a t
==
5
commutes
implies
is t h e o c t a h e d r a l
vo- by a l l permut:.fions and s i g n changes of t h e
The p r o j e c t i o n of wi - ~ . ( ji #, 0 ) on V6
is
J
.
or
G = SO2,
Bm
( a combination of
BCm
( s p l i t form,
char k
n
group d e f i n e d b y t h e form
f
=
21
C x.x i
l'i
r x
g r a p h autonorphism t o b e
il*.
=
J lol
l
(a=[l..*l\,al:l
form f i x e d by elements of
fixes
Gr
- x-l
=
0
f
1
-
(xl
=
2)
.
1).
.
Cm)
if
We t a k e t h e
lie w i l l t a k e t h e
'3
is
-
f
?
x -1)
Thecorresponding
n
=
2
2
2 L: xix
-i + xl + xml
i=
2
and hence t h e hyperplane
.
on t h i s h y p e r p l a n e i s t h e group
GO-
c
If we now combine
f
with
n
i1
f T i s r e p l a c e d by
Gc
f
and
=
.Z
i=2
Cm
a a x T - l a-1 -1
1 .
al
-1
x1
the roots are
If e i t h e r i = 0 o r j = 0 : t h e p r o j e c t i o n of
i s 31
Y ct ( k > 0 )
The p r o j e c t e d s y s t e m i s of t y p e
n = 2m
Thus,
, then
i s t h u s spanned by
Wr
E
H
i s t h e weight on
ai
Vr
with t h e
.
( k > 0)
&Ii
.
1
basis
w
Now
with
.
- 1, m
) .l
wi
If
->
a,
-ki
~=
T
-1
(rn
.
n = 2m
diag(a-m,
Wi
-
m ,
and u s e t h e i n d i c e s
t
->
a s i n ( a ) , t h e form
--
-
..-"
+ x-1.z.)
+ x1x1
1
+
X-lX-l
'
If we make t h e change of c o o r d i n a t e s
xl
xl + tx-l
r e p l a c e d by
r e p l a c e d by xl + F X - ~( t E k , t # F) , we s e e t h a t f i s
n
2
2
r e p l a c e d by 2 C ~
~ + x
2(x1 - + ~
+ bxWl)
and f
i s ren i=2
X-l
I!
, where
+ 2bx-1 1)
a = t +
forms have t h e same m a t r i x ,
v e r s i o n of
n
-1
.
f
That i s ,
which h a s index
Example:
group ( r e
If
n =
2
f = xl
corresponds t o
4,
and
G~
G,
n
over
x lil
, we
.
S i n c e t h e s e two
.
k
ko =
2
- x2 - x32 - x4)
.
A1
tF
over k
r e t h e new
0
i s sozn
i s SOZn(ko) f o r a form of index
a , and
k =
2
b =
R,
GT
i s t h e Lorentz
If we observe t h a t
see that
SL2(a) and t h e
D2
O-com-
ponent of t h e L o r e n t z group a r e isomorphic over t h e i r c e n t e r s .
Thus,
SL2(c)
i s t h e u n i v e r s a l covering group of t h e connected
Lorent z group.
E x e r c i s e : Work out
-
i n t h e same way.
D3
A3
For o t h e r examples s e e E. C a r t a n , Oeuvres compl\etes, No. 38,
e s p e c i a l l y a t t h e end.
Aside from t h e s p e c i f i c f a c t s worked out i n t h e above examp l e s we s h o u l d n o t e t h e f o l l o w i n g .
I n the single root length
c a s e , t h e f i x e d p o i n t s e t of a graph automorphism y i e l d s no new
group, o n l y a n imbedding of one Chevalley group i n a n o t h e r ( e . g .
Spn
or
SOn
in
SLn)
.
To g e t a new group ( e . g .
Sun)
we must
u s e a f i e l d automorphism a s w e l l .
Now t o s t a r t o u r g e n e r a l development we w i l l c o n s i d e r f i r s t
172
t h e e f f e c t of t w i s t i n g a b s t r a c t r e f l e c t i o n groups and r o o t systems.
I
Let
V
be a f i n i t e dimensional r e a l Euclidean v e c t o r space and
let
C
be a f i n i t e s e t of nonzero elements of
1
(1) a
(2)
E
w,Z
Z
implies
for a l l
= C
1
ca
a
C
E
c
if
C
>
t h e hyperplane orthogonal t o
TT)
spectively
V
P
and l e t
(re-
C
r e l a t i v e t o t h a t ordering.
of
V
which perriutes t h e p o s i t i v e m u l t i p l e s of t h e elements of
Let
.
, and
Z, P
Suppose
u-
i s an automorphism
have t h e same l e n g t h .
C
be t h e corresponding permutation of t h e r o o t s .
J'
denote t h e f i x e d p o i n t s i n
and
V
(f3,z)=
vr i s
(p,a)
for a l l
-a .
Theorem 32:
Let
C,
P,
p
E
n,
W
.
.
Let
cr - o r b i t of
Vc
.
6
e t c . be a s above.
The r e s t r i c t i o n of
(b)
WgIVc
i s a r e f l e c t i o n group.
(c)
If
denotes t h e p r o j e c t i o n of
,X
Cc
If
9
a
E
Cr
]
is
, then
to
generates
However, (1)may f a i l f o r
Zc
a
on
is f a i t h f u l .
Vc
Z
on
WrI Vg
.
and
, then
Vg
i s t h e corresponding liroot systemii; i.e .
I wvcr
~l
W6
Hence t h e p r o j e c t i o n of
(a)
Wg
a
Note t h a t
and
V6
respectively.
W
t h e average of . t h e elements i n - t h e
,
c fix C
It is not r e q u i r e d t h a t
b- i s of f i n i t e o r d e r and normalizes
I
is the reflection i n
be t h e p o s i t i v e ( r e s p e c t i v e l y s i m p l e ) elements
although it w i l l i f a l l elements of
I
.
of
each of
i
1
.
a
(See Appendix I ) . We p i c k a n o r d e r i n g on
-
#
0, c
wu
where
satisfying
V
,
wECr
-
Cg
.
F
173
_L
-,_
j 1
i s t h e p r o j e c t i o n of
If
(d)
on
Vc,
nb
then
i s t h e c o r r e s p o n d i n g 99simpie systernX9;i.e , if m u l t i p l e s a r e c a s t o u t
-ITc),
( i n c a s e (1)f a i l s f o r
% are
t h e p o s i t i v e e l e m e n t s of
a
G $ O
then
a e C ,
on
Vc
by
-
; indeed
some r o o t
3
a
>
Let
vi-
.
So
wlvCf
be a ~ o - o r b i t 02 s i i r ~ p l er o o t s , l e t
11r
be the
w e Wr,
If
Thus,
a e Pr
root
.
m r d l ~ r= Pr
Si n c e
-
P,
wr
p r o j e c t i o n s on
;
then
u- wro- -1
.'
Vr
,
then
a=-
I ~r-
0 = (v,p) = (v,p)
for
I
.
wr V~ =
and
=
wr
0;'
7i
WE/ V~
a
.
p e
if
.rr
.
for
wr
(vb - wE(vg
owrdl
c W,
Plr
f o r any
.
Since
wr e Wc
i n a single orbit, the
P,
1
be t h e c o r r e s p o n d i n g
are all positive
I t f o l l o : ~t h~a t , i f
--
0
by u n i q u e n e s s , and
of t h e el.emeiits of'
m u l t i p l e s cf each othc:;,
pr
Pr
then
a>
To s e e t h i s , f i r s t c o n s i d e r
perm-tes the elelants
J
,
then
be t h e unique element of
wr e W6
Then
0
w # l
wa(a e r ) , l e t
,.
a
G = z <0 and
s e t of p o s i t i v e r o o t s , and l e t
wrPT -
Thus,
-
0
group g e n e r a t e d by a l l
so t h a t
,
.
a
G - o r b i t of
0 .
<
If
<
a >
implies
a > O
w a
.
0
This
- (-a) < 0
Proof of ( a ) .
(2)
.
=
is also positive.
o
-> 7
v
Wb.
i s p o s i t i v e , so a r e a l l v e c t o r s i n t h e
t h e i r average
Thus
V
6 and w i t h a l l e l e m e n t s of
(1) If
of
p o s i t i v e l i n e a r c o m b i n a t i o n s of
Denote t h e p r o j e c t i o n of
commutes w i t h
If
i s l i n e a r l y independent a n d
TT, .
e l e m e n t s of
Proof :
ng
then
v c Vg
with
Bence
wv = v
r
i s any element
(v,aj = 0
.
,
then
.
.
.
(4)
If
e l e m e n t s of
for
(5)
wa
<
0
(i.e.,
.
a
9- o r b i t
w
with
Let
a
P
Appendix 11.17).
of
Wo
.
Z
.
$ E PT)
- Pa .
If t h e
a
.
Hence,
Now
,n
wPTqs
where each
and (6) )
a o 13
wi
.
,
, we
N(w)
o-wodlp
wiwi+l
... wra
i.e.,
a
part.
Now assume
<
0
0
.
Now
a,
a , $ e Pa
woa
P-orbit
wi = w,
If
i s i n some p a r t .
We may assume
.
so t h a t
i
N(wa)
0
permutes
(see
-
woP =
-
P
P
,
a n d t h e uniqueness
a, p
<
c
0
.
To prove ( 7 ) ,
wo = w1w2
and
0
of s i m p l e r o o t s
...
w ~ + ~wra
,
>
belong t o t h e
then
Similarly, if
>
o
a
Then
a
and
... wr
(by ( 5 )
and
...
w ~ + ~w.r a e PT ;
a
<
0
,
a
i s i n some
belong t o t h e same p a r t , s a y t o
.
<
a p - o r b i t of s i m p l e r o o t s ] i s a p a r t i t i o n
for some
wn
Choose
=
=
wPn
wn
and
-
such t h a t
-
>
.
may t h u s show t h a t
are called parts, then
a
By ( 4 ) ,
wwnPa>O
same p a r t if and o n l y if a = c p f o r some
we c o n s i d e r
.
a
N(wwl,) = N(w)
T h i s f o l l o w s from
Wg
generates
be a s i m p l e r o o t such t h a t
.
w,qs
{wpalw E Wg
(7)
of
.
and l e t
1
i s t h e element of W
wo
wo E Wr
of simple r o o t s ]
Using i n d u c t i o n on
w i s a p r o d u c t of
(6) If
wra =r w a
T h i s f o l l o w s from
be t h e 9 - o r b i t c o n t a i n i n g
t h e elements of
then
#
for all
wp<O
then a l l
Wr.
B
{wala
w E ,W
Let
w e Wg
of r o o t s and
have t h e same s i g n .
wv
,w
a e X
g- o r b i t
is a
v
-f3
wPa
.
are positive multiples
of each o t h e r , a s h a s been noted i n ( 3 ) .
( 8 ) Eo
and
16 such t h a t
c o n s i s t s of a l l
w c !lo
a , i s a r o o t whose support l i e s i n a simple
Now
.
Q-orbit.'
J
11
h a s i t s support i n n and hence so does
simple r o o t s not i n
.a
Conversely, assume
a l s o contains
8 = ca- , c > 0
(8)
f3 ,
t o p o s i t i v e m u l t i p l e s of simple r o o t s n o t i n
n
W
e see then t h a t
@ e P,
,
and t h a t any p a r t c o n t a i n i n g
P
The p a r t s a r e j u s t t h e s e t s of
a
such t h a t
and hence f o r m a p a r t i t i o n .
( 6 l w s Wb, a
roots) = z
a maps
since
h a s support i n a p - o r b i t of simple
~ .
(9) P a r t s (b) and ( c ) f o l l o w from ( 3 ) , ( 5 ) , and ( 8 ) .
(10)
Proof of ( d ) .
and form f a ]
.
We s e l e c t one r o o t
a
from each 9 - o r b i t
T h i s s e t , c o n s i s t i n g of elements whose s u p p o r t s i n
a r e d i s j o i n t , i s independent s i n c e
is.
If
a
>
0
then
.
it i s a p o s i t i v e l i n e a r combination of t h e elements of
-a
i s a p o s i t i v e l i n e a r combination of t h e elements of
Remark:
Hence
if,.
To achieve c o n d i t i o n (1) f o r a r o o t system, we can
s t i c k t o t h e s e t of s h o r t e s t p r o j e c t i o n s i n t h e v a r i o u s d i r e c t i o n s .
( a ) Ear
Cn
(b)
b
.
For
type
of o r d e r 2 ,
For
W
of t y p e
o- of order 2 ,
BnWl
W
of t y p e
A2,,
W
AZn-l
we g e t
of type
Dn
Wg
,
, we
get
of t y p e
we g e t
Wg
of t y p e
BCn
.
Wb of
( c ) For
r of o r d e r 3 ,
type
.
G2
of t y p e
W
To s e e t h i s l e t
r o o t s with
< E , F > = - 1 , < 8, a >
=
n
(d)
For
r
type
of o r d e r
F,+
2,
W
P,
a,
connected w i t h
6
D4
a,
-
3
of t y p e
,
y, F
be t h e simple
p , and
,
Wr of
we g e t
y
.
Then
W6
giving
of t y p e
Wc
E6, we g e t
of
.
Wr
(e)
For
CI-
of o r d e r 2 ,
W
of t y p e
C 2 , we g e t
(f)
For
b of o r d e r 2 ,
W
of t y p e
G2
,
we g e t
Wr
W
of t y p e
F4
,
we g e t
Wc o f
( t h e d i h e d r a l g r o u p of o r d e r 1 6 ) .
To s e e
type
(g)
For
r of o r d e r 2 ,
Dl6
this let
r a
b e t h e Dynkin
0--@&9---0
C
B
=v2
6
Y
-
(2
between
+ \/2)
a
.
and
diagram of
6
, 6 8 = Gy
-P = l / 2 ( ~+ f l y ) ,
=
oftype
A1.
of
A1
type
and
G2
we have
.
Since
< p,z>
Hence
A l t e r n a t i v e l y , we n o t e t h a t
Wr
'
a = l / 2 ( a + d T 6)
=
-
1,<a
T h i s c o r r e s p o n d s t o an a n g l e of
.
F4
i s of t y p e
makes
"0
3>
7n/8
B16 .
s i x positive
r o o t s n e g a t i v e and t h a t t h e r e a r e 24 p o s i t i v e r o o t s i n
,
= -
8 - 1 and
I
0:
P
.
wo = (w-v,r-)&
a P
a l l , So t h a t
W,
i s of t y p e
2
2
Hence,
=
8
WF
Note
t h a t t h i s i s t h e o n l y c a s e of t h o s e we have c o n s i d e r e d
i n which
f a i l s t o be c r y s t a l l o g r a p h i c (See Appendix V ) .
Wc
In ( e ) , ( f ) , (g)
we a r e assuming t h a t m u l t i p l e s have been
c a s t out.
The p a r t i t i o n of
equivalence r e l a t i o n
p o s i t i v e m u l t i p l e of
Letting
C/R
C
R
i n ( 7 ) above can be used t o d e f i n e an
on
C
where
a r
by
a
p
-a
i f and o n l y i f
a
i s t h e p r o j e c t i o n of
is a
on
Vc
d e n o t e t h e c o l l e c t i o n of e q u i v a l e n c e c l a s s e s we have
t h e following :
Corollary:
If
C
a n element of
C/R
i s c r y s t a l l o g r a p h i c and indecornposable, t h e n
i s t h e p o s i t i v e system of r o o t s of a system of
one of t h e f o l l o w i n g t y p e s :
(b)
A2 ( t h i s o c c u r s o n l y i f
(c)
C2
( t h i s occurs if
or
Fq )
(dl
p
i s of t y p e
i s of t y p e
AZn)
.
C2
G2
Now l e t
istic
C
Z
.
G
Let
be a Chevalley group over a f i e l d
r be an automorphism of
G
k of c h a r a c t e r -
which i s t h e product
of a graph automorphism and a f i e l d automorphism
8
of
k
and
.
such t h a t i f
i s t h e c o r r e s p o n d i n g p e r m u t a t i o n of t h e r o o t s
9
then
(1) if f
preserves lengths, then order
p
(2) if
doesn't preserve l e n g t h s , then
->
x
i s t h e map
p
xP)
8 = order
$ -
p82 = 1 (where
.
(Condition (1) f o c u s e s o u r a t t e n t i o n on t h e o n l y i n t e r e s t i n g c a s e ,
Observe t h a t
9
Condition ( 2 )
8 = i d . i s allowed.
= id.,
could be r e p l a c e d by
€I=
2p
thereby extending t h e
development t o f o l l o w , s u i t a b l y modified, t o i m p e r f e c t f i e l d s
We know t h a t
G
p = 2
i s of t y p e
G2
if
.
i s of t y p e
G
Recall a l s o t h a t
lal
if
-
C2
or
F4
and
k. )
p =3
if
rxa(t)
2 IgaI
where
( e tpe)
€a = +
- 1 and
Theorem 29. )
Jal
if
<
I
5 a
ea = 1 if
i s simple.
( S e e t h e proof of
r p r e s e r v e s 'U, H , B , U-, and N ,
Now
The a c t i o n t h u s induced on
of t h e r o o t s .
Since
J
W
and hence
N/H
W
i s concordant w i t h t h e permutation
.
J=
p r e s e r v e s a n g l e s , it a g r e e s up t o p o s i t i v e
m u l t i p l e s w i t h an i s o m e t r y on t h e r e a l space g e n e r a t e d by t h e r o o t s .
Thus t h e r e s u l t s of Theorem 32 may b e a p p l i e d .
t h a t if
n
i s t h e o r d e r of
t h e l e n g t h of each
9- o r b i t
g,
is
n = 1, 2 , o r 3 ,
then
1 or
Also we observe
n
.
so t h a t
U e , = 1 over each
Lemma 60:
Proof:
cs"
Since
9- o r b i t of l e n g t h
Xa ( 2 u s i m p l e )
a c t s on each
automorphism, it d o e s s o on a l l of
Lemma 61:
Proof:
k / ~,
a e
If
a e a
Choose
t + te = 0
we s e t
ea
if
#
,
1
x = x,(t)
.
whence t h e lemma.
j3 e a
s o t h a t no
ea = 1
if
,
as a field
ya ,r # 1
then
If t h e o r b i t of
y i e l d another root.
x = xa(l)
G
.
n
Then
a
can be added t o it t o
h a s l e n g t h 1, s e t
Xajr.
x e
,
t e k
with
t
#
0
and
,
n
If t h e l e n g t h i s
x = y 0 b y . r 2y . - . o v e r t h e o r b i t , and u s e
y = x c ( l ) , then
Lemma 60.
Theorem 33 :
Let
G , 6-, e t c . be a s above,
( a ) For each
cr
by
(b)
(c)
If
<
%,
t h e group
Uw = U n w-lu-w
is fixed
.
For each
nW e
w e
,
w e W
,
Ur,U>>
,
nw(weWr)
u
G0- = w d i r
BO-
there exists
so t h a t
n$
nw e Nr
=w
,
indeed
.
isasin(b),then
nwUw, O- w i t h u n i q u e n e s s of e x p r e s s i o n
on t h e r i g h t .
Proof:
roots
(a)
This i s c l e a r since
(b)
We may assume t h a t
7~
.
and
U
w
By Lemma 61, choose
=
w
x
7T
E
c,
a r e f i x e d by
W-'U-W
f o r some p - o r b i t o f simple
$-a,cr
c;
#
1
,
where
a
E
C/R
corresponds t o
w
some
W
F
where
n,e
<
r w
Using Theorem 4? we may w r i t e
=
,
w
.
UCT,U&>
(c)
b e B
and
.
Wr
P
UW
P
v e U
'
and
since
,
and
.
for
W
x =6 x
Now
.
w F Wr,
and
w=
x
.
Since
say
BwB
F
w e W
Applying
0-
we g e t
Thus,
we have
'
a
w7r
c(BwB) =B6wB
x = bnwv w i t h
a s i n ( b ) : and w r i t e
nw
.
crv = v
and
Bc
b F
and
.
v F Uw , r
The c o n c l u s i o n s of Theorem 33 a r e s t i l l v a l i d i f
?
G,
F
0
GT= 403 U> and Br= G 0- T\ Bc.
00
0
we can r e p l a c e Hr by H r - G-r T \ H r . ,
a r e r e p l a c e d by
8,
nwH = w
and
w < O ,
Choose
W
,
Uniqueness f o l l o w s f r o m Theorem 49
Corollary:
,
c u =u
wf 1
Since
x e Gc,
Let
w
we have
v
c n w = nW
forsomeaev,
wa<O
,
u F U
x = un v
and by Theorem 4 and t h e u n i q u e n e s s i n Theorem 41,
= cru*crnwo-v
we have
.
n
Br=U$,,
Lemma 6 2 : L e t a g e n e r i c a l l y d e n o t e a c l a s s i n
a union of c l a s s e s i n
a C S
such t h a t i f
C/R
C/R
.
Also
Let
S
be
which i s c l o s e d under a d d i t i o n and
then
-a
%
S
.
Then
ys,r=
a LS
a,c
w i t h t h e p r o d u c t t a k e n i n a n y f i x e d o r d e r and t h e r e i s u n i q u e n e s s
of e x p r e s s i o n on t h e r i g h t .
In particular,
U,
=
a> 0
U
=
WpO-
Proof:
a>o
,
for a
F
cr and
3
WE.
We a r r a n g e t h e p o s i t i v e r o o t s i n a manner c o n s i s t e n t w i t h
t h e o r d e r of t h e ,aVs;
etc.
w
Ha
Now
ES =
a>o
fa
i.e.,
those r o o t s i n t h e first
a
are first,
i n t h e o r d e r j u s t d e s c r i b e d and w i t h
u n i q u e n e s s o f e x p r e s s i o n on t h e r i g h t by Lemmal7.
Hence
a> 0
i n t h e g i v e n o r d e r and a g a i n w i t h uniqueness of e x p r e s s i o n on
the right.
1
5
The lemma f o l l o w s by c o n s i d e r i n g t h e f i x e d p o i n t s of
on both s i d e s of t h e l a s t e q u a t i o n .
If
corollary:'
(Xa,
a r e c l a s s e s i n ,Y/R
a, b
xb)C TT Kc ,
7
that
by
c
a
#
+b ,
then
where t h e r o o t s on t h e r i g h t a r e i n t h e c l o s e d
a
subsystem g e n e r a t e d by
The c o n d i t i o n on
with
and
b
,
a
t h o s e of
and
b
excluded,
can be s t a t e d a l t e r n a t e l y , i n t e r m s of
Xr,
i s i n t h e i n t e r i o r of t h e ( p l a n e ) convex cone g e n e r a t e d
6 ,
and
Remark:
The e x a c t r e l a t i o n s i n t h e above c o r o l l a r y can be q u i t e
complicated b u t g e n e r a l l y resemble t h o s e i n t h e Chevalley group
whose Weyl group i s
6
Wc
.
For example, if
i s of o r d e r 2 , s a y a
0-01)
+
( a + f3, j3
( t E ke)
we g e t
,
y]
,
d =
P Y '
{ a + P + y]
xb(u) = xa(u)x
Y
(U
8
)
,
a =
,
(U 6
i s of t y p e
G
b = (a,y!
and i f we s e t
k)
,
,
Aj
c =
xa(t) = xB(t)
and s i m i l a r l y f o r
8
( x a ( t ) , x b ( u ) ) = x c ( + t u ) x d ( + t uu )
.
and
In
c
and
C2, t h e corres-
ponding r e l a t i o n i s
n
I
(
If
G
i s of t y p e
i s of t y p e
"X
.
2 ~ 3
.
E.g.,
X
and
T is
of o r d e r
n
,
we s a y
Gc
t h e group c o n s i d e r e d i n t h e above remark i s
The group of t y p e 2 ~ 2i s c a l l e d t h e Suzuki group
and t h e g r o u p s of t y p e 2G2 and 2 ~ 4a r e c a l l e d Ree groups. We
of t y p e
d,
wrue
Lemma 63:
Let
.
G c n~
and
G@X
Y
a
,
Z/R
be a c l a s s i n
then
has t h e
a70-
following s t r u c t u r e :
(a)
If
a-A1,
(b)
If
a m ~ y , then
(c)
If
a&A2,
3a 7 r
then
= [ x a ( t ) It
= {x. o x . . . Ix = x , ( t ) ,
a,6
a = {a,p,u
+ p],
then
8
% a , r = { x a ( t ) x g ( t )xa++u) ltte
If
(t,u)
t
0
+
t
ue =
01
P
,
u
+
u
-
P
+ P,
.
8
0
+
2?]
t t )
,
then 2 82 = l
a = [a,p,c
and
Xa
8
(tl*
= [xa(t)xg(t ) x ~ + ~ ~ ( u
) x ~+ +u8~) I t , u
7
(7
(t,u)
If
39
= 1 and
( t ,u,v)
(t,u,v)!t
P
0
%?a,('-
.
k)
0
( t , u ) (t ,u )
t2@t9)
+
30, 2a
+ 3p], then
8
8
l+e)
= ( x a ( t ) x g ( t ) x ~ + ~ ~ ( u ) x ~- +t ~ ( u
d e n o t e s t h e g i v e n element t h e n
t
,U
,v
V
v
) = ( t + t , u -u
Note t h a t i n ( a ) and ( b ) ,
k
9
+
P
t t3@,v+ v
1
P
- t u + t Zt3Q J
i s a one parameter group f o r t h e
a7c
and
9
E
.
+
a = [ a , p , a + P , a + 2p, c
a-427
2
c
d e n o t e s t h e g i v e n element,
0
kg
+
k]
E
= 1 and
a-C2,
= ( t + t , u + u
fields
u
2
If
If
(e)
+
€3
a e a, t
d e n o t e s t h e g i v e n element, t h e n
( t , u ) ( t ,u ) = (t
(d)
kg]
E
respectively.
*
( a ) and ( b )
Proof:
For ( c ) ,
a r e e a s y and we omit t h e i r proofs.
Fa+*SO t h a t N a,P = 1 , Then
0
Q
x ( t ) = x a ( t ) , and o - x ~ + ~ ( u=) x
B
a+b ( - u ) .
normalize t h e p a r a m e t r i z a t i o n of
r X a ( t )
Write
€3
1, r
= xg(t
x
E:
2a , r
as
x = xa(t)x (v)x
(u)
B
a+$
c o e f f i c i e n t s on b o t h s i d e s of
x = o-x
( d ) i s similar t o t h a t of ( c ) .
and compare t h e
t o get (c).
The proof o f
For ( e ) , f i r s t normalize t h e s i g n s
a s i n Theorem 28, and t h e n complete t h e proof a s i n ( c ) and ( d ) .
Exercise:
Remark:
Complete t h e d e t a i l s of t h e above proof.
SL2
The r o l e of t h e group
by t h e g r o u p s
SL2(kg)
,
, SU3 ( k , @ )( s p l i t
SL2(k)
Suzuki group, and Ree group of t y p e
Exercise:
i n t h e untwisted case i s taken
the
G
is
.
G2
Determine t h e s t r u c t u r e of
,
form)
i n t h e case
Hr
universal.
Lemma 64:
If
G
is universal, then
except p e r h a p s f o r t h e c a s e
U-r
Proof:
P
Let
Gr
=
<
U 0- ,UL>
Gcw
i s g e n e r a t e d by
G,
2G2
with
(")
Hr
of
[
.
- Hr
kala
Since
G
is universal,
Hc
C Gr
o- e x a c t l y as t h e r o o t s a r e .
.
.
t o prove ( a ) when t h e r e i s a s i n g l e o r b i t ; i . e ,
of t h e t y p e s
SL2, 2 ~ 2 . ,2c2 3
or
.
2 ~ 2
For
; i.e.,
i s a d i r e c t product
H
These groups
s i m p l e ] ( s e e t h e c o r o l l a r y t o Lemma 28).
a r e permuted by
By t h e
3
c o r o l l a r y t o Theorem 33, i t s u f f i c e s t o show
9
.
H,_A G6
=
and
infinite.
0
Y
HE
and l e t
k
,U
Hence i t i s enough
, when
SL2,
Gc
i s one
t h i s i s clear.
x e Ug-
(1) For
[ l ] , write
Y
n = n ( x ) e N ~ G ~ Then
.
and
a f i x e d c h o i c e of
ated.
U&.
m u l t i p l i c a t i o n by
Uzc
I1
Gr=
it
Hr be t h e g r o u p s o g e n e r -
To s e e t h i s l e t
UkHp
V?'
Consider
.
x]
0
Hr
x = u n u
w i t h u . a U- i = 1 , 2
1 2
1
Cr
i s g e n e r a t e d by [ n ( x ) n ( ~ ~ ) - ~ l x ~
.T h i s
FP
n ( x o ) U&
s e t i s c l o s e d under
It i s a l s o c l o s e d under r i g h t m u l t i p l i c a -
.
t i o n by n ( x o ) - l
T h i s f o l l o w s from n ( x o ) - '
-1
-1
-1 =
= n ( x o ) n ( x o ) n ( x o ) and n ( x o ) u @ x o )
U,
x ~a )U ~ -~ [ I )
x = u , ( n ( ~ ) n ( ~ ~ ) - ~ )f onr ( ~
- Gb
Gr 0
11
,
(2)
P
Her - Hc .
0
c
G,
.
77
since
We s e e t h a t
F?
whence
If
0-
= n(x-l)
fl
a and
A2, C2,
a r e t h e s i m p l e r o o t s of
or
l a b e l e d a s i n Lemma 63 ( c ) , ( d ) , o r ( e ) r e s p e c t i v e l y , t h e n
.b
i s isomorphic t o
I
I
(3)
R
let
Let
h
b e a r e p r e s e n t a t i o n of
vC
weight of
->
<
be t h e weight such t h a t
shifting the coefficients t o
let
cp: t
kSr v i a t h e map
xk
and l e t
11) ,
f(x) # 0
x e Ur-
write
Then
and
v-
k ) having
Let
p
let
xvand
x a
m
ur-
of
i s isomorphic under
++
= n(x)v
k q g e n e r a t e d by a l l
[l]
, < h,fl >
,
= O
2f by
be t h e lowest
For
f o r lower weights.
c
p
@.I
subgroup
= 1
be a c o r r e s p o n d i n g weight v e c t o r .
+ -+ t e r m s
9
>
Hr
a s h i g h e s t weight and
h
xv- = f ( x ) v
Hp
h,a
( o b t a i n e d from one of
be a c o r r e s p o n d i n g weight v e c t o r .
R
.
h , ( t ) h g ( t 8)
G2
f(x)f(xo)-l
i n (2) to the
.
To prove ( 3 ) ,
and w r i t e
x = u n ( x ) u 2 a s i n (1). We s e e
1
+
t e r m s f o r lower w e i g h t s , s o n ( x ) v - = f ( x ) v
n ( x ) n ( x o ) - l v t = f ( x ) f (xo)-'v+
t h e n by t h e c h o i c e of
.
If
h,f(x)f(xo)-'=t
8
n ( x ) n ( x o ) - I = ha ( t ) h p ( t )
( s e e Lemma 1 9 ( c ) ) .
(3)
,
t h e n f o l l o w s from (1).
.I.
( 4 ) The c a s e
Gcd
Here
2 ~ ..2
f ( x ) = -ue
is
[
->
x
(-u8)
elements
.
.I,
u o kq.
u
~f
u + t t 8-
t
by Lemma 63 ( c ) . .
1
to
0
1
m
Thus,
v a l u e s of
8
of
k-
# u
,
set
.
4,
m = k"'
is
k
2,
+
=
X
.
-
.
-
ue)-l
,
and i f
ul
.
0
o r l ) ,so t h a t
Here
2 ~ 2
G
;
Then
uul
and
R
while
a+B
can b e t a k e n a s
v
-
=
X -a
i n t h e expression f o r
.
f*
see that
(6)
+
t 3+3e u
m = k"'
Q
,
= u
are
u e m
and
u2@ + t u
-B
.
xk
Letting
a c t i n g on t h i s i d e a l ,
x =
.
G p 2 ~ 2
t 1+3e~38+ tv3'
by a l l v a l u e s of
f(x)
and w r i t i n g
f (x)
.
v = ( v8) 2 8
By t a k i n g
we
.
The c a s e
+
u
Let
u1
+
f ( x ) = t 2+2e
x a ( t ) xP ( t 8 ) x ~ + ~ ~ he
( ~ +) txI+@)
~ + ~we can d e t e r m i n e
t = 0
8
u
To s e e t h i s , f i r s t n o t e t h a t s i n c e t h e c h a r a c t e r i s t i c
k .
t h e r e i s an i d e a l i n
lvsupportedi' by s h o r t r o o t s .
The r e p r e s e n t a t i o n
v
ul = ( u
ul8 -
so t h a t
( 5 ) The c a s e
-
(ttB)
whose t r a c e s a r e norms
.
.Ir
m = k-'.
and
tte =
i s t h e group g e n e r a t e d by r a t i o s of
(their traces are
f
+
f(x) = u
Thus,
-41
ul o k.'.
choose
.
-
.I,
and
To
R : X ~ - > S La n~d i(f ~x ~= x a ( t ) xP ( te ) xa+$ ( u )
then
of
.
of ( 3 ) i n t h i s c a s e
R
s e e t h i s , we n o t e t h a t t h e r e p r e s e n t a t i o n
m = kq.
and
f
-
f o r which
Here
tuv
.
f ( x ) = t 4+6@ -
The group
(t,u,v)
#
rn
1+3@-
2
i s generated
( 0 7 0 , 0 ) , and it c o n t a i n s
k32
and -1 ; hence
representation
on
gk,
m =k
k
if
is f i n i t e .
Here t h e
can be t a k e n t o be t h e a d j o i n t r e p r e s e n t a t i o n
R
and
= '2a+3@
V+
* ,
.
v- = X-2a-38
Letting
i n Lemma 63 ( e l , and working modulo t h e i d e a l i n
.
x
Xk
be a s
"supportedw
by t h e s h o r t r o o t s , we can compute f ( x )
Setting t = u = 0 9
we see t h a t - v 2 E m
hence - 1 e m and k*2 C m
If k i s
.
,
m =k
finite
I
t2 = - 1 with
,
t =0
so
t e k .
3e2 = 1
Since
,
0
To show ( * )
t o = +- t
so
.
But
, suppose
te 2 = t
and
t3 = t 2t =
-t ,
T h i s proves t h e lemma.
2 ~ 2w i t h
k
d
~~ = k*/m
H
~w i t h
~ = HG
,
t2'=- 1
0
is universal, then
G
.
k*2
t = ( te 2) 3 = t 3
we s e e
except p o s s i b l y f o r
G
Then
a contradiction.
If
Corollary:
-16
f o l l o w s from ( * )
0
Go_ and
Gr=
Hr=
H,
i n f i n i t e i n which c a s e
as i n ( 6 ) above.
m
J-
Remarks:
(a)
One can make t h e changes i n v a r i a b l e s
u
->
u
- t1+3e t o
convert t h e form
-
v2
+
t 2u 2
.
+
m = kq. always i f
It i s not known whether
tvje
->
v
f
in
v
+
Gg
tu
and t h e n
4+6 e - u1+38
(6) t o t
Both b e f o r e and a f t e r t h i s s i m p l i f i c a t i o n
t h e form s a t i s f i e s t h e c o n d i t i o n of homogenity:
f ( t , u , v ) = t4+6ef ( l , u / t 1+3e, v / t 2+3e)
(b)
d i r e c t proof i n c a s e
I
t
+ .
0
A c o r o l l a r y of ( 3 ) above, i s t h a t t h e forms i n ( 5 ) and
(6) are definite, i.e.,
Suppose
if
f = 0
f
implies
t = u (= v ) = 0
.
A
i s a s i n ( 5 ) can be made as f o l l o w s :
O = f ( t , u ) = t 2+28
+
u2@ + t u
w i t h one of
t, u
.
2 ~ 2
nonzero.
.
. We s e e
f ( t , u ) = t 2+2e f ( l , u / t 2w1) u s i n g
2e2 = 1 . Hence we may assume
t = 1 . Thus, 1 + u 2e + u = 0 o r (by applying 8 ) ue = 1 + u .
Hence u e 2 = l + u e = u and u = u 2e2
2 . Thus, u = 0 o r 1, a
t =0
If
,
then
,
u =0
t
s o we have
#
0
= U
contradiction.
A d i r e c t proof i n c a s e
i s as ( 6 ) a p p e a r s t o be
f
q u i t e complicated.
(c)
The fo'rm i n ( 5 ) l e a d s t o a geometric i n t e r p r e t a t i o n of
. Form t h e graph
f ( x ) . Imbed k3 i n
adding t h e p l a n e a t
direction
Q
(0,0,1)
+ t u i n k3 of t h e form
v = t 2+2e + u2'
2 ~ 2
p3(k)
,
co
,
projective
3-space over
and a d j o i n t h e p o i n t a t
,
by
o i n the
t o t h e graph t o o b t a i n a s u b s e t
i s t h e n an ovoid i n
k
Q
of
p3(k)
.
p3(k) ; i.e.
(1) No l i n e meets
i n more t h a n two p o i n t s ,
Q
( 2 ) The l i n e s through any p o i n t of
Q
not meeting
Q
again
always l i e i n a plane.
The group
2 ~ 2i s t h e n r e a l i z e d a s t h e group of p r o j e c t i v e
P3 ( k )
t r a n s f o r m a t i o n s of
fixing
Q
.
For f u r t h e r d e t a i l s a s
w e l l a s a corresponding geometric i n t e r p r e t a t i o n o f
J. T i t s , Sdminaire Bourbaki, 210 (1960).
of
2 ~ 2,
2 ~ 2s e e
For an e x h a u s t i v e t r e a t m e n t
53
e s p e c i a l l y i n t h e f i n i t e c a s e , s e e Luneberg, S p r i n g e r
Lecture Notes 1 0 (1965).
Theorem 34:
Let
G
Excluding t h e c a s e s :
(d)
and
c
be a s above with
( a ) ' ~ ~ ( 4, ) ( b )
2 ~ 4 ( 2 ), we have t h a t
P
Gd
G
2 ~ 2 ( 2 ),
universal.
(c)
' ~ ~ ( 3,)
i s simple over i t s c e n t e r .
Sketch of p r o o f :
Using a c a l c u l u s of double c o s e t s r e
B6
,
which
can be developed e x a c t l y a s f o r t h e Chevalley groups w i t h
in
p l a c e of
,
W
and
Wr
( s e e Theorem 3 2 ) ) i n p l a c e of E
( o r ,Z
R
and Theorem 33, t h e ' p r o o f can be reduced e x a c t l y a s f o r t h e Chevalley
P
groups t o t h e proof o f :
P
9
Xa , =
on
Hr
-
DG; .
can be used t o show
t a k e s care o f n e a r l y everything.
If
t h e commutator r e l a t i o n s w i t h i n t h e
used.
If
k
has
Igenough"
by t h e C o r o l l a r y t o Lemma 64 and t h e a c t i o n
Hr
elements, so does
of
Gr
a,'='-
f l ~ ;.
This
h a s !?fewn e l e m e n t s t h e n
k
xaqs
and among them can be
T h i s l e a d s t o a number o f s p e c i a l c a l c u l a t i o n s .
The d e t a i l s
a r e omitted.
Remark:
The g r o u p s i n ( a ) and ( b ) above a r e s o l v a b l e .
The g r o u p
i n ( c ) c o n t a i n s a normal subgroup of i n d e x 3 isomorphic t o
A1(8).
The group i n ( d ) c o n t a i n s a Stnewtss i m p l e normal subgroup of i c d e x 2,
( S e e J. T i t s , ;?Algebraic and a b s t r a c t s i m p l e g r o u p s , f v Annals of
Math. 1964.)
Exercise:
C e n t e r of
Gr
9
= ( C e n t e r of
GIr.
We now a r e going t o d e t e r m i n e t h e o r d e r s of t h e f i n i t e
be a f i n i t e f i e l d of
b e minimal such t h a t 8 = pa (1.e . ,
Chevalley g r o u p s of t w i s t e d t y p e .
characteristic
p
.
Let
a
Let
k
a
t e = tP
such t h a t
f o r a l l t P k ) . Then 1 kl = p2a f o r
2~
ja f o r 3 ~ 4
; and ( k t = p 2a+l
2
= p
A n 3 n 7 2E6 ;
f o r 2 ~ 2, 2F q , 2 G 2 . We c a n w r i t e r % ( t
=)
x ( eat s ( a ))
ikl
Ya
where
q(a)
i s Some power of
geometric a v e r a g e of
when
Let
l e s s than
' F ~,
be t h e automorphism of
V
the roots a s
g
permutes t h e r o o t s .
we s e e t h a t
o"
a c t s on t h e space
under
0
Since
W .
is the
q = pa
except
2 ~ 2i n which c a s e
be t h e r e a l Euclidean s p a c e g e n e r a t e d by t h e r o o t s and
V
0
or
q
If
(kl
over each 9 - o r b i t t h e n
2 ~ 2,
i s of t y p e
GT
let
q(a)
p
permuting t h e r a y s t h r o u g h
Since
'5
normalizes
W
,
of polynomials i n v a r i a n t
I
r0 a l s o a c t s o n t h e subspace of
I
of
homogeneous e l e m e n t s of a g i v e n p o s i t i v e d e g r e e , we may choose
the basic invariants
roI
that
j
base f i e l d
,
= a .I
J j
Ij
on
V
of
r0 on V
f o r some
e
( h e r e we have extended t h e
e
j
of Theorem 27 s u c h
A s b e f o r e , we l e t
ifoj 1j
we a l s o have t h e s e t
Theorem 35:
.
Let
= 1,
We r e c a l l a l s o t h a t
C
dj
be t h e d e g r e e
Since
..., 41
a;,
of e i g e n v a l u e s
d e n o t e s t h e number of
N
w , q , N,
e
j
and
'
dj
be a s above,
We have
The o r d e r of t h e c o r r e s p o n d i n g simple group i s
1 Gr I
o b t a i n e d by d i v i d i n g
C
acts
.
is universal.
G
(b)
..., 4 ,
j = 1,
and t h e s e a r e u n i q u e l y determined.
positive roots i n
assume
,
%? t o Q ) .
of
,
Ij
i s t h e c e n t e r of
G
,
by
I Cr I
where
and
r ,H , - U , etc,
~emma.65: Let
by
i s t h e number of p o s i t i v e r o o t s i n
N(w)
where
( a ) It s u f f i c e s t o show t h a t
a e C/R
I,H I
contribution t o
Since
corresponding t o
( b ) follows.
Corollary :
U,
Lemma 66:
Proof:
(c)
a
= qIal
(b)
for
Let
,
r h a ( t ) = hpa(tq(a))
be a
a
the
J
Hr
made by e l e m e n t s of
nq(a))= qm a ea
1
(xa,rl
T h i s i s s o by Lemma 63.
by Lemma 62.
4 - o r b i t o f simple r o o t s .
(
made n e g a t i v e
C
.
w
proof:
is
be a s above.
1 if
"supportedsv by
m = la1 , S i n c e t h e
e
0j
X"
a r e t h e r o o t s of t h e polynomial
vs
-
1
,
T h i s f o l l o w s from ( a ) , ( b ) , and Theorem 33.
i s a p-Sylow subgroup.
We have t h e f o l l o w i n g f o r m a l i d e n t i t y i n
t:
We modify t h e proof of Theorem 26 a s f o l l o w s :
(a)
T
t h e r e i s r e p l a c e d by
r o here.
(b)
X
t h e r e i s r e p l a c e d by
Xo h e r e , where
i s t h e s e t of u n i t v e c t o r s i n
same d i r e c t i o n s of t h e r o o t s .
V
Zo
which l i e i n t h e
a
Only t h o s e s u b s e t s
(c)
a
rr
of
%
f i x e d by
a r e con-
sidered.
(d)
(-lln
number of' ro o r b i t s i n
(e)
W(t)
(-ilk
i s now d e f i n e d t o be
where
is the
k
.
a
i s n o w d e f i n e d t o be
C
we w,
tN(w)
With t h e s e m o d i f i c a t i o n s t h e proof proceeds e x a c t l y a s b e f o r e
through s t e p ( 5 ) .
(6')
in
K
For
S t e p s ( 6 ) - ( 8 ) become:
a C
rr ,
congruent t o
Da
w
under
~ ( - 1 ) ~ N ~ = wd .e t ( H i n t :
?
c u t by
,
on
V
V?
of t h e c e l l s o f
0
(7 )
Let
r e s t r i c t i o n of
(8' )
K
Let
,
let
W
and f i x e d by
w
and
K
W
6
V
If
?
= V
K
f i x e d by
X t o < Wa,ro
be a
of skew i n v a r i a n t s under
i n v a r i a n t s under
Wn
Wo-
t h e n t h e c e l l s of K
w%
be a c h a r a c t e r on
M
NT
.
<
W,
W
>
?O
be t h e number of c e l l s
.
?
i s t h e complex
a r e t h e i n t e r s e c t i o n s with
.)
<
W,%
>
<
induced up t o
% > module, l e t
, and l e t I a ( M )
d.
and
W,ro
The
e
j
qs
the
>
.
Then
A
I(M)
be t h e space
be t h e s p a c e of
Then
The remainder of t h e proof proceeds a s b e f o r e .
Lemma 6 7 :
Then
form a permutation of t h e
e
0j
's.
I
I
S e t t = 1 i n Lemma 66.
Proof:
among t h e
as among t h e
e Os
j
Then (::)
e
.
vs
0j
1 h a s t h e same m u l t i p l i c i t y
T h i s i s so s i n c e o t h e r w i s e
t h e r i g h t s i d e of t h e e x p r e s s i o n would have e i t h e r a r o o t o r a
2 = 1 and a l l
pole at t = 1
Assume
1 , t h e n e i t h e r c0
.
e q s not
1 are
cube r o o t s of
-1
e v s not
1, coming i n c o n j u g a t e complex p a i r s
1 are
since
( a ) f o l l o w s from Lemmas 6 5 , 66, 67.
Proof of Theorem 35:
C
= 1 and a l l
c0 is
Thus i n a l l c a s e s ( + ) i m p l i e s t h e lemma.
real.
0
%3
or else
be t h e c e n t e r
Clearly
Gr.
C
0
3
C
Now l e t
Using t h e c o r o l l a r y
0-
t o Theorem 33 and an argument similar t o t h a t i n t h e proof of
C o r o l l a r y l(b) t o Theorem 4
v
a c t s v T d i a g o n a l l y , we have
Corollary:
The v a l u e s of
,
C
0
we s e e
C C
I Gr I
,
and
C
0
c
.
HrC H
hence
C
( cr
I
IG,
I
0
=
= C
Since
0-3
I om
H
_
proving ( b ) .
*
( L ~ / L ~)r
, ~(
a r e a s follows:
1
G
r
e . q #~ 1
lcr I
d
Chevalley
group ( s = 1)
>
2 ~ n ( n 2)
None
(
q N ~ ( jW1)
q
I H O ~ ( L ~ / Lk*)
~ ,1
-1 i f d . i s odd
3
d
Replace q j-1 by
d.
d.
q J-(-l)
J in (a)
same a s
Same change a s 2~ n
(3, q+l)
-1 f o r one d 3.=n
Replace one qn-1 by
qn+l i n (>::)
( 4 , qn+l)
a , u2 f o r d
12 2
6
9 ( q - 1 N q -1)
Same change; i , e.
( n + l ,q + l )
2
E6
2~
n
= 494
2~ n
j
( 9 +q 4+1)
1
e.?s# 1
Go-
-1 f o r d
2c2
2
-1 f o r d
G2
-1 f o r d
* ~ 4
j
j
j
1
6 2
6
q ( 9 - U ( q +1)
6
= 6,12
q
I
ICr
4 ( q2 - 1 ~ ~ 4 + 1 )
= 4
=
I
IGT
1
24 ( q2-1)( q6+ l ) ( q8-1)
1
( q12+1
Here
d
d e n o t e s a p r i m i t i v e cube r o o t of 1
I Cg I ) :
R o o f (except f o r
We c o n s i d e r t h e c a s e s :
We f i r s t n o t e
*A n
-
(::)
use t h e standard coordinates
%
t i o n s of
-> -
wi
i s g i v e n by
[ai],
-1 e
Wn+2-i
.
.
W%
{wi( 1 <
-1 o W o;
we s e e
.
i
<
-
n
Since
.
($1
To prove
+
W
1) f o r
we
.
An
Then
a c t s v i a a l l permuta-
Alternatively, since
W
is
t r a n s i t i v e on t h e simple systems (Appendix I I . 2 4 ) , t h e r e e x i s t s w,
w
such t h a t
-1 e Vir0
or
-
.
14 W
.
- =
.
= 1 or
o- ; i . e . , -1 e W
0
S i n c e t h e r e a r e i n v a r i a n t s of odd d e g r e e ( d i = 2 , 3 , . . . ) ,
(n)
By
Hence, - w o ( - 1 )
e W
%
f i x e s t h e i n v a r i a n t s of even d e g r e e and
changes t h e s i g n s of t h o s e o r odd d e g r e e .
.
2
E6
t h e case
2~2n+1
15 i
(:::)
in
2 ~ nmay be used h e r e , and t h e same c o n c l u s i o n h o l d s .
*Dn ( n
[vil
The second argument t o e s t a b l i s h
< n)
even o r o d d ) .
,
Relative t o t h e standard coordinates
t h e basic i n v a r i a n t s a r e t h e first
n-1 e l e m e n t a r y
2
fvi]
symmetric polynomials i n
together with
vi
,
v i a a l l p e r m u t a t i o n s and even number of s i g n changes.
can b e t a k e n t o be t h e map
vi
->
v i (1
<i
n
-
1)
The d e g r e e s of t h e i n v a r i a n t s a r e
Lemma 6 7 , t h e f i t s a r e 1, 1, ,
w2
2
W
must occur i n t h e same dimension.
.
V
i n t h e u s u a l formula b y
Since
2 ~ 2,
2
<
> is
W,ro
form, s o t h a t
2 ~ .4
the
G2 ,
E:
j
-
d
1, 1, -1, -1
I
.
ru
2 , 4, 6 , and 4.
By
5
i s r e a l , w and
4 2
Thus, we r e p l a c e ( q -1)
Since
e i q s a r e 1, -1 by Lemma 67.
j
= 2
.
.
and
A s before t h e r e i s a quadratic
Consider
I = C
a' +
a long root 0 short root
i s a n i n v a r i a n t of d e g r e e d f i x e d by
t h e r e i s a q u a d r a t i c i n v a r i a n t f i x e d by
The f i r s t p a r t i s c l e a r s i n c e
lvi 1 i = 1, 2 , 3 , 41
%
and
which d o e s n o t d i v i d e
W and
and permute t h e r a y s t h r o u g h t h e r o o t s .
choose c o o r d i n a t e s
n
a f i n i t e g r o u p , i t f i x e s some nonzero q u a d r a t i c
= 1 for
i n v a r i a n t f i x e d by
.
->-v
w2 ) = q8 + q 4 + 1 .
I n both c a s e s t h e
J
I
v
The d e g r e e s of t h e i n v a r i a n t s a r e 2 , 6 , 8, 1 2
eiqs are
We c l a i m t h a t
( q4- w ) ( q4
ro
Here
,
acts
W
ro.
Hence, o n l y t h e l a s t i n v a r i a n t changes s i g n under
' D ~.
-
and
4
preserve lengths
To s e e t h e second p a r t ,
so t h a t t h e long r o o t s
( r e s p e c t i v e l y , t h e s h o r t r o o t s ) a r e t h e v e c t o r s o b t a i n e d from
2vl,
v1 + v2 + v3 +
V4
t i o n s and s i g n changes.
(respectively,
vl + v 2 )
by a l l permuta-
The q u a d r a t i c i n v a r i a n t i s
2
v2 + v2
1
+
v32
+ v 42
,
I
To show t h a t t h i s d o e s n o t d i v i d e
c o n s i d e r t h e sum of t h o s e
which i n v o l v e o n l y v
and v2 and n o t e t h a t t h i s
1
2
i s n o t d i v i s i b l e by vl2 + v2
Hence, I can be t a k e n a s one of t h e
terms i n
I
.
b a s i c i n v a r i a n t s , and
Remark:
(2
e
j
= 1 if
d
~ 2 )i s n o t d i v i s i b l e by 3
j
== 8 , .
.
Aside from c y c l i c g r o u p s
of prime o r d e r , t h e s e a r e t h e o n l y known f i n i t e s i m p l e g r o u p s
with t h i s property.
As
Now we c o n s i d e r t h e automorphisms of t h e t w i s t e d groups.
f o r t h e u n t w i s t e d groups d i a g o n a l automorphisrns and f i e l d a u t o morphisms can b e d e f i n e d .
Theorem 36:
Let
subgroup of
G
that
6
G
and
( o r Gr
be a s i n t h i s s e c t i o n and
0-
g e n e r a t e d by
)
is not t h e identity.
Ur
and
G,
UL .
P
the
A ssume
Then e v e r y automorphism of
G,
9
i s a p r o d u c t of a n i n n e r , a d i a g o n a l , and a f i e l d automorphism.
Remark:
Observe t h a t graph automorphisms a r e missing.
Thus t h e
t w i s t e d g r o u p s cannot t h e m s e l v e s be t w i s t e d , a t l e a s t n o t i n t h e
s i m p l e way we have been c o n s i d e r i n g .
S k e t c h of p r o o f :
A s i n s t e p (1) of t h e proof of Theorem 30, t h e
,
automorphism, c a l l i t
morphism s o t h a t i t f i x e s
may be normalized by a n i n n e r a u t o -
UC
and
U&
( i n t h e f i n i t e c a s e by
S y l o w l s theorem, i n t h e i n f i n i t e c a s e by arguments from t h e t h e o r y
of algebraic groups).
the
Xaqs
3a , T )
( a simple,
and a l s o t h e
Then it a l s o f i x e s
a E Z/R;
X -a
v
Hr, and i t permutes
h e n c e f o r t h we w r i t e
for
a
r s a c c o r d i n g t o t h e same p e r m u t a t i o n ,
i n a n a n g l e p r e s e r v i n g manner ( s e e s t e p ( 2 ) ) i n t e r m s of t h e
corresponding simple system
-
1.
i
nr
of
V
.
0-
By checking c a s e s
one s e e s t h a t t h e permutation i s n e c e s s a r i l y t h e i d e n t i t y :
(XalV S
i s f i n i t e , one need o n l y compare t h e v a r i o u s
o t h e r , w h i l e if
if
k
w i t h each
i s a r b i t r a r y f u r t h e r argument i s n e c e s s a r y
k
q
s are
(one can, f o r example, check which C
Abelian and
which a r e n o t , t h u s r u l i n g o u t a l l p o s s i b i l i t i e s except f o r
2
2 ~ 3,
E6
,
and
3 ~ 4, and t h e n r u l e o u t t h e s e c a s e s ( t h e f i r s t
two t o g e t h e r ) by c o n s i d e r i n g t h e commutator r e l a t i o n s among t h e
.
$,?s)
AS i n s t e p ( 4 ) of t h e proof of
Theorem 30, we need
?
o n l y complete t h e proof of o u r theorem when
groups
Ga =
of t h e t y p e s
< Xa , X-a > ,
A1,
-
1
i n o t h e r words, when
or
excluded, b u t n o t
A1(2), A1(3),
assume.
A1
The c a s e
2 ~ 2(with
i s one of t h e
0
Gb
2 ~ 2 ( 2 ) and
i s of one
2~2(3)
2 ~ 2 ( 4 ) ) ,which we h e n c e f o r t h
or
having been t r e a t e d in
5
1 0 , we w i l l t r e a t
o n l y t h e o t h e r c a s e s , i n a sequence of s t e p s .
We w r i t e
or
a s given i n
x(t,u,v)
Lemma 63 and
I
"C2
2 ~ 2 ,
Go_
f o r t h e g e n e r a l element of
d(s)
for
ha.( s ) h g ( s e )
(1) We have' t h e e q u a t i o n s
.
U,
x ( t,u)
T h i s f o l l o w s from t h e d e f i n i t i o n s and Lemma 2 0 ( c )
( 2 ) Let
t =0
setting
U2
U1,
,
for
is
i f t h e type i s
Exercise:
.
Then
2 ~ 2o r
,
.. ) d ( s ) - '
Consider
g
then
s
...
)) 3
3 U1 3 U2 3 1
Ur
* ~ * ( 4 ) i s excluded, t h e n
= x ( g ( s ) t,...)
a homomorphism whose image g e n e r a t e s
that
while
2 ~ 2
the
Prove t h i s .
d(s)x(t,.
g(s) = s
,(Ur ,,U
ia
.
2G2
( 3 ) If t h e c a s e
Proof:
o b t a i n e d by
U T 3 U1 3 U2 = 1
2 (Ur, Ur ) 3 ( Uc
Ug
if t h e t y p e i s
Ur
u =0
then a l s o
lower c e n t r a l s e r i e s
Ur
be t h e subgroups of
.
s e ke
for
.
.
2 ~ 2
,
'-',
g(s) = s
Ck: kg] = 2
Since
t a k e s on a v a l u e o u t s i d e of
s
3
.
ke
E kg
ks ->
g:
st.
kq'
additively.
k
By (1) we have
= ( s2-e)6 s o t h a t
with
,
,
so t h a t
we need o n l y show
Now i f
for a l l
doesn't,
g
s e kS
,
whence we e a s i l y conclude ( t h e r e a d e r i s asked t o supply t h e
proof) t h a t
k
h a s a t most 4 e l e m e n t s , a c o n t r a d i c t i o n .
2 ~ 2t h e proof i s s i m i l a r , but e a s i e r .
2 ~ 2 and
(4)
The automorphism
v
cp ( o f Gr
)
can b e n o r n a l i z e d by
a d i a g o n a l and a f i e l d automorphism t o b e t h e i d e n t i t y on
Proof:
on
Since
.
ug/ul
f: k
For
->
k
cp
fixes
U
,
it a l s o f i x e s
U1,
hence a c t s
Thus t h e r e i s an a d d i t i v e isomorphism
such t h a t
cp x ( t
,... )
= x(f(t),
...I
UT/ul
.
.
By m u l t i p l y i n g
f(1) = 1
:
kC
.
->
cp
Since
by a d i a g o n a l automorphism we may assume
rp
9
,
HT
fixes
t h e r e i s an isomorphism
.
Q
.
k-l-
cpd(s) = d ( i ( s ))
such t h a t
.
Combining t h e s e
e q u a t i o n s w i t h t h e one i n ( 3 ) we g e t
.I.
s e k."
f(g(s)t) = g(i(s))f(t) for a l l
t = 1 , we g e t
Setting
f(g(s)t) = f(g(s))f(t)
f
(
.
i s m u l t i p l i c a t i v e on
2 ~ 2( 4 )
.
t e k
,
f(g(s)) =g(i(s))
If t h e c a s e
k
,
so that
is excluded, t h e n
by ( 3 ) , hence i s a n automorphism.
same c o n c l u s i o n , however, h o l d s i n t h a t c a s e a l s o s i n c e
0
and
1 and permutes t h e two e l e m e n t s of
k
not i n
fixes
f
kg
The
.
O u r o b j e c t now i s t o show t h a t once t h e n o r m a l i z a t i o n i n
( 4 ) h a s been a t t a i n e d
(5)
w
some
is necessarily the identity.
f i x e s each element of
cp
1
e Gc
u1/u2
and
,
U2
and a l s o
which r e p r e s e n t s t h e n o n t r i v i a l element of t h e Weyl
group.
Proof:
The first p a r t e a s i l y f o l l o w s from ( 2 ) and ( 4 ) , t h e n
t h e second f o l l o w s a s i n t h e proof of Theorem 3 3 ( b ) .
( 6 ) If t h e t y p e i s
Proof:
Consider t h e t y p e
the fact that
f =1,
2 ~ 2o r
2 ~ 2.
we g e t
cp
cpx(t , u ) = x ( t , u
+
f i x e s every
; ( t ) ) with
.!
,
then
1 + F, t h
d(s)
j
is the identity.
From t h e e q u a t i o n
g(s) = g(i(s))
= i ( s ) 2 - 2 e, and t h e n t a k i n g t h e
i n o t h e r words
2G2
.
,
power,
(b)
i.e.,
of ( 4 ) and
s2-28
s = i(s) ;
By ( 4 ) and ( 5 ) ,
an a d d i t i v e homomorphism.
Conjugating t h i s e q u a t i o n by
d(s) = qd(s)
,
u s i n g ( l ) ,and
comparing t h e new e q u a t i o n w i t h t h e o l d , we g e t j(s2-2et)
1+0 , j ( s t ) = s1+2ej(t)
= s2'j ( t) , and on r e p l a c i n g s by s
s
Choosing
0 , 1, which i s p o s s i b l e because
e x c l u d e d , and r e p l a c i n g
s by
s+l
s i n c e o t h e r w i s e we would have s + s2' = ( s
.
s
c o n t r a r y t o t h e c h o i c e of
Ur
f i x e s e v e r y element o f
Thus
+
.
Ur
then
.
I n o t h e r words
j(t) = 0
2 ~ 2i n s t e a d ,
0
Gc
Since
of ( 5 ) ,
w
and t h e element
s = s2
s2e)2'
t h e argument i s similar, r e q u i r i n g one e x t r a s t e p .
i s g e n e r a t e d by
s + s2@
$; 0
Now
If t h e t y p e i s
,
h a s been
1 and combining
and by
( s + s2') j ( t ) = 0
t h e t h r e e e q u a t i o n s , we g e t
rp
*c2(2)
.
is the
identity.
The preceding argument, s l i g h t l y m o d i f i e d , b a r e l y f a i l s f o r
.
2 ~ 2, i n f a c t f a i l s j u s t f o r t h e smallest c a s e
The
2~2(4)
proof t o f o l l o w , however, works i n a l l c a s e s .
( 7 ) If t h e t y p e i s
Proof:
P
0
w x ( t , u ) w- l = x n x
calculation i n
depending on
we w r i t e
with
SL3
w
is the identity.
then
a s i n ( 5 ) a n d , assuming
w
Choose
2 ~ 2,
$I.
0
,
write
P
n e Hr
e Ur,
x,x
shows t h a t
b u t n o t on
u
t
w .
x = x(at5-',
or
cpx(t,u) = x ( t , u + j ( t ) )
u
,
.
:::)
A simple
f o r some
(Prove t h i s . )
apply
rp
a
If now
t o t h e above
-I
e q u a t i o n , a n d u s e ( 4 ) and ( 5 ) , w e g e t
so t h a t
j(t) = 0
tc-'=t(u+
4
j(t))
and we may complete t h e proof a s b e f o r e .
,
k
;'
,
,
It i s a l s o p o s s i b l e t o determine t h e isomorphisms among t h e
v a r i o u s C h e v a l l e y g r o u p s , b o t h t w i s t e d and u n t w i s t e d .
We s t a t e
t h e r e s u l t s f o r t h e f i n i t e groups, omitting t h e proofs.
( a ) ' Among t h e f i n i t e simple C h e v a l l e y g r o u p s , t h e i r
Theorem 37 :
t w i s t e d a n a l o g u e s , and t h e a l t e r n a t i n g g r o u p s q ( n
list of isomorphisms i s g i v e n a s f o l l o w s .
.
(1)
Those independent of
(3)
J u s t s i x o t h e r c a s e s , of t h e i n d i c a t e d o r d e r s .
k
A1(4) -- Al(5) ~0~
A1(7)
A2(2)
A1(9)1.
46
~
(
~
60
168
360
2
)
~
a
~
20160
> 5)
,
a complete
(b)
I n a d d i t i o n t h e r e a r e t h e f o l l o w i n g c a s e s i n which t h e
Chevalley group j u s t f a i l s t o be simple.
The d e r i v e d group of
360
B2(2) N
The i n d i c e s i n t h e o r i g i n a l group are 2, 3 , 2, 2 , r e s p e c t i v e l y . .
Remarks:
(a)
The e x i s t e n c e of t h e isomorphisms i n ( 1 ) and ( 2 )
i s e a s y , and i n ( 3 ) is proved, e . g . ,
i n Dieudonne' (Can. J. Math.1949).
There a l s o t h e f i r s t c a s e of ( b ) , c o n s i d e r e d i n t h e form
B2(2)
(symmetric group) i s proved.
S6
(b)
It i s n a t u r a l t o i n c l u d e t h e simple g r o u p s
in
t h e above comparison s i n c e t h e y a r e t h e d e r i v e d g r o u p s of t h e
Weyl groups of t y p e
Anml
and t h e IfJeyl g r o u p s i n a s e n s e form
t h e s k e l e t o n s of t h e corresponding Chevalley groups.
l i k e t o p o i n t o u t t h a t t h e Weyl g r o u p s
W(En)
a r e a l s o almost
simple and a r e r e l a t e d t o e a r l i e r g r o u p s a s f o l l o w s .
Proposition:
We have t h e isomorphisms:
~ W ( E w~B 2) ( 3 )
N
~ ~ w ( E ~ )N/ D4(2)
c
2~3(4)
,
with
C
W
e would
t h e c e n t e r , of o r d e r 2.
1
Proof :
The proof i s s i m i l a r t o t h e proof of
given n e a r t h e beginning of
3
S6 = W ( A 5 )
PJ
B2(2)
1Q.
Aside from t h e c y c l i c g r o u n s of prime o r d e r and t h e g r o u p s
c o n s i d e r e d above, o n l y 11 o r 1 2 o t h e r f i n i t e s i m p l e g r o u p s a r e
a t p r e s e n t ($qag,1368) known. We w i l l d i s c u s s them b r i e f l y .
(a) The f i v e Mathieu g r o u p s
Mn ( n = 11, 1 2 , 22, 23, 2 4 ) .
These were d i s c o v e r e d by Mathieu a b o u t a hundred y e a r s ago and p u t
on a f i r m f o o t i n g by W i t t ('Hamburger Abh. 1 2 ( 1 9 3 8 ) ) . They a r i s e
as h i g h l y t r a n s i t i v e p e r m u t a t i o n g r o u p s on t h e i n d i c a t e d numbers
of l e t t e r s .
(b)
Their orders a r e :
The f i r s t Janko g r o u p
J1
d i s c o v e r e d by Janko
( J . Algebra 3 ( 1 9 6 6 ) ) a b o u t f i v e y e a r s ago.
It i s a subgroup of
G 2 (11) and can be r e p r e s e n t e d a s a permutation
g r o u p on 266 l e t t e r s .
I
Its order i s
The r e m a i n i n g g r o u p s were a l l uncovered l a s t f a l l , more o r l e s s .
of Janko. The e x i s t e n c e of
j 2 1/2
J2 was p u t on a f i r m b a s i s f i r s t by H a l l and Wales u s i n g a machine,
and t h e n by T i t s i n t e r m s of a 9sgeometry.7? It h a s a subgroup of
(c)
The g r o u p s
J2
and
i n d e x 1 0 0 isomorphic t o & ~ ~ ( 2,W) 2 ~ ( $2 )
.
,
and i s i t s e l f of i n d e x
h a s n o t y e t been p u t on a f i r m
J 2 1/2
b a s i s , and it a p p e a r s t h a t it w i l l t a k e a g r e a t d e a l of work t o
416
j
n
G2(4)
The g r o u p
do s o ( b e c a u s e i t d o e s n o t seem t o have any 'IlargeF7s u b g r o u p s ) ,
b u t t h e e v i d e n c e f o r i t s e x i s t e n c e i s overwhelming.
(d)
of
G.
The group
Higman.
H
qf
The o r d e r s a r e :
D. Higman and Sims, and t h e group
The f i r s t g r o u p c o n t a i n s
M2*
H
0
a s a subgroup of index
100 and was c o n s t r u c t e d i n t e r m s of t h e automorphism group of a
g r a p h w i t h 100 v e r t i c e s whose e x i s t e n c e
of S t e i n e r systems.
depends on p r o p e r t i e s
I n s p i r e d by t h i s c o n s t r u c t i o n ,
G. Higman
t h e n c o n s t r u c t e d h i s own group i n t e r m s of a v e r y s p e c i a l geometry
invented f o r t h e occasion,
The two g r o u p s have t h e same o r d e r ,
and everyone seems t o f e e l t h a t t h e y a r e isomorphic, b u t no one
h a s y e t proved t h i s .
(e)
The o r d e r i s :
The ( l a t e s t ) group
S
of Suzuki.
This contains
G2(4)
a s a subgroup of i n d e x 1782, and i s c o n t r u c t e d i n t e r m s of a g r a p h
whose e x i s t e n c e depends on t h e imbedding
J2 C G 2 ( 4 ) .
It p o s s e s s e s
a n i n v o l u t o r y automorphism whose s e t o f f i x e d p o i n t s i s e x a c t l y
J2
Its order i s :
M
( f ) The g r o u p
o f McLaughlin.
i n terms of a graph and c o n t a i n s
275.
This group i s constructed
' ~ ~ ( 9 a)s a s u b g r o u p o f i n d e x
Its order i s :
Theorem 38:
Among a l l t h e f i n i t e s i m p l e g r o u p s above ( i . e . ,
all
t h a t a r e c u r r e n t l y known), t h e o n l y c o i n c i d e n c e s i n t h e o r d e r s
which do n o t come f r o m isomorphisms a r e :
Bn(q)
and
Cn(q)
(b)
A2(4)
and
~ ~ ( 2 ) ~ ( 2 8
(c)
H
and
H
3'
for
n
23
(a)
.
and
q
odd
.
'
if t h e y a r e n v t i s o m o r p h i c .
That t h e g r o u p s i n ( a ) h a v e t h e same o r d e r and a r e n o t
i s o m o r p h i c h a s b e e n proved e a r l i e r .
The o r d e r s i n ( b ) a r e . b o t h
e q u a l t o 20160 by Theorem 2 5 , and t h e g r o u p s a r e n o t i s o m o r p h i c
since r e l a t i v e t o t h e normalizer
B
of a 2-Sylow s u b g r o u p t h e
f i r s t g r o u p h a s s i x d o u b l e c o s e t s and t h e second h a s 24.
The p r o o f
t h a t ( a ) , ( b ) and ( c ) r e p r e s e n t t h e o n l y p o s s i b i l i t i e s d e p e n d s on
a n e x h a u s t i v e a n a l y s i s of t h e g r o u p o r d e r s which can n o t b e
undertaken here.
.
2
.
R e p r e s e n t 3 t i . n ~ . I n t h i s s e c t i o n we c o n s i d e r t h e i r r e d u c i b l e
A s we s h a l l s e e ,
r e p r e s e n t a t i o n s of t h e i n f i n i t e Chevalley g r o u p s .
h e r e t h e t h e o r y i s q u i t e complete.
A 1 1 r e p r e s e n t a t i o n s a r e assumed
t o be f i n i t e - d i m e n s i o n a l and t h e s t a n d a r d t e r m i n o l o g y i s u s e d .
1
particular
1 must a c t a s t h e i d e n t i t y , and t h e t r i v i a l
s i o n a l (but not t h e t r i v i a l
In
0-dimen-
1-dimensional) r e p r e s e n t a t i o n i s ex-
cluded fr'om t h e l i s t of i r r e d u c i b l e r e p r e s e n t a t i o n s .
We s t a r t
w i t h a g e n e r a l lemma.
Lemma 6 8 ;
Let
K
be a n a l g e b r a i c a l l y c l o s e d f i e l d ,
a s s o c i a t i v e algebras with
(a)
If
(B ,V)
B
i b l e modules f o r
for
and
(Y,W)
C
, then
and
, and
A = B @C
.
a r e (finite-dimensional) irreduc(u.,U ) = ( B @ Y, V @W)
Conversely, every i r r e d u c i b l e
i s one
A-module
( a, U )
is
r e a l i z a b l e , uniquely, a s a .tensor product a s i n ( a ] .
Proof:
(a)
By B u r n s i d e l s Theorem ( s e e , e . g . ,
i n A b s t r a c t a l g e b r a , Vol. 2 ) ,
I
K
C
A .
(b)
I
1 over
and
B
aA = End U
and
(b)
exist since
(a,U)
is f i n i t e - d i m e n s i o n a l .
into
of
under t h e r u l e
c4, = a ( c ) o 4
V
i r r e d u c i b l e submodule.
->
f (v)
End V
and
, whence
%C = End W
is irreducible.
B-homomorphisms
v @f
=
V be an i r r e d u c i b l e
Let
U
BB
Jacobson, Lectures
.
U
,
The map
B-submodule of
Let
-
-,-,*,
.,cw..~.~...-,--...:r--.
.; .,. . -:
. ,..,
<
_i ' .
...
U
. ...
.
(Check t h i s . )
: V @W
->
Such
b e t h e s p a c e of
T h i s i s nonzero and is a
i s e a s i l y checked t o be a n
i s i r r e d u c i b l e by ( a ) , and
L
U
Let
U
(Y,W)
be an
d e f i n e d by
A-homomorphism.
i s by assumption.
C-module
V O W
Hence by Schurrs
Lemma ( s e e l o c , c i - t . )
9
is a n isomorphism,
a
If
=
1
B @
X
f
is
a s e c o n d decolllposition o f t h e r e q u i r e d form, t h e n r e s t r i c t i o n t o
B
P @1 , i . e . m u l t i p l e s of
?
p Q1
yields
i s o m o r p h i c , s o t h a t by t h e Jordan-Holder
@
1
and
are elso.
Similarly
b. a n d
P
1
and
o r Krull-Schmidt
d
are
theorems
a r e i s o m o r p h i c , which
pr.oves t h e u n i q u e n e s s i n ( b ) .
( a ) If
Corollary:
Tci
G =
K
i s an a l g e b r a i c a l l y c l o s e d f i e l d a n d
i s a d i r e c t ' p r o d u c t of a f i n i t e number o f g r o u p s , t h e n
t h e t e n s o r product
V
reducible
,
KG-module
of i r r e d u c i b l e
KGi-modules
and every i r r e d u c i b l e
Vi
i s a n ir-
KG-module i s u n i q u e l y
r e a l i z a b l e i n t h i s way.
(b)
algebras over
K
S i m i l a r l y f o r a d i r e c t sum
k
= C
i
of L i e
.
We a p p l y Lemma 68, e x t e n d e d t o s e v e r a l f a c t o r s , i n ( a ) t o
Proof:
.?
group a l g e b r a s , i n ( b ) t o enveloping a l g e b r a s .
E x e r -.c i s e :
P
over
K
-
lf t h e d i r e c t p r o d u c t above i s one o f a l g e b r a i c g r o u p s
!of t o p o l o g i c a l group:
,
of L i e groups,.
.
. .) , t h e n
r a t i o n a l (conlir,uoi;s , ~ ~ a . 1 - y l ; .i c. , j i f a n d o n l y i f e a c h
Remark:
V
Vi
is
is.
If we a r e i n t e r e s t e d i n t h e i r r e d u c i b l e r e p r e s e n t a t i o n s
of a C h e v a l l e y group
G
, we
may a s w e l l assume i t i s u n i v e r s a l .
The c o r o l l a r y t h e n i m p l i e s t h a t we may a s w e l l a l s o assume t h a t
G
( i. e . t h a t
C)
i s indecomposab3.e.
T h i s we w i l l do whenever it
is convenient.
Now we t a k e up t h e s t u d y of r a t i o n a l r e p r e s e n t a t i o n s f o r
I
I
I
I
J
Chevalley groups o v e r a l g e b r a i c a l l y c l o s e d f i e l d s viewed a s a l g e b r a i c groups.
I n s u c h r e p r e s e n t a t i o n s t h e c o o r d i n a t e s of t h e
r e p r e s e n t a t i v e m a t r i x a r e r e q u i r e d t o b e r a t i o n a l f u n c t i o n s of
t h e original coordinates.
Whether t h i s r e q u i r e m e n t i s t o b e t a k e n
l o c a l l y ( e . g . a s i n t h e proof of Theorem 7 , Car. 1) o r g l o b a l l y
i s i m m a t e r i a l , i n view of t h e f o l l o w i n g r e s u l t .
Lemma 69:
Let
G
be a C h e v a l l e y group viewed a s a n a l g e b r a i c
->
f :G
group a s above, a n d
k
a function.
Then t h e f o l l o w i n g
conditions a r e equivalent.
I
(a)
f
is expressible a s a r a t i o n a l function l o c a l l y .
(b)
f
is expressible a s a r a t i o n a l function globally.
(c)
f
i s e x p r e s s i b l e a s a polynomial.
Proof:
I
It w i l l b e enough t o show t h a t ( a ) i m p l i e s ( c ) .
b e t h e a l g e b r a of polynomial f u n c t i o n s on
e x i s t s a n open c o v e r i n g
[Ui]
of
G
G
, which
.
such t h a t
f
=
gi/hi
and
hi
ii)
+
, and
0 on
may- b e t a k e n f i n i t e
G
elements
Ui
A
By a s s u m p t i o n t h e r e
by t h e maximal c o n d i t i o n on t h e open s u b s e t s of
by H i l b e r t ' s b a s i s theorem i n
Let
for a l l
(which h o l d s
g i , hi
i
.
in
A
Since t h e
hi d o n ' t a l l v a n i s h t o g e t h e r , by H i l b e r t ts N u l l s t e l l a n s a t z t h e r e
. h i on G
Let
e x i s t elements a i i n 11 s u c h t h a t 1 = Z a 1
U = fi U i ; it i s nonempty , i n f a c t dense., s i n c e G i s i r r e d u c i b l e .
.
0
On Uo
we have
f = Z aifhi
s i t y a l s o on each
Ui
=
a n d on
Z aigi
G
, as
,
a p o l y n o m i a l , hence by den-
required.
P r e s e n t l y we w i l l need t h e f o l l o w i n g r e s u l t .
Lemma 70:
The a l g e b r a
of polynomial f u n c t i o n s on
A
is inte-
G
g r a l l y closed ( i n i t s quotient f i e l d ) .
Proof:
We o b s e r v e f i r s t t h a t
il
is an i n t e g r a l domain ( s i n c e
G
i s i r r e d u c i b l e a s an a l g e b r a i c s e t , t h e polynomial i d e a l d e f i n i n g
it i s p r i m e ) , s o t h a t it r e a l l y has a q u o t i e n t f i e l d .
f = p1/p2
(pi
f o r some
ai E
the
pi
E
and
ordinates
i s i n t e g r a l over
A)
.
ai
f n + alf n-1
A:
On r e s t r i c t i o n t o t h e open s e t
Assume
+
... + an =
U-HU
of
0
G
become, by Theorem 7 ( b ) , polynomials i n t h e co-
.
-1]
(ta,ti,ti
S i n c e such polynomials form a u n i q u e
f a c t o r i z a t i o n domain, we s e e by t h e above e q u a t i o n t h a t
i s such a polynomial, on
U-HU
t h e t r a n s l a t e s of
i s a polynomial on
f
itself
The same b e i n g t r u e on each of
by elements of
, i. e .
G
.
U-HU
f
, we
G
is i n
A
, by
conclude t h a t
f
Lemma 69.
Two more lelnmas and t h e n t h e main theorem.
Lemma 71:
The r a t i o n a l c h a r a c t e r s of
a r e j u s t t h e elements of t h e l a t t i c e
H
L
weights of t h e r e p r e s e n t a t i o n d e f i n i n g
Proof:
Let
be a c h a r a c t e r .
h
d i a g o n a l elements of
i.e.
H
E
, then
t i o n defining
h
G
g e n e r a t e d by t h e g l o b a l
G
.
Then it i s a polynomial i n t h e
a l i n e a r combination of elements of
L
L
.
L
.
(Prove t h i s . )
Being m u l t i p l i c a t i v e ,
Conversely, i f
i s a power product of weights i n t h e r e p r e s e n t a -
, and
a l l exponents may be t a k e n p o s i t i v e s i n c e
t h e product o f t h e l a t t e r weights ( a s f u n c t i o n s ) i s
h
k*)
( w r i t t e n a s a group of d i a g o n a l m a t r i c e s ) ,
it e q u a l s some element of
h
(hqrnomorphisms i n t o
is a polynomial on
H
.
1
, so
that
1
and t h e c o r r e s p o n d i n g w.eight s p a c e s
A
,
Vh
relative t o
,
H
.
in
t h e obvious way.
Let
Lemma
--
V
'72:-
element of
I:
vi
G-module,
a weight,
h
v
a
a root.
Then t h e r e e x i s t v e c t o r s
= 1 , 2 , ...)
so that
x a ( t ) v = v + P tivi
vA
Vh+ia(i
E
,
be a r a t i o n a l
and
an
for all
t E k .
Proof
x,(t
:.
Since
V
->
xu($)
t : x a ( t ) v = C t ivi
i s a n isomorphism,
.
i s a polynomial i n
)
t
is r a t i o n a l and
If we a p p l y
h
t o t h i s e q u a t i o n and compare t h e r e s u l t w i t h t h e e q u a t i o n g o t by
x,(t)
replacing
t
Setting
0
=
by
hxa[t)h-'
we g e t
.,
v
=
v
= xu(a(h)t)
0
, whence
Theorem 39 (Compare w i t h Theorem 3 ) :
over an a l g e b r a i c a l l y closed f i e l d
group o v e r
(a)
element
by a l l
x
Further
A
-
+
E
( i. e . a sernisirnple a l g e b r a i c
G-module
c o n t a i n s a nonzero
V
h
E
L
and i s f i x e d
.
assume
diin V
be a C h e v a l l e y group
G
which b e l o n g s t o some weight
U
Vi Vh+ia,
*
assume t h e n o t a t i o n s a s above.
Every nonzero r a t i o n a l
v
(b)
, and
k)
"
get
t h e lemma.
Let
k
, we
h
V = kGv
=
1
,
+ w i t h v + a s i n ( a ) . Then V
e v e r y weight
C a (a p o s i t i v e r o o t ) , a n d
(c)
In (a)
(d)
If
V
Q,a>
E
V
z+f o r
on
= C
V
P
V
kvC
-+ .
kU v
h a s t h e form
.
every p o s i t i v e r o o t
i s i r r e d u c i b l e , t h e n t h e weight
weight1') and t h e l i n e
=
h
a
.
( t h e "highest
of ( a ) a r e u n i q u e l y d e t e r m i n e d .
210
(e)
h
Given any c h a r a c t e r
on
satisfying ( c )
H
e x i s t s a unique i r r e d u c i b l e r a t i o n a l
V
G-module
, there
i n which
h
is r e a l i z e d a s i n ( a ) .
Proof:
(a)
The p r o o f i s t h e same a s t h a t of Theorem 3 ( a ) w i t h
Lemma 72 i n p l a c e of Lemma 11.
(b)
Since
i s dense i n
U B
r a t i o n a l , a n y l i n e a r f u n c t i o n on
a l s o v a n i s h e s on
f
Gv
.
Thus
V
V
which v a n i s h e s on
ku-Bv
=
(Theorem 7 ) and
G
+
-+ .
kU v
=
V
is
U-BV+
The o t h e r
a s s e r t i o n s of ( b ) folLow from t h i s e q u a t i o n and Lemma 72.
(c)
wh
,
i s a weight on
weight v e c t o r ) .
t h a t <h,a>
E
Since
Z+
(d)
wh,
A
=
V
(with
-a,ax ,
wav
f
a corresponding
it f o l l o w s f r o m ( b )
.
This f o l l o w s from t h e s e c o n d and t h i r d p a r t s of ( b ) .
( e ) We w i l l u s e t h e correspondence between l o c a l weights
(on L )
in (e)
and g l o b a l w e i g h t s (on
.
By Lemma 7 1 ,
, so
i n g weight on
Let
(y,V )
t
weight,
v
h
L
E
.
(see p. 6 0 ) .
G)
Let
h
Z'
E
for a l l
be a n i r r e d u c i b l e L-module w i t h
f
c h o i c e of t h e l a t t i c e
k
M
G
v') .
Since
h
G
, it
1 for a l l
a
0
and
->
9 :. G
t
,
.
t
G
such t h a t
..
a correspond-
is i n the l a t t i c e
k
used t o
f o l l o w s (Theorem 7 , Cor. 1) t h a t t h e r e e x i s t s a
r a t i o n a l homomorphism
or
>
a
and some
g e n e r a t e d by t h e weights of t h e r e p r e s e n t a t i o n of
construct
1
( c o n s t r u c t e d from
in
be a s
as its highest
a c o r r e s p o n d i n g weight v e c t o r , and
i n g C h e v a l l e y g r o u p over
A
a l s o d e n o t e t h e correspond-
h(Ha) = a , a >
that
Let
x c l ( t ) ->
i n the usual notation.
?
x,(t)
The r e s u l t i n g
I
r e p r e s e n t a t i o n of
on
G
need n o t be i r r e d u c i b l e {and i t s
V'
l e a s t it c o n t a i n s t h e v e c t o r
f i x e d by every
v
.
U
E
, and
f
e r a t e d by
I
x
V
1' f
(b)
.
Check t h i s . )
The
+
.
and
V = kGv
.
the
G-homomorphism
+
V,v
Since
+
v2
V
f
vl
generates
C
- V2 A V , which
V1
, and
1
Let
v
is onto.
because
4-
(v' '
+
ku-v
of
+
Vi , v i ( i
f
vl +
+
v2 (: V
1,2)
=
+
v2
E
.
Consider
;V 2 ,
V1
on t h e first fac-
Since a l s o
i s i r r e d u c i b l e and
V2
Thus
, so that
V2
.
11
=
=
i s an isomorphism.
similarly t o
gen-
V'
meets t h e e x i s -
, projection
pl
0
pl
in (e).
V
V1,
is
it f o l l o w s t h a t
isomorphic t o
->
V
'/v 1 1 1
t ?
by ( b ) , s o t h a t
Vh = kv
pl:
V
For t h e u n i q u e n e s s , l e t
+
Then
V = V'
G-module
V
at
and i s
h
be t h e submodule of
a maximal submodule of
t e n c e r e q u i r e m e n t s of ( e )
k e r pl
I
' 1
i s i n f a c t unique a s f o l l o w s from t h e e q u a t i o n
tor.
I
+ which i s o f weight
V
Let
s a t i s f y t h e c o n d i t i o n s on
I
v
, but
M)
r e p r e s e n t a t i o n Class may vary w i t h t h e c h o i c e o f
Y1
V
is
and
V2
a r e isomorphic, a s r e q u i r e d .
Acomplement:
V
I
1
If
char k = 0
is i t s e l f irreducible, i.e.
,
t h e n i n t h e e x i s t e n c e proof above
1
V = V
o t h e r words, if t h e Chevalley group
irreducible
g-module. V
G
and a f i e l d
1t
and
V
1 1 1
. . .
In
= O .
i s c o n s t r u c t e d from a n
k
of c h a r a c t e r i s t i c
0
,
t h e n a s a l i n e a r group it i s i r r e d u c i b l e .
Proof:
I
Recall t h a t
V
by a s s u m p t i o n ) , and t h a t a l a t t i c e
M
t h e n used t o s h i f t t h e c o e f f i c i e n t s t o
ducible r e l a t i v e t o
f a -module
was o r i g i n a l l y a n
.
(irreducible
a s i n Theorem 2 , Cor. 1 was
k
.
It f o l l o w s t h a t
Clearly
Vk
VQ
is irre-
is irreducible
I
i
relative t o
Lk: o t h e r w i s e
subspace
, excluding
V1
nonzero
v
E
writing
v
= C
over
9)
V1
+
kv
such t h a t
t im i (mi
E
and choosing
t h e r e would be a p r o p e r i n v a r i a n t
a
since'
Xuv
M, t i
=
0 for a l l
E
k
so that
1
Xa
r e c o v e r each
from
t
s e v e r a l v a l u e s of
that
Vk
G
by u s i n g
X,'s
and t h e
is i r r e d u c i b l e f o r
G
CL
>
=
0
.
I
and
V'
"
$ 0
that
S i n c e we can
,
generate
...
for
we conclude
.
$ 0
char k
o r when
i s l a r g e i1comparedh9
to
char k
, then
i n general and t h e exact s i t u a t i o n is not
a t a l l u n d e r s t o o d , e x c e p t i n a few s c a t t e r e d c a s e s ( t y p e s
B2
t
$ V"
, so
x,(t ) = 1 + t X a +
I n contrast t o t h e case j u s t considered, if
V'
0
t h e n some
and l i n e a r l y independent
Xami $ 0 f o r some i , we
Z tiX,mi
would a r r i v e a t t h e c o n t r a d i c t i o n
I.
+ n Vg f 0 ,
kv
h)
.
A1,
A2,
However, t h e
following is t r u e .
J
E x e r c i s e :-
(a)
For t h e l a t t i c e
M
V
of Theorem 2 , Cor. 1 ( w i t h
c v + n M
t h e r e assumed t o b e i r r e d u c i b l e ) assume t h a t
i s pre-
I
scribed.
I
Prove t h a t t h e r e i s a u n i q u e minimal c h o i c e f o r
( c o n t a i n e d i n a l l o t h e r s ) a n d a u n i q u e maximal c h o i c e ,
Assume ilow a s i n t h e complement, e x c e p t t h a t
(b)
I-
M
If
H i s maximal, t h e n
V"
If
M
V
=
0
char k
,
i.e.
$ 0
V
.
11
irreducible.
(c)
Example :
If
i s minimal, t h e n
i s of t y p e
:Il
,
char k
r e p r e s e n t a t i o n is used, then ( b ) holds f o r
and ( c ) holds f o r
DLin
=
a,H ,
Y>
, but
t
=
= V
? 1.
.
, ai-id t h e a d j o i n t
M
H/2, Y>
lila x = a,
2
not v i c e versa.
is
21 3
The proof g i v e n above f o r t h e e x i s t e n c e i n Theorem 3 9 ( e )
b r i n g s o u t t h e c o n n e c t i o n between t h e r e p r e s e n t a t i o n s of
%
t h o s e of
G
and
and shows t h a t e v e r y i r r e d u c i b l e r a t i o n a l r e p r e s e n -
t a t i o n of a C h e v a l l e y group i n c h a r a c t e r i s t i c
s t r u c t e d by t h e r e d u c t i o n
mod p
of a group i n c h a r a c t e r i s t i c
0
p f 0
can b e con-
of a c o r r e s p o n d i n g r e p r e s e n t a t i o n
.
It depends, however, on t h e
% ,
e x i s t e n c e of r e p r e s e n t a t i o n s of
which we have n o t proved
h e r e , thus i n i t s e n t i r e t y is very long.
We s h a l l now develop an
a l t e r n a t e , more i n t r i n s i c , p r o o f .
G-=module V
We s t a r t w i t h t h e c o n n e c t i o n between a
all
,
V"
dual
x
on which
G, f
E
, and
V"
E
(xf)( v )
a c t s by t h e r u l e
G
v
V
E
.
and i t s
= f (x-lv)
wo
We r e c a l l t h a t
for
is t h e ele-
ment of t h e Weyl g r o u p which makes a l l p o s i t i v e . r o o t s n e g a t i v e .
Lemma 73:
V
Let
highest weight,
and
ff
and
v
be a n i r r e d u c i b l e r a t i o n a l
v
+
V
as
v
a c o r r e s p o n d i n g weight v e c t o r ,
=
+
then
f (v) = c
v":
-
cv
defined thus:
Then
h'
and
v.'.
h i g h e s t weight v e c t o r f o r
I n t h e d e f i n i t i o n of
= dim V h = 1
dim Vw
.
A". = -woh
i f we w r i t e
-
v
f + a r e h i g h e s t weight and
.
f + we have u s e d t h e f a c t t h a t
Here, and a l s o i n s i m i l a r s i t u a t i o n s
0
l a t e r , we e x t e n d
b = uh ( u
v
E
V
E
U, h
to
h
E
H)
,
=
,
w 0v
+ terms of o t h e r (hence h i g h e r ) w e i g h t s ,
.
I
,
Proof:
its
h
.
I
t h e element of
E
G-module,
B
by t h e r u l e
h(h)
.I,
and s i m i l a r l y f o r
h"'
=
.
h(h)
if
If we w r i t e
a s i n t h e lemnia and u s e Lemma 7 2 , we s e e t h a t
b v = c (woh) ( h ) v r - + h i g h e r t e r m s .
Since
c = ff ( v )
and
+
,
(w0h) ( h ) = hi"(b-l)
b
by
b-l
we g e t
-
Theorem 40:
tions
b
E
a
B
For
on
G
, made
b f + = A* ( b ) f f
A
If
let
such t h a t
into a
(a)
L
E
.
f + ( b v ) = ~ ' " (-b1 ) f + ( v )
we have
, as
required.
.iA be t h e s p a c e of polynomial func-
a(yb)
=
a(y)h(b) for a l l
V q ->
x E G
A
is a
(b)
A
y
a r e a s i n Lemma 71, t h e n t h e map
V , A , vf
d e f i n e d by
( ~) (fx ) = f ( x v + )
for
f
E
v3'
Conversely, i f
is such t h a t
A
G-submodule.
,iA
$
, then
0
.
I
.
exercise
.
(b)
A
h
The l a t t e r is f i n i t e -
dimensional and r a t i o n a l and i t s h i g h e s t wieght i s
(a)
and
G-isomorphism i n t o .
contains a unique i r r e d u c i b l e
Proof:
G,
E
G-module i n t h e obvious way.
J,
'P ::
On r e p l a c i n g
hq.
.
The p o i n t s t o be - c h e c k e d h e r e w i l l b e l e f t a s an
We o b s e r v e f i r s t t h a t a s a
f inite-dimensional
G-module
is l o c a l l y
Ah
( i n f a c t , it i s f i n i t e - d i m e n s i o n a l
,
b u t we s h a l l
n o t prove t h i s ) , s i n c e t h e s e t of polynomials of a g i v e n degree
is.
Thus t h e r e e x i s t i r r e d u c i b l e submodules and a l l of them a r e
f i n i t e - d i m e n s i o n a l and r a t i o n a l .
any one of them and
have
(")
a+(bxbl) =
a
Let
p
b e t h e h i g h e s t weight of
+
a c o r r e s p o n d i n g nonzero weight v e c t o r . . We
-1 +
I
1.
p(b ) a (x)A(b ) f o r a l l x E G , b , b E B
.
+
+
Bw B i s dense i n G , a ( w o )
0 . Since a l s o
0
+
-1
= a (wo.wO bwo) , we g e t from t h e above e q u a t i o n t h a t
.
Since
=
by
~ ( w z ' b w ~ ), s o t h a t
A
, the
function
+
a (bwo)
p(b-l)
. 0 .I
',
p = A
a
+
Since
/I
i s u n i q u e l y determined
is determined by i t s v a l u e a t
wo
by
21 5
x = w
(+) with
BwoB
and t h e d e n s i t y of
0
in
G
, proving
the
uniqueness i n ( b ) .
Remarks:
(a)
0
In characteristic
it e a s i l y f o l l o w s f r o m t h e
Ah
theorem of complete r e d u c i b i l i t y t h a t
(b)
Therepresentationof
i t s e l f is irreducible.
B
on
G
is,, i n t h e context
h
of polynomial r e p r e s e n t a t i o n s . , t h e one i n d u c e d by t h e c h a r a c t e r
h
.
B
on
The f a c t t h a t it c o n t a i n s a r e p r e s e n t a t i o n o f h i g h e s t
,
.I.
weight
i s , i n view o f Theorem 3 9 ( a ) , a form of F r o b e n i u s
reciprocity.
Lemma 74:
Let
b e a s i n Lemma 73 and
f+
i n Theorem 4 0 ( a ) s o t h a t
b e t h e s t a b i l i z e r of
xv
in
A
+
+
= v
+
.
w e have
x
Then i f
a+(x)
=
~"(h-')
=
0
A = 0
,
then
wv
then
w
+
w
if
E
W
E
+
= v
w
has weight
=
hw wv
o
w
,
that
+
+
A
E
1
0
.'
G
E
a s above i s always p o s s i b l e : :
wh
=
h
,
G =
BG ,
while i f
hence i s a i n u l t i p l e o f
H
.
If
Wh
E
a+(x)
.
Wh
+
,
0
then
,
then
a + ( x ) = h"(hml)
w
fixes
kv+
if
+
h
v
+
0
,
,
so
we can a c h i e v e
From t h e d e f i n i t i o n s and Lemma 72 we have
while i f
w
assume t h a t t h e
.
w
WA
Let
( s e e Theorem 4 )
uhw wul
t h a t by modifying it by a s u i t a b l e element of
wv
Wh
as
.P
has been chosen s o t h a t
G
is written
G
~ f +w i t h
=
+ higher terms.
w
and f o r
E
is t r i v i a l since
V
+
+
otherwise.
A choice f o r
Proof::
E
,
W
corresponding r e p r e s e n t a t i v e
wv
+
a (x)wov
=
a
+
+
a (x)wov
by t h e c h o i c e of
by t h e e q u a t i o n s o
21 6
This b r i n g s us t o t h e
- - ----.-.
of t h e--e
Second proof
Proof:
Let
m
o
r
- a > . - - -
I t w i l l b e enough
be a s i n Theorem 3 9 ( c ) ,
h
n 3 9- (-e l :
e
t.0
prove
t h a t t h e f u n c t i o n d e f i n e d by t h e l a s t e q u a t i o n s of Lemma 74 i s
r a t i o n a l on
G
.
The e x i s t e n c e w i l l t h e n f o l l o w from Theorem 4 0 ( b )
\I.
with
i n p l a c e of
A"'
h
.
By Lemma 70 any power of t h i s f u n c t i o n
w i l l do, s o t h a t by Lemma 7L+ it w i l l b e enough t o c o n s t r u c t a n
i r r e d u c i b l e r e p r e s e n t a t i o n whose h i g h e s t weight i s some p o s i t i v e
power ( p o s i t i v e m u l t i p l e if we w r i t e c h a r a c t e r s on
of
.
h
additirely)
T h i s we w i l l do, u s i n g t h e f o l l o w i n g i n t e r e s t i n g r e s u l t .
Lemma 75 ( C h e v a l l e y ) :
Let
a line
L
Proof:.
Let
and
t h e ideal defining
in
whose s t a b i l i z e r i n
V
ii
be a l i n e a r a l g e b r a i c group and
G
Then t h e r e e x i s t s a r a t i o n a l
a c l o s e d subgroup.
I
H
is
G
G-module
P
and
V
.
P
be t h e a l g e b r a of polynomials i n t h e m a t r i c e n t r i e s
.
P
By H i l b e r t f s b a s i s theorem
I
is g e n e r a t e d by a f i n i t e number of i t s e l e m e n t s , s o t h a t t h e r e
e x i s t s a finite-dimensional
G-invariant
generates
I
.
following equivalent conditions:
x
E
that
f
E
B fi I
I, y
E
, say
C
,
P; x I C I ; xCCC
-
is t h e product i n
-
V
.
L
For
C
We resume t h e proof of e x i s t e n c e .
p a r a b o l i c subgroup of
G
For
such
we have t h e
G
E
of
--
is exactly
b e t h e s e t of s i r ~ p l er o o t s .
x
B
P ; f ( x l y )= 0 f o r a l l
If now
of a b a s i s f o r
t h a t t h e s t a b i l i z e r of
subspace
i = 2
c
=
,
and
.
P
Let
,
corresponding t o
R B ,v
dimC, V =
L
=
-
II
. .,
=
kv
it f o l l o w s
.
{u1 , a 2 , . . ,aG)
let
-
,
[ai)
Pi
be t h e
( s e e Lemma
Li = kv
0
i
t h e c o r r e s p o n d i n g l i n e of L e m a 75,
c o r r e s p o n d i n g r a t i o n a l c h a r a c t e r on
H
, and
Vi
=
.
kGvi
, then
j f i
If
, hence
Pi
w
.
pi
the
B
a l s o on
and
is represented i n
Pi
does n o t f i x
j
S i n c e wi
J
by c h o i c e , it f o l l o w s from p a r t s ( b ) and ( c ) of Theorem
so that
p
i
=
pi
and
<,u.,a.>= 0
1
Li ,
applied t o
If now
Vi
that
<pi,ai>
is a p o s i t i v e i n t e g e r , s a y
is a s b e f o r e s o t h a t
A
Q,ui>
= c
dh = Z e . p
with d = K d i and
1 i
e
e
t h e t e n s o r product
, then n v i i
that
ei
uvii
dh
, so
B
for
i
=
E
, it
Z'
cid/'di
.
39
di
.
follows
If we form
i s a v e c t o r of weight
t h a t we may e x t r a c t a n i r r e d u c i b l e component whose
h i g h e s t weight is
dh
, and
t h u s complete our s e c o n d e x i s t e n c e
proof.
We a r e i n d e b t e d t o G . D. Mostow f o r t h e proof j u s t g i v e n .
Remark:
The e x t r a problems t h a t a r i s e when
char k
0 a r e compen-
s a t e d f o r by t h e f a c t t h a t o n l y a f i n i t e number of r e p r e s e n t a t i o n s
h a s t o c o n s i d e r e d i n t h i s c a s e , a s we s h a l l now s e e .
-
Lemma 76:
Assume
char k = p f 0
.
Let Fr ( f o r Frobenius ) denote
t h e o p e r a t i o n o f r e p l a c i n g t h e m a t r i c e n t r i e s of t h e e l e m e n t s of
th
G by t h e i r p
i s an i r r e d u c i b l e r a t i o n a l reppowers.
If
then s o is
r e s e n t a t i o n of G
o Fr
If t h e h i g h e s t weight of
P
,
9
is
Proof:
,
t h a t of
.
9
f
Fr
is
-
Exercise.
Theorem 4.1:
Assume t h a t
G
above i s u n i v e r s a l ( i . e .
s i m p l y c o n n e c t e d a l g e b r a i c g r o u p ) , and t h a t
char k = p
G
is a
$ 0
.
Let
,
2
b e t h e s e t of
p
G
i r r e d u c i b l e r a t i o n a l r e p r e s e n t a t i o n s of
f o r which t h e h i g h e s t w e i g h t
(ai
.
simple)
satisfies
h
can b e w r i t t e n u n i q u e l y
@ fj o F r J
J=O
pj
2
E
S k e t c h of p r o o-f -: , We o b s e r v e f i r s t t h a t s i n c e
L
=
L1
,
so that a l l
with a l l
A's
<A
=
Ofj
o FrJ
Let
A
)
is universal
G
4-
, L ~ >E
occur a s h i g h e s t
.
weights, i n p a r t i c u l a r t h o s e used t o define
JI
Consider
be t h e h i g h e s t weight of
j
h i g h e s t w e i g h t v e c t o r of w e i g h t
pjts
n,
in
A =
X
e x a c t l y once.
A
9 "
Lemma 76.
If we
we o b t a i n , i n view o f t h e u n i q u e n e s s of
t h e e x p a n s i o n of a number i n t h e s c a l e of
h i g h e s t weight
, by
pj h j
The
Sj
p r o d u c t of t h e c o r r e s p o n d i n g w e i g h t v e c t o r s y i e l d s f o r
vary t h e
<p -1
Then e v e r y i r r e d u c i b l e r a t i o n a l r e p r e s e n t a t i o n o f
m
G
<
- Q,ai>
0
G
n e e d o n l y show t h a t e a c h
p
,
each p o s s i b l e
Thus t o p r o v e t h e t h e o r e m we
above is i r r e d u c i b l e .
P
The p r o o f of
t h i s f a c t depends e v e n t u a l l y on t h e l i n e a r i n d e p e n d e n c e of t h e
d i s t i n c t automorphislns
F
(j = O,,..
.)
of
k
.
We omit t h e
d e t a i l s , r e f e r r i n g t h e r e a d e r t o R . S t e i n b e r g , Nagoya Piath. J
. 22
(1963) , o r t o P. C a r t i e r , Sem. B o u r b a k i 255 ( 1 9 6 3 ) .
/
Corollary;
assume t h a t one of t h e s p e c i a l s i t u a t i o n s of Theorem
28 h o l d s .
Let
by
=
<A,ui>
0
(resp.
fL
for a l l
Then e v e r y e l e m e n t of
Proof:
Given f
E
i
f
/?
-)
2
b e t h e s u b s e t s of
such t h a t
a
defined
is long (resp. s h o r t ) .
can b e w r i t t e n u n i q u e l y
k? , w r i t e
/C
96 OfS
with
t h e corresponding h i g h e s t weight
A
Rs
and
and ps a r e i n
is irreducible.
I f we d e f i n e
I
and
G"
cp
We have t o show t h a t
.*,
pi
= cp o p4 ,
i n t erclianged, a n d s e t
.
I
.
t o show t h a t t h e r e p r e s e n t a t i o n
.
.
. .
.
26 b u t w i t h
a s i n Theorem
ap;
,
I
.
=
w
0
yS ,
p-t BPS
G
we haqe
-8.
of
G?.
is irreducible.
S i n c e t h e corresponding h i g h e s t weights s a t i s f y
.b
<A;,av>
. .
if
= QL?a>
0
=
<h:,a">
u,
i f not
p a s ,a> i f
=
0
=
a.
is long
if not,
.
I
we s e e t h a t
is short
/
and
hi
2 "' , s o
Gel- .
c o r r e s p o n d t o e l e m e n t s of
.
I
t h a t t h e c o r o l l a r y f o l l o w s f r o m Theorem 4 1 a p p l i e d t o
Examples :
Pi
(a)
(i= 0 1 ,
SL2
. ,
=
.
Here t h e r e a r e
1) i n
R,the
p
it h
s p a c e of p o l y n o m i a l s homogeneous of d e g r e e
I
(b)
.
o
If
V
V
=
Sp4, p = 2
.
0 9
b e i n g r e a l i z e d on t h e
i
over
k2
.
Here t h e r e a r e 4 r e p r e s e n t a t i o n s i n
i s t h e g r a p h autornorphisrn of Theorem 2 8 t h e n
f
s o t h a t by t h e above c o r o l l a r y , t h e s e 4 a r e , i n
t e r m s of t h e d e f i n i n g r e p r e s e n t a t i o n
J"P
representations
p ,
just 1 ( t r i v i a l ) ,
, a n d p B ( ~ o .90) .
The r e s u l t s we have o b t a i n e d can e a s i l y be e x t e n d e d t o t h e
I
I
case t h a t
k
i s i n f i n i t e ( b u t perhaps n o t a l g e b r a i c a l l y c l o s e d ) .
We c o n s i d e r r e p r e s e n t a t i o n s on v e c t o r s p a c e s over
b r a i c a l l y closed f i e l d containing
k
,
K
, some
alge-
a n d c a l l them r a t i o n a l i f
I,
t h e c o o r d i n a t e s of t h e s o u r c e .
The p r e c e d i n g t h e o r y i s t h e n a p p l i -
c a b l e a l m o s t word f o r word b e c a u s e of t h e f o l l o w i n g two f a c t s b o t h
coming f r o m t h e d e n s e n e s s of
(a)
k
GK
G
ex-
G~
On r e s t r i c t i o n t o
s e n t a t i o n of
Exercise:
with
G
Every i r r e d u c i b l e polynomial r e p r e s e n t a k i o n of
t e n d s u n i q u e l y t o one of
(b)
( t h i s is
GK
.
K)
extended t o
in
G
every i r r e d u c i b l e r a t i o n a l repre-
G
remains i r r e d u c i b l e .
Prove ( a ) and ( b ) .
The s t r u c t u r e of a r b i t r a r y i r r e d u c i b l e r e p r e s e n t a t i o n s i s
g i v e n i n t e r m s of t h e p o l y n o m i a l ones by t h e f o l l o w i n g g e n e r a l
theorem.
write
cp
Given a n isomorphism
Y
of
k
f o r t h e n a t u r a l isomorphism of
o b t a i n e d from
G
, we
shall also
onto t h e group
VG
.
by r e p l a c i n g
k
by
Theorem 42 ( B o r e 1 , T i t s ) : L e t
G
b e an indecomposable u n i v e r s a l
G
Yk
K
into
C h e v a l l e y g r o u p over a n i n f i n i t e f i e l d
k
, and
let
D-
be an
LI
a r b i t r a r y (not necessarily r a t i o n a l ) irreducib1.e r e p r e s e n t a t i o n
6
cally closed f i e l d
f
g
of
on a f i n i t e - d i m e n s i o n a l
G
K
.
t h e r e e x i s t finitely-many
vector space
Assume t h a t
c
isomorphisms
V
over a n a l g e b r a i -
is n o n t r i v i a l .
iPi
of
k
corresponding i r r e d u c i b l e r a t i o n a l r e p r e s e n t a t i o n s
over
K
such t h a t
=
@pi o Y i .
1
,
into
Ji
Then
K
of
and
YiG
imbeddable a s a s u b f i e l d of
K
.
a r e s u c h t h a t no s u c h imbedding e x i s t s , e . g . i f
t h e n e v e r y i r r e d u c i b l e r e p r e s e n t a t i o n of
vector space over
K
k
I n o t h e r words, i f
char k
K
and
+
on a f i n i t e - d i m e n s i o n a l
G
is n e c e s s a r i l y t r i v i a l .
(Deduce t h a t t h e
If
same i s t r u e even if t h e r e p r e s e n t a t i o n i s n o t i r r e d u c i b l e . )
k
,
char K
i s f i n i t e , t h e s e statements a r e , of course, f a l s e .
(b)
The t h e o r e m c a n b e c o m p l e t e d by s t a t e m e n t s c o n c e r n i n g
t h e u n i q u e n e s s of t h e d e c o m p o s i t i o n a n d t h e c o n d i t i o n f o r i r r e d u c i b i l i t y if t h e f a c t o r s a r e p r e s c r i b e d .
Since these statements a r e
a b i t c o m p l i c a t e d we s h a l l omit them.
(c)
The t h e o r e m was c o n j e c t u r e d by u s i n Nagoya Math.
(1963). The p r o o f t o f o l l o w i s b a s e d on a n a s y e t unpub-
J . 22
l i s h e d p a p e r by A. Bore1 a n d J . T i t s i n which r e s u l t s of a more
general character a r e considered.
Lemma 77:
fields
and
Let
k, kt
r: G
with
>
(a)
G, G
1
b e indecomposable C h e v a l l e p g r o u p s o v e r
k
i n f i n i t e and
T h e r e e x i s t s a n isomorphism
9
If
G
of
G
(a)
into
is universal, then
t o t h e u n i v e r s a l c o v e r i n g g r o u p of
Proof:
1
algebraically closed,
a homomorphism s u c h t h a t
G
a r a t i o n a l hoinomor~~hism
(b)
k
G
1
of
3
G'
9
i s dense i n
o-G
1
into
k
such t h a t
k
G
= P o " -
can b e l i f t e d , u n i q u e l y ,
.
h e w i l l o b s e r v e t h a t what i s shown t h e r e i s t h a t
o-x,(t)
=
xar ( ~ ~ $ q( (t a)) )
.
and
If t h e r e a d e r w i l l examine t h e p r o o f of Theorem 31
malized s o t h a t
1
with
can b e n o r -
oa
->
T
a
an
angle-preserving
k
onto
that
k
t
of
limp
, and
1
q(a) = 1 on
is dense i n
o-G
on
1
G
t
,
1
,
=
.
p
+1,V
a n isomorphism of
S i n c e we a r e assuming o n l y
not t h a t
0-G
= G
, the
1
p r o o f of t h e
c o r r e s p o n d i n g r e s u l t ' i n t h e p r e s e n t c a s e i s somewhat h a r d e r .
e v e r , t h e main i d e a s a r e q u i t e s i m i l a r .
We omit t h e p r o o f .
H
t h e a b o v e e q u a t i o n s a n d t h e c o r r e s p o n d i n g ones on
f r o m Theorem 7 t h a t
(b)
G
t
Proof of Theorem 42:
GL(V)
of Theorem 8 , t h a t
, uniquely
G
since
- , the
Let
A = O-ti
o-G
containing
.
Theorem 30, s t e p ( 1 2 ) ,
l
I
I
I
R
theorem
.
BG
We c l a i m
is a connected
:i
A s i n t h e p r o o f of
__IC
b e i n g g e n e r a t e d by t h e s e g r o u p s , i s a l s o .
h a s w e i g h t s on
o-U-
is connected, and s i m i l a r l y f o r
a c o n n e c t e d s o l v a b l e normal s u b g r o u p o f
, finite
V
.
Let
R
be
By t h e Lie-Kolchin
a
i n nu.nzber.
permutes t h e
c o r r e s p o n d i n g w e i g h t s p a c e s , a n d , b e i n g c o n i l e c t e d , f i x e s them a l l .
Since
since
R = 1
i s i r r e d u c i b l e , t h e r e i s o n l y one s u c h s p a c e a n d it i s
V
a l l of
1
I
I
,
o-U
=
can b e l i f t e d
S
s m a l l e s t a l g e b r a i c sub-
s e m i s i m p l e g r o u p , hence a C h e v a l l e y g r o u p .
A
it f o l l o w s
h a s t h e f o r m of ( a ) .
cr
) ( 3 ), (C)
(
t o a n y c o v e r i n g of
so that
From
From t h e s e e q u a t i o n s we s e e a l s o , e . g . by c o n s i d e r i n g
the relations
g r o u p of
,
How-
,
V
A
=
, so
so that
21
that
,
R
so that
A
c o n s i s t s of s c a l a r s , of d e t e r m i n a n t
R
is f i n i t e .
Since
is semisimple, a s claimed.
b e t h e u n i v e r s a l c o v e r i n g g r o u p of
i t s i n d e c o m p o s a b l e components,
iio
i s connected,
R
Let
A1
=
n~~~
w r i t t e n a s a p r o d u c t of
d
=
1
Tiiio t h e
f a c t o r i z a t i o n of t h e a d j o i n t group, and
a, B, $'
corresponding
=
rxi
the
,
c o r r e s p o n d i n g n a t u r a l maps a s shown:
G
0
86
By Lemma 77 we can l i f t
6: G
->
k
phism o f
.
Ail
of t h e form
A1
We have
phism o f
G
K
illto
a6
=
6(x) = roicpi(x)
and
cr
conpsnentwise t o g e t a homomorphism
with each
a r a t i o n a l hcmomor~hismof
E~
a
i n t o t h e c e n t e r of
of
Ail
.
,
=
@ai
with
i
On s e t t i n g
cpiG
into
s i n c e o t h e r w i s e we would have a homomor-
.
By Lemma
6G, Cor. ( a ) , u
i n t e r p r e t e d a s an i r r e d u c i b l e r a t i o n a l r e p r e s e n t a t i o n of
be factored
a n isomor-
cpi
ai
A1
,
, may
an i r r e d u c i b l e r a t i o n a l r e p r e s e n t a t i o n
a.E
Si =
1
i
, we
see that
o-
=
u6
=
Om1
.~ 1
.cp
i
as required.
i
CorollarE
(a)
Every a b s o l u t e l y i r r e d u c i b l e r e a l r e p r e s e n t a t i o n
o f a r e a l C h e v a l l e y group
1
(b)
G
is r a t i o n a l .
Every holomorphic i r r e d u c i b l e r e p r e s e n t a t i o n of a
s emisimple coraplex L i e group i s r a t i o n a l .
(c)
Zvery c o n t i n u o u s i r r e d u c i b l e r e p r e s e n t a t i o n o f a
s i m p l y c o n n e c t e d s e m i s i m p l e complex L i e group i s t l i e t e n s o r p r o d u c t
of a holomorphic one and a n a n t i h o l o m o r p h i c one.
t r a n s i t i o n t o t h e nonuniversal case i s an easy e x e r c i s e .
(b)
The p r o o f i s s i m i l a r t o t h a t of ( a ) .
(c)
The o n l y c o n t i n u o u s isomorphisms of
into
are
t h e i d e n t i t y a n d complex c o n j u g a t i o n .
Prove t h a t t h e word ~ t a b s o l u t e l y lii n ( a ) a n d t h e words
Exercise.:
i T s i m p l y c o n n e c t e d B 1i n ( c ) may n o t b e removed.
Now we s h a l l t o u c h b r i e f l y on some a d d i t i o n a l r e s u l t s .
I
Characters.
-
A s i s customary i n r e p r e s e n t a t i o n t h e o r y , t h e c h a r a c -
t e r s ( i. e . t h e t r a c e s of t h e r e p r e s e n t a t i v e m a t r i c e s ) p l a y a v i t a l
role.
We s t a t e t h e p r i n c i p a l r e s u l t s i n t h e form of a n e x e r c i s e .
Exercise:
(a)
P r o v e t h a t two i r r e d u c i b l e r a t i o n a l
G-modules
i s o m o r p h i c i f a n d o n l y i f t h e i r c h a r a c t e r s a r e equ.al.
I
. t h e c h a r a c t e r s on
(b)
1
I
H
are
(Consider
.)
lissume t h a t
char k
= 0
a n d t h a t t h e t h e o r e m of
c o m p l e t e r e d u c i b i l i t y h a s been p r o v e d i n t h i s c a s e .
Prove ( a ) f o r
r e p r e s e n t a t i o n s which n e e d n o t b e i r r e d u c i b l e .
(c)
:issum
char k
=
b e a s i n Theorem 3 9 ( e ) , X
one-half
S
=
A
0
.
t h e corresponding c h a r a c t e r , and
t h e sum o f t h e p o s i t i v e r o o t s , a c h a r a c t e r on
C det w.w(h + 6)
w c'iJ
,
a sum o f f u n c t i o n s on
(1) ~ ( h =) S h ( h ) / ' s 3 ( h ) a t a l l
(2)
dim V =
na
u>o
(Hint.
P r o v e WeylTs formnu-las : L e t
+ 6,(r>/<d,a>
h
E
H
where
H
.
So(h)
H
.
V, A
6
Set
Then
+0.
.
u s e t h e c o r r e s p o n d i n g f o r m u l a s f o r L i e a l g e b r a s ( s e e ,- e .g..
J a c o b s o n , L i e A l g e b r a s ) a n d t h e complement t o Theorem 3 9 ) .
,.
225
Remark;-
The forinula (1) d e t e r m i n e s
o u t t h a t t h e e l e m e n t s of
H
of
f o r which
So
The u n i t a r i a n t -r i c k .
.
G
X
u n i q u e l y s i n c e it t u r n s
which a r e c o n j u g a t e t o t h o s e e l e m e n t s
$ 0 form a d e n s e open s e t i n
.
G
The b a s i c r e s u l t s a b o u t t h e i r r e d u c i b l e com-
.
p l e x r e p r e s e n t a t i o n s of a compact s e m i s i m p l e L i e g r o u p
a maximal compact s u b g r o u p of a complex C h e v a l l e y g r o u p
8 8 , can be deduced f r o m t h o s e of
43 ( b ) and
K
(
portant f a c t :
45
(K
proof i s a n easy e x e r c i s e .
K
to
.
in
G
is g e n e r a t e d by t h e g r o u p s
SU2
is Zariski-dense
By
( )
G
as i n
b e c a u s e of t h e f o l l o w i n g i m -
is Zariski-dense
down t o t h e f a c t t h a t
r e s e n t a t i o n s of
G
, i.e.
K
.
Because of Lemmas
rp2SU2)
in
t h i s comes
S L ~ ( ,~ whose
)
t h e r a t i o n a l i r r e d u c i b l e rep-
r e m a i n d i s t i n c t a n d i r r e d u c i b l e on r e s t r i c t i o n
G
T h a t a complete s e t of c o n t i n u o u s r e p r e s e n t a t i o n s of
K
i s s o o b t a i n e d t h e n f o l l o w s from t h e f a c t t h a t t h e c o r r e s p o n d i n g
c h a r a c t e r s f o r m a complete s e t of c o n t i n u o u s c l a s s f u n c t i o n s on
K
.
The p r o o f of t h i s u s e s t h e f o r m u l a f o r Haar measure on
K
a n d t h e o r t h o g o n a l i t y and completeness p r o p e r t i e s of complex exp o n e n t i a l ~ ,and y i e l d s a s a by-product Weyl f s c h a r a c t e r formula
itself.
T h i s i s how - \ i e y l proved h i s f o r m u l a i n 14ath. Z e i t . 24
( 1 9 2 6 ) and i t i s s t i l l t h e b e s t way.
The theorem of complete r e -
d u c i b i l i t y can b e p r o v e d a s f o l l o w s .
Given any r a t i o n a l r e p r e s e n -
t a t i o n space
V
by a v e r a g i n g o v e r
of
V
onto
V
f
for
K
G
and a n i n v a r i a n t s u b s p a c e
, relative
V
I
, we
can,
t o Haar measure, any p r o j e c t i o n
and t a k i n g t h e k e r n e l of t h e r e s u l t , g e t a corn-
plementary subspace i n v a r i a n t under
K
, thus
a l s o i n v a r i a n t under
G
It is t h e n n o t d i f f i c u l t t o r e p l a c e t h e complex
because of ( + ) .
f i e l d by any f i e l d of c h a r a c t e r i s t i c
-.
Invariant b i l i n e a m m s
Chevalley g r o u p ,
G
0
.
denotes an indecoinposable i n f i n i t e
an i r r e d u c i b l e r a t i o n a l
V
G-module, and
A
i t s h i g h e s t weight.
Lemma 7 8 :
The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t .
(a)
There e x i s t s on
(b)
V
and i t s d u a l
V
a ( n o n z e r o ) i n v a r i a n t b i l i n e a r form.
$ a r e isomorphic.
Proof : E x e r c i s e ( s e e Lemma 7 3 ) .
Exercise:
except
Prove t h a t
-w0 i s t h e i d e n t i t y f o r a l l s i m p l e t y p e s
An(n 2 2 ) , D2n+l,
and f o r t h e s e t y p e s it comes from
E6,
i n v o l u t o r y automorphism of t h e Dynkin diagram.
( H i n t : f o r a l l of
DZn t h e Dynkin diagram h a s no sym-
t h e u n l i s t e d c a s e s except f o r
metry. )
E x e r c i-se:
If t h e r e e x i s t s a n i n v a r i a n t b i l i n e a r form on
V
, then
it i s unique up t o m u l t i p l i c a t i o n by a s c a l a r and i s e i t h e r symm e t r i c o r skew-symmetric.
--
Lemma 79 : Let
(a)
h
(b)
If
symmetric if
h =
(Hint:
I (ha(-1 J , t h e
i s i n t h e c e n t e r of
V
product over t h e p o s i t i v e r o o t s
G
and
hz
=
1
.
p o s s e s s e s a n i n v a r i a n t b i l i n e a r form t h e n it is
h(h) = 1
skew-symmetric i f
u s e S c h u r r s Lemma.)
,
~ ( h =) -1
.
.
( a ) Since
Proof:
by a l l elements of
ha(-1) = h -U (-1) (check t h i s ) ,
.
W
This i m p l i e s t h a t
a s e a s i l y f o l l o w s from Theorem 4
we have
h2 = 1
(b)
with
on
v+
1
.
Since
fC
.
t
1
V
+
+
0
(wov )
h
is i n t h e c e n t e r ,
2
ha(-1)
= h,(l)
= 1 ,
4.
VeP, v
->
+
->
f+
a s i n Lemma 73; t h e corresponding b i l i n e a r form
i s g i v e n by ( v , v ) = (cpv)(v )
-1 +
( X V + , ~ V + ) = f C ( x yv )
f o r a l l x, y
= f
is f i x e d
.
We have an isomorphism 9 : V
and
h
by t h e d e f i n i t i o n of
It f o l l o w s t h a t
G
E
f+
.
+ ,wov + )
4+
-1 +
(wov , v ) = f ( w o v )
T~US
, and
(V
.
is a minimal product of s i m p l e r e f l e c t i o n s i n
a @ ?I***
W , t h e n f o r d e f i n i t e n e s s we p i c k wo = w a ( l ) w B ( l )
in G , so
If
W o = W W W
...
that
wil
.
=
,
and s i m i l a r l y f o r
w0
... .
and b r i n g i n g a l l t h e
t i o n by
wfs
the l e f t
.
, we
h's
get t o the right
+
h
h
,
by Appendix 11 ( 2 5 ) and t o
which i s j u s t
Thus
+
wil
4-
A(h)f (wov ) = ~ ( h ) ( v,wov )
Observation:
= wU(l)ha(l)
t o t h e r i g h t , by r e p e a t e d con juga-
w(l)w(l)w(l)
3.
wU(-1)
S u b s t i t u t i n g i n t o t h e expression f o r
be proved i n t h e l a s t s e c t i o n .
becomes
We have
w - 1 ) - w - 1
,
=
wo
woh
by a lemma t o
, and
+ +
(wov , v )
as required.
a s i n Lemma 79 i s 1 i n each of t h e f o l l o w i n g
c a s e s , s i n c e t h e c e n t e r is of odd o r d e r .
is a d j o i n t .
(a)
G
(b)
Char k = 2
(c)
G
Exercise:
.
is of t y p e
Aen,E6,
E8, F 4 , G 2 .
I n t h e remaining c a s e s f i n d
over t h e s i m p l e r o o t s .
h
, as
a product
n
ma(-1)
(X.
Example
Assume
.
SL2
:.
For e v e r y
char k = 0
e x a c t l y one
, so
-
i
,
i
.
1
I n v a r i a n t Hermitean forms.
Then t h e i n v a r i a n t form i s symmet-
t h e automorphism of Theorem 1 6 ,
irnal compact subgroup,
i s d e f i n e d by
xv
+
+
(b)
V x V
v
G
i s complex,
i s
o-
i s t h e corresponding max-
+ a r e a s b e f o r e , and
->
f: G
t e r m s of o t h e r w e i g h t s .
+
f(o-x-')
t h e n u s e t h e d e n s i t y of
K = G,
and
V
= f (x)v
Prove t h a t
i s even.
i
Assume now t h a t
..
(a)
v i z . t h e s p a c e of polynomials
i s odd, skew-symmetric i f
i
i = l , , . . t h e r e is
t h a t f o r each
of dimension
V
homogeneous of degree
r i c if
t h e r e i s an i n v a r i a n t b i l i n e a r form.
V
.
=
in
U-HU
U-HU
( F i r s t prove it on
,
G .)
Prove t h a t t h e r e e x i s t s a unique form
(
,)
from
to
which i s l i n e a r i n t h e second p o s i t i o n , c o n j u g a t e
+ +
-1
l i n e a r i n t h e f i r s t , a n d s a t i s f i e s (xv ,yv ) = f ( c x . y ) , a n d
t h a t t h i s form i s Hermitean.
(c)
under
K
Prove t h a t
(
,
i s p o s i t i v e d e f i n i t e and i n v a r i a n t
)
.
Dimensions.
finite field
Assume now t h a t
k
, that
V
i s a Chevalley group o v e r a n in-
G
and
h
a r e a s before, and t h a t
vz
i s t h e u n i v e r s a l a l g e b r a of Theorem 2 , w r i t t e n i n t h e form
215 35%
(a)
of page 16.
Prove t h a t t h e r e e x i s t s an antiautomorphism
vZLsuch t h a t
(b)
6X,
=
X-,
and
Define a b i l i n e a r form
o-Ha
( 0 ,u
= Ha
t
)
for a l l
from
q2!
r
a
to
of
.
a%
thus:
write
ru.u
T
i n t h e above form and t h e n s e t e v e r y
Xu
= 0
.
Prove t h a t t h i s forin i s symmetric.
vz
( c ) Now d e f i n e a b i l i n e a r form from
(
,
that
=
h o (
h(H,)
E
,
ZL+
'
(interpreting
a
for a l l
>
0)
V
.
2 thus:
%-
h
a s a l i n e a r form on
.
Assuming now t h a t t h i s form
i s r e d u c e d modulo t h e c h a r a c t e r i s t i c of
is j u s t t h e dimension of
to
k
, prove
su
t h a t its rank
3
1 .
Representations continued.
In t h i s section t h e irreducible
r e p r e s e n t a t i o n s of c h a r a c t e r i s t i c
base f i e l d
( t h e c h a r a c t e r i s t i c of t h e
k ) of t h e f i n i t e C h e v a l l e y g r o u p s and t h e i r t w i s t e d
a n a l o g u e s w i l l be c o n s i d e r e d .
Theorem 43:
p
Let
The main r e s u l t i s a s f o l l o w s .
be a f i n i t e u n i v e r s a l C h e v a l l e y g r o u p o r one
G
of i t s t w i s t e d a n a l o g u e s c o n s t r u c t e d a s i n 811
p o i n t s o f a n automorphism o f t h e form
the
a s t h e s e t of f i x e d
->
.
x
Then
(+ t d a ) )
Pa
i r r e d u c i b l e polynomial r e p r e s e n t a t i o n s of t h e i n c l u d x,(t)
J
ll-q ( a )
a simple
i n g a l g e b r a i c g r o u p ( g o t by e x t e n d i n g t h e b a s e f i e l d
a l g e b r a i c c l o s u r e ) f o r which t h e h i g h e s t w e i g h t s
0
5 <
h,a
> <_
-
q(a)
1 f o r a l l simple
and d i s t i n c t on r e s t r i c t i o n t o
a
h
t o its
k
satisfy
remain i r r e d u c i b l e
and form a complete s e t .
G
By Theorem 41 we a l s o have a t e n s o r p r o d u c t theorem w i t h t h e
C3
product
s u i t a b l y t r u n c a t e d , f o r example t o .
0
G
i s a
0
C h e v a l l e y g r o u p o v e r a f i e l d of
Exercise:
n3if
pn
elements.
Deduce from Theorem 43 t h e n a t u r e of t h e t r u n c a t i o n s
f o r t h e v a r i o u s t w i s t e d groups.
I n s t e a d of p r o v i n g t h e above r e s u l t s ( s e e Nagoya Math. J 22
( 1 9 6 3 ) ) , which would t a k e t o o l o n g , we s h a l l g i v e a n a p r i o r i
development, s i m i l a r t o t h e one of t h e l a s t s e c t i o n .
We s t a r t w i t h a group of t w i s t e d r a n k 1 ( i . e . o f t y p e
A1,
2 ~ 2 72 ~ 2 , o r
2G 2 ,
not being excluded).
t h e degenerate cases
The s u b s c r i p t
c on
Gr,
A1(2), A1(3),
Wb
,
...
...
will
h e n c e f o r t h be o m i t t e d .
X , Y , w, K
We w r i t e
for
ya(a
t h e n o n t r i v i a l element of t h e Weyl g r o u p r e a l i z e d i n
a l g e b r a i c a l l y closed f i e l d of c h a r a c t e r i s t i c
U =X
and
U-
= Y
i n t h e present case.
X
d e n o t e t h e sum o f t h e e l e m e n t s of
Definition:
An e l e m e n t
v
in a
p
.
G
>
,
and a n
We o b s e r v e t h a t
In addition,
in
X-a ,
o),
will
.
KG
V
KG-module
i s s a i d t o be a
h i g h e s t w e i g h t v e c t o r i f i t i s n o n z e r o and s a t i s f i e s
(a) x v = v
(b)
(c)
hv=h(h)v
-
Xwv = ;l;v
The c o u p l e
Remark:
C.
W.
forall
(A,)
U .
X E
h e H
for a l l
f o r some
p
K
E:
and some c h a r a c t e r
h
onH.
.
i s ' c a l l e d t h e corresponding weight.
The r e f i n e m e n t
(c)
o f t h e u s u a l d e f i n i t i o n i s due t o
FP
C u r t i s (111. J . Math. 7(1963) and J . f u r Math.
219 ( 1 9 6 5 ) ) .
That s u c h a r e f i n e m e n t i s needed i s a l r e a d y s e e n i n t h e s i m p l e s t
case
G = SL2(p)
.
Here t h e r e a r e
p
r e p r e s e n t a t i o n s r e a l i z e d on
t h e s p a c e s of homogeneous p o l y n o m i a l s of d e g r e e s
i = 0 , 1,
w i t h t h e h i g h e s t weight i n t h e u s u a l sense being
ihl
t h e group
(p
-
l)hl
H
p - 1
i s c y c l i c of o r d e r
a r e i d e n t i c a l on
H
,
,
t h e weights
.
...,
Since
0
and
h e n c e do n o t d i s t i n g u i s h t h e
c o r r e s p o n d i n g r e p r e s e n t a t i o n s from e a c h o t h e r .
Theorem 44:
(a)
Let
G
( o f r a n k 1) and t h e n o t a t i o n s be a s above.
Every n o n z e r o
weight v e c t o r , v
.
KG-module
For such a
v
V
contains a highest
we have
KGv = KYv
=
KU-v
.
p
-
1,
If
(b)
i s i r r e d u c i b l e , t h e n it d e t e r m i n e s
V
t h e unique l i n e of
V
f i x e d by
,
U
Kv
as
hence it a l s o d e t e r m i n e s t h e
c o r r e s p o n d i n g h i g h e s t weight.
(c)
Two i r r e d u c i b l e
KG-modules a r e isomorphic if and o n l y
i f t h e i r highest weights a r e equal.
The proof depends on t h e f o l l o w i n g two lemmas.
Lemma
-
80:
Let
and
Y
K
K
be any f i n i t e p-group and
(a)
Every nonzero
i n v a r i a n t under
Y
any f i e l d of c h a r a c t e r i s t i c
KY-module
Every i r r e d u c i b l e
(c)
Rad ICY = I C c ( y ) y
Proof:
(a)
t h e subspace
By i n d u c t i o n on
I
Cc(y) =
in
V1
V1
,
IYI .
Choose
i s t h e unique maximal
It i s n i l p o t e n t .
Assume
By ( a ) .
Yl
y e Y
KY
.
IYI >
1
0
.
.
Since
of i n d e x
on
V
Y
and
i s nonzero by t h e
t o generate Y / Y ~
(1 - y ) P v = 0
Then
(1 - y ) r v
whence ( a ) .
(b)
.
KY
of i n v a r i a n t s of
and nonzero.
maximal so t h a t
Y
01
it h a s a normal subgroup
i n d u c t i v e assumption.
v
c o n t a i n s nonzero v e c t o r s
V
i s t h e unique minimal i d e a l of
p-group,
.
KY-module i s t r i v i a l .
( o n e - s i d e d o r t w o - s i d e d ) i d e a l of
I
p
.
(b)
(d)
Y
be a s above, o r , more g e n e r a l l y , l e t
.
,
and
Now choose
r
The r e s u l t i n g v e c t o r i s f i x e d by
(c)
Rad KY i s a n i d e a l , maximal b e c a u s e
By i n s p e c t i o n
is
KY
i t s codimension i n
1
.
Bring t h e k e r n e l of t h e t r i v i a l
r e p r e s e n t a t i o n , which i s t h e o n l y i r r e d u c i b l e one by ( b ) , it i s t h e
u n i q u e maximal l e f t i d e a l ; a n d s i m i l a r l y f o r r i g h t i d e a l s .
On e a c h
f a c t o r of a composition s e r i e s f o r t h e l e f t r e g u l a r r e p r e s e n t a t i o n
of
KY
as
0
,
f
For
(a)
Proof:
=
x
and
If
g
.
1
f(x) = 1
,
Similarly f o r
(b)
( X W ) ~
,
write
=
wxw
=
f(x)'h(x)wg(x) with
.
a r e permutations of
0
acts
is nilpotent.
h(x) e H
and
f (x ) = f (x)
w h i l e if
v
-
x e X
(x), g(x) e X
(a) f
x
Rad KY
so t h a t
Lemma 81:
a c t s t r i v i a l l y b y ( b ) , h e n c e Rad KY
Y
on i t s e l f
1
,
we g e t t h e c o n t r a d i c t i o n xw e B ,
x w E B , so that
we s e e t h a t w-lx-1
,
'.
.
g
Xx2
-
X
+
C
Xf(x)h(x)wg(x)
.
xex-1
Here
f (x)
g e t s absorbed i n
and
h(x)
norrnzlizes
X
,
whence
(b).
P r o o f o f Theorem
44:
U =X
on
p o i n t s of
(a)
V
By Lemma 8 0 ( b )
i s nonzero.
Since
t h e space of f i x e d
H
i s A b e l i a n , t h a t s p a c e c o n t a i n s a nonzero v e c t o r
normalizes
v
U
and
such t h a t ( a )
and ( b ) o f t h e d e f i n i t i o n o f h i g h e s t w e i g h t v e c t o r h o l d .
Let
zwv = vl
.
If
v1 = 0 , t h e n ( c ) h o l d s w i t h
replace
v
by
vl
with
/L =
.
.
C h ( h ( x ))
I f n o t , we
Then ( a ) and ( b ) of t h e d e f i n i t i o n s t i l l h o l d
i n p l a c e of
wh
.
p = 0
h
,
-
a n d by Lemmagl(b) s o d o e s ( c ) w i t h
KGv = KYv
Now t o p r o v e t h a t
because o f , t h e decornposition
G = YBuwB
,
By t h e two p a r t s of Lemma 81 we may w r i t e
it i s enough,
wv e KYv
t o prove t h a t
-
Xwv
,
= !:v
.
a f t e r some
s i m p l i f i c a t i o n , i n t h e form
(: )
wv
+
,
h ( ~ h - ' w ) ~ ( x ) v= j s v
C
xeX-1
with
' y ( x ) = wxw-1 e Y
(b)
Let
V
F
,
= Kv
and
Lemma 8 O ( c ) t h a t t h e sum
t h e r e e x i s t s some
vl
whence o u r a s s e r t i o n . .
,"
that
H
vl e V "
i s Abelian.
y e Y
any
+
vl
V
2v
,
It f o l l o w s from
is direct.
f i x e d by
X
Now assume
.
hl = wYvl
= 0
,
'
since
weight
( h , , ) uniquely.
are irreducible
v2
then
2
and
=
V = KGv = I(Yv
cv+v
V2
11
dtr) c
get that
By ( a ) ,
.
KG-module d e t e r m i n e s i t s h i g h e s t
C o n v e r s e l y , assume t h a t
V1
and
KG-modules w i t h h i g h e s t w e i g h t v e c t o r s
of t h e same w e i g h t
E
.
(x,,u)
.
Now
v2
K
and
v
c =0
and
Set
4V
11
E
fbr
Y(1-y) = 0
.
By ( b ) a n i r r e d u c i b l e
since
H
-
Thus vl i s a h i g h e s t w e i g h t v e c t o r .
.,I
V = KYvl C KYV" = V , a c o n t r a d i c t i o n , whence ( b )
(c)
We may assume
i s an eigenvector f o r
-
We have
.
= Rad KY-v
4 V'
1
and a l s o t h a t
FP
9
V =V
V, v
E
V
,
v = v1
E
V1
+
vl
and
V2
and
s i n c e o t h e r w i s e we c o u l d w r i t e
Fiad KY. v
c = 1
+ v2
V2
,
2nd t h e n p r o j e c t i n g o n V1
a contradiction.
Thus we
may c o m p l e t e t h e p r o o f as i n t h e p r o o f o f Theorem 3 9 i e ) .
Theorem 4 5 :
Let
--A
b e t h e h i g h e s t w e i g h t of a n i r r e d u c i b l e
(
V.
KG-,module
~ f ' h S l ,t h e n
(b)
Every w e i g h t a s i n ( a ) c a n b e r e a l i z e d .
(HI.
+ 1.
Proof:
In
KG
(1) Proof of
-
xi3hw% and
=
( b
v = XFh.
H
wh,h,wh
-
Thus t h e
C h ( h c * lh) ,
=
then
~TEH
A s i n t h e p r o o f of Theorem 44 ( a ) , a
-~ w ( u + c v )
=
on t h e r e s p e c t i v e t e r m s .
i z e t h e weight
c =
HA
let
, u = ~
h ( h ( x )) u + C Z W H ~ Z *
XEX-1
a c t s , from t h e l e f t , a c c o r d i n g t o t h e c h a r a c t e r s
simple c a l c u l a t i o n y i e l d s
Here
,
If
r = O .
number of p o s s i b i l i t i e s i s
u
& = I ,t h e n
(a)
(h,O)
Ch(h(x))
and we g e t
$;
by t a k i n g
instead.
=
-
Thus i f
c
=
Finally if
1 by t a k i n g
c
=
wh
f
0.
If
h
1, t h e n
0.
=
we may r e a l - -
h,
we t a k e
wh = h,
~ h ( h ( x ) )=
-
1
( h , , ' ~ ) i n an
To a c h i e v e
K G ( U + C V )modulo a maximal
i r r e d u c i b l e module we s i m p l y take
submodule.
Since
dim V
If
(2)
X
and
Y
=
1, t h e n
V
i s t r i v i a l and
( h , ~ )= ( 1 , O ) .
a r e p--groups, t h e y a c t t r i v i a l l y by Lemma g O ( b ) ,
whence ( 2 ) .
(3) If
dim V ,6 1, t h e n
h
determines
a s i n t h e p r o o f o f Theorem 4 4 ( b ) .
fixes
vi7
wv,
Write
V" f 0 .
V
=
Since
i t f i x e s some l i n e i n i t , u n i q u e l y d e t e r m i n e d i n
by Theorem 44 ( b ) w i t h
fixes
We have
,
Y
X.
i n place of
we c o n c l u d e t h a t
wv
&
V
17
.
Since
Projecting
Y
T
V
+
Y
V
71
::-
V,
clearly
(::.Io f t h e p r o o f
2 h(wh-1w)
(
v'.,
44 ( a ) o n t o
o f Theorem
we g e t
= ,!5,
whence ( 3 ) .
4
Combine (1) , ( 2 ) and (3
Proof o f ( a ) .
o Theorems
(a)
solutions
If
C h ( h ( x )) = 0.
The number o f
XEX- 1
h ( x ) = h w i t h h g i v e n i s , modulo p ,
of
independent o f
4.4 and 45 :
l
h
n(h)
.
then
i n p a r t i c u l a r f o r each
h,
is a t least
h
1
( c f . Lemma 64, S t e p ( 1 ) ) .
(b)
The i r r e d u c i b l e r e p r e s e n t a t i o n of weight
b e r e a l i z e d i n t h e l e f t i d e a l g e n e r a t e d by
c
=
1 f o r the t r i v i a l representation
=
0
for
(c)
If
L
C
(dl
Dim V
=
~ wHi t h ~
(1,0)
K
is a splitting field for
H,
i t i s one
+
1x1
1x1
( h , , : ~=) (1,-1)
if
i f not.
(el
The number of p - r e g u l a r conjugacy c l a s s e s o f
(a)
If
is
G
1.
C l (h(x))
=
0
h
by
Cn(h)h(h)
$
(
=
0
1,
then
9
were
a contradiction.
0
by Theorem 45 (a), s o t h a t
f o r every
h $ 1.
H
0
f o r some
h,
h
r e p l a c e d by
wh
.
By t h e o r t h o g o n a l i t y
(which a r e v a l i d s i n c e
n ( h ) , a s an element o f
we conclude t h a t
n(h)
=
above a p p l i e d w i t h
r e l a t i o n s f o r t h e c h a r a c t e r s on
If
can
G.
Proof:
Then
C
(h,,u)
otherwise.
(
]HI
-X H ~ W X+
p $ / hI ) ,
K, i s i n d e p e n d e n t o f
we w o u l d g e t
h.
) H ( = c ~ ( ~ ) =(mod
o
p),
(b)
Let
V
y
G, b
E
We c o n s i d e r , a s i n Theorem 40
h , .
p of
V"
a : G
space of f u n c t i o n s
for a l l
+
i n t o t h e induced r e p r e s e n t a t i o n
K
V = KW+ + Rad K X ~ W V ' ,
V.
a ( y b ) = a ( y )h'"(b)
such t h a t
(cpf) ( x )
B d e f i n e d by
E
a h i g h e s t weight v e c t o r f o r
=
f (xv')
we may d e f i n e
by t h e e q u a t i o n s of Lemma 7 4 , w i t h
Coverting f u n c t i o n s . on
rJ
C a (x)x,
whence
(b)
we s e e t h a t
.
a+
submodule i n c a s e
h
l
0,
then
dim V
(c)
By
(b).
(dl
If
/I =
<
IX 1
so that
KG
V
;(L =
i n t h e u s u a l way,
(b),
may b e r e a l i z e d
a s one o f two i n c a s e
v
.
h
a s t h e unique i r r e d u c i b l e
G,
-
Xwv
=
0,
whence
by Theorem 44 ( a ) . C o n v e r s e l y i f
t h e n t h e a n n i h i l a t o r of
yf+ i s given
=
becomes t h e e l e m e n t o f
.--+
Bh;:
a'
i n p l a c e of
k t t h e same t i m e we s e e t h a t
i n t h e i n d u c e d module
v+
f C a s i n Lemma 73, p r o v e
t o elements o f
G
with
Using t h e d e c o m p o s i t i o n
t h a t i t i s a h i g h e s t w e i g h t v e c t o r , and t h a t
a
V"
b e a n i r r e d u c i b l e module whose d u a l
has t h e h i g h e s t weight
t h e isomorphism
23 7
in
KY
contains
h = 1.
Yv = 0
and
1x1,
dim V
Y b y Lemma 8 0 ( d ) ,
0.
By a c l a s s i c a l t h e o r e m o f B r a u e r a n d N e s b i t t
(el
( U n i v e r s i t y o f Toronto S t u d i e s , 1 9 3 7 ) t h e number i n q u e s t i o n
e q u a l s t h e number o f i r r e d u c i b l e
IHI
+
KG-modules,
hence equals
1 by Theorem 4 5 ( b ) .
Example: G
Remarks.
=
(a)
SL2(q).
Here
(HI
=
q
-
1,
/HI+
so t h a t
We s e e t h a t t h e e x t r a c o n d i t i o n
-
Xv
=
rn
two p u r p o s e s . F i r s t i t d i s t i n g u i s h e s t h e s m a l l e s t module
1= q .
serves
(I,o)
23 8
-
from t h e l a r g e s t
relation
(U-B
i t t a k e s t h e p l a c e of t h e d e n s i t y argument
KGv = KU?
dense i n
(b)
Secondly, i n t h e proof o f t h e key
(1, 1
used i n t h e i n f i n i t e case.
G)
The p r e c e d i n g development a p p l i e s t o a wide c l a s s o f
doubly t r a n s i t i v e permutation groups
H o f two p o i n t s ) , s i n c e
of a point,
facts that
H
i s A b e l i a n and h a s i n
which i s a p-Sylow s u b g r o u p o f
(with
B
the stabi'lieer
i t depends o n l y on t h e
B
a n o r m a l complement
U
G.
Now we c o n s i d e r g r o u p s o f a r b i t r a r y r a n k .
W (
=
w ~ )w i l l
b e g i v e n t h e s t r u c t u r e o f r e f l e c t i o n g r o u p a s i n Theorem 3 2 w i t h
Z/R
( s e e p . 177), p r o j e c t e d i n t o
Vo
a n d s c a l e d down t o a s e t
of u n i t v e c t o r s , t h e c o r r e s p o n d i n g r o o t s y s t e m .
root
a,
we w r i t e
t o represent
wa
for
Y,
in
W.
If
and choose
X-a
w
E
W
w
f l e c t i o n s , 'and s e t
=
wawb
o f t h e minimal e x p r e s s i o n chosen.
in
cXa
, Ya>
i s a r b i t r a r y , we choose a
.. . a s a p r o d u c t
- -w = wawb . . . . Then F
minimal e x p r e s s i o n
wa
For each s i m p l e
o f s i m p l e re-
i s independent
We p o s t p o n e t h e p r o o f of t h i s
f a c t , which c o u l d ( a n d p r o b a b l y s h o u l d ) have b e e n g i v e n much
e a r l i e r , t o t h e end o f t h e s e c t i o n s o a s n o t t o i n t e r r u p t t h e
p r e s e n t development.
Lemma 82 : If
Proof:
on
Since
w
-
w
=
=
A s a c o n s e q u a n c e we h a v e :
w,wb
. ..
i s a n y minimal e x p r e s s i o n ,
--
. ..
t h i s e a s i l y follows by i n d u c t i o n
wawb
then
~ ( w ) o r b y Appendix 11.25.
We e x t e n d t h e e a r l i e r d e f i n i t i o n of h i g h e s t w e i g h t v e c t o r
23 9
b y t h e new r e q u i r e m e n t :
(c)
--
(pa
XaWa v = pav
Theorem 4 6 :
Let
K)
E
f o r every simple r o o t
a.
b e a ( p e r h a p s t w i s t e d ) f i n i t e Chevalley
G
g r o u p (of a r b i t r a r y r a n k ) .
Same as
a , ( b ( c
(d)
Let
Ha = H
of Theorem 44.
( a , b , c
n <Xa,
.
Ya>
,ua = 0
weight o f some i r r e d u c i b l e module t h e n
and
ira = 0
(e)
~ J H ,
if
+ 1,
= 1.
Every weight a s i n ( a )
irreducible
Proof:
~IH,
o r -1 i f
i s the highest
(
If
c a n b e r e a l i z e d on some
KGmodule.
We s h a u prove t h i s theorem i n s e v e r a l s t e p s .
(all
There e x i s t s i n
This i s proved as i n Theorem 44 ( a ) .
(a21
v
If
is as i n
v
a nonzero e i g e n v e c t o r
V
then so is
(al),
B.
for
--
vl = Xa wa v
( a s i m p l e ) , u n l e s s it i s 0.
P r o o f : Let
E
Xa
x
b e any element o f
t
xa
and
xa
components a r e
?
E
1.
Xa
,
U.
x
Write
=
t
x a xa
t h e subgroup o f e l e m e n t s o f
We r e c a l l t h a t
Xa
with
U
Fa n o r m a l i z e
and
Xa
whose
t
Xa
-
I-
xvl = x a X a wa v = v l , s i n c e U v = v .
S i n c e H n o r m a l i z e s Xa and i s n o r m a l i z e d by
we s e e t h a t
wa
vl i s a l s o an e i g e n v e c t o r f o r H.
( s e e Appendix 1.11). Thus
(a31
Choose
v
i s maximal s u b j e c t t o
weight v e c t o r .
as i n
-
(all,
then
vl = ~ ' i S v 0.
w
Then
E
vl
W
SO
that
~ ( w )
is a highest
240
Proof:
By Lemma 82 and ( a 2 ) ,
Let
be any simple r o o t .
a
-
XaFawl
0
=
i s an e i g e n v e c t o r f o r
> 0,
w-la
If
-
-=
-
by Appendix 1 1 . 1 9 , s o t h a t
and
vl
'waw waw
w
may choose a minimal e x p r e s s i o n
w Xw
Xa a w
w.
by. t h e c h o i c e o f
then
If w-la
=
N (w) + l
by Lemma 82,
< 0,
...
wawb
=
N (waw)
B.
t h e n we
s t a r t i n g with
,
wa.
Then
-
XaGavl
=
(Val2 XbFb
. .v
By
Proof:
[al!
and
We have
(a31
G = GoG
-
w
enough t o show each
w
=
wa
. .. v ,
Tava?tb Fb
= ,!L
a simple.
with
by Lemma 82
,u
with
by Lemma 8 l ( b )
K
E
C
-
U % U-B by Theorem 4.
,
:Y
fixes
KU-IT,
Assume
y
Theorem 44 ( a )
(bl)
t h a t Gayv
so
applied t o
If
Proof : Write
=
V"
a s before, then
y
=
and f o r this we may assume
U-.
E
.
Write
zaijayv =
Thus
-x , ~ ~ ( y - l ) =v
with
t
yaya
xeXa
U- Gayav
E
V
?
=
Kv
and
-
XaTj,
with
y(x)
P?
xeXa
ya
=
t
yaya
and
C
- KU-v
as
t$
by
V" = R a d KU-v
.
as before.
y(x)xijav
C
y
.
c X a ,Ya>
vT+V
Then
t
Say,yav
i s f i x e d by every
Z
Then
(b2)
=
91
V
Thus i t i s
W
above, b u t u s i n g n e g a t i v e r o o t s i n s t e a d .
normalize
(a).
we have t h e f i r s t s t a t e m e n t i n
(y(x)-l)xijav
Proof of ( b ) .
E
V
E
U-.
.
If t h i s i s f a l s e , t h e r e e x i s t s
I
1
vl
V"
E
, vl
vl
0,
a s an eigenvector f o r
1
4
1:
3
X.
f i x e d by
e i g e n v e c t o r f o r each
H,
-XaGa
24 1
A s u s u a l we may c h o o s e
and t h e n by
Then, a s b e f o r e ,
V
=
KGvl
>
.
a l s o an
= KU-V~
C
-
V
ft
a contradiction.
(c)
Same p r o o f a s f o r Theorem
(dl
By Theorem 4 5 ( a )
(el
Let
n
44 ( c ) .
<
applied t o
Xa,Ya
a = 0,
The r e a d e r s h o u l d h a v e
be t h e s e t o f simple r o o t s
P
and
W,
t h e corresponding subgroup o f
W.
no t r o u b l e i n p r o v i n g t h a t t h e l e f t i d e a l o f
-
C
IJHh %Zw i s a n i r r e d u c i b l e
wnwo
i s (h,,ua)
"
KG
such t h a t
L
g e n e r a t e d by
KG- module whose h i g h e s t w e i g h t
(a)
A splitting field for
(b )
Let
dim V = ) U 1
Then
if
(c)
KGmodules
Proof:
w0 - wawb
Cor.
, or,
(h,pa)
,
a
=
G.
(h,,ua).
-1.
Ha(a
of s i m p l e r o o t s , l e t
p
n).
E
H,
be t h e
Then t h e number o f i r r e d u c i b l e
e q u i v a l e n t l y , o f p - r e g u l a r con j u g a c y c l a s s e s o f
Clear.
(b)
Write
UG,V
= X v
wo
a s i n Lemma 8 2 ,
( d l t o Theor(c)
i s a l s o one f o r
.
For each s e t
(a)
.. .
h = 1 and a l l
not
g r o u p g e n e r a t e d by a l l
H
b e i r r e d u c i b l e , of h i g h e s t w e i g h t
V
< 1 ~ 1i f
of
a
.
Corollary:
5
and (a31
(bl)
vl
...
v , with
a - a b b
and t h e n p r o c e e d as i n t h e p r o o f
=
44 a n d 45.
The g i v e n sum c o u n t s t h e number o f p o s s i b l e w e i g h t s
according t o t h e s e t
n.
of simple r o o t s
a
such t h a t
,
E x e r c i s e. : I f
i s u n i v e r s a l , t h e above number i s
G
.
-
I4
a simple ( I H a I +
l),=
43. If i n a d d i t i o n
~ ( a ) , i n t h e n o t a t i o n o f 'Theorem
i s n o t t w i s t e d , t h e n t h e number i s
G
q*
.
It remains t o prove t h e f o l l o w i n g r e s u l t used ( i n e s s e n t i a l l ~ )
i n t h e proof o f Lemma 82.
(a)
Lemma $3:
w
w
If
W, t h e n any two minimal e x p r e s s i o n s f o r
&
a s a p r o d u c t o f s i m p l e r e f l e c t i o n s can b e t r a n s f o r m e d i n t o
each o t h e r b y t h e r e l a t i o n s
(*)
. .
n = order
..
=
wbwawb
wawb
,
with
(b)
Assume t h a t f o r each s i m p l e r o o t
wawbwa
expression f o r
w
and
-
terms on each s i d e ,
b
d i s t i n c t simple r o o t s ) .
wa
of
G
i Xa,X-a
>.
Let
c o r r e s p o n d i n g element
chosen t o l i e i n
a
(n
E
W.
Then
w
=
a
the
(any Chevalley g r o u p )
w = wawb
--
wawb
...
...
is
b e a minimal
i s independent o f t h e
minimal e x p r e s s i o n chosen.
Proof:
(a)
relations
w,
T h i s i s a r e f i n e m e n t o f Appendix IV.38
=
1 a r e not required.
since the
It i s a n e a s y e x e r c i s e t o
c o n v e r t t h e proof o f t h e l a t t e r r e s u l t i n t o a proof o f t h e former,
which we s h a l l l e a v e t o t h e r e a d e r .
(b)
Because of
( a ) we o n l y have t o prove
h a s t h e form o f t h e two s i d e s o f
(
(b)
when
For t h i s we can r e f e r t o
t h e proof o f Lemma 56 s i n c e t h e e x t r a r e s t r i c t i o n s t h e r e ,
G
i s u n t w i s t e d and t h a t
wa
=
wa(l)
w
f o r each
a,
a r e not
that
e s s e n t i a l f o r t h e proof.
Remark:
It would b e n i c e i f someone c o u l d i n c o r p o r a t e i n t h e
e l e m e n t a r y development j u s t g i v e n t h e t e n s o r p r o d u c t theorem
mentioned a f t e r Theorem 43 o r a t l e a s t a p r o o f t h a t e v e r y
i r r e d u c i b l e K-module f o r
G
c a n b e extended t o t h e i n c l u d i n g
a l g e b r a i c g r o u p , hence a l s o t o t h e o t h e r f i n i t e Chevalley g r o u p s
c o n t a i n e d i n t h e l a t t e r group.
814. R e p r e s e n t a t i o n s conc1,uded.
Now we t u r n t o t h e complex
r e p r e s e n t a t i o n s of t h e groups j u s t considered4
i s i n poor shape.
Only
Here t h e t h e o r y
(Green, T4A.M.S. 1955) and a few
GLn
groups of low r a n k have been worked o u t c o m p l e t e l y , t h e n o n l y i n
Here we s h a l l c o n s i d e r a few g e n e r a l
terms of t h e c h a r a c t e r s .
r e s u l t s which may l e a d t o a g e n e r a l t h e o r y *
Henceforth
w i l l d e n o t e t h e complex f i e l d *
K
(one--dimensional) c h a r a c t e r
r e a l i z e d on a s p a c e
module f o r
Vh
,
on a subgroup
h
G
Vh
we s h a l l w r i t e
T h i s may b e d e f i n e d by
G.
'
Given a
of a group
B
G,
f o r t h e induced
G
Vh = KG %Vh
(this
d i f f e r s from o u r e a r l i e r v e r s i o n i n t h a t we have n o t s w i t c h e d t o
a s p a c e of f u n c t i o n s ) , and may b e r e a l i z e d i n
i d e a l g e n e r a t e d by
form).
Bh =
biB
h(b-')b
I t s dimension i s
KG
i n the left
(and w i l l b e used i n t h i s
(G/B].
E x e r c i s e : Check t h e s e a s s e r t i o n s .
Lemma 84.:
,
,!L
Let
B, C
b e subgroups of a f i n i t e g r o u p
b e c h a r a c t e r s on
i n g modules f o r
(a)
G
Vh
and l e t
, vG
,u
G,
E
Bh x C
then
P
in
KG
t o m u l t i p l i c a t i o n by a nonzero s c a l a r by t h e
t o which
(b)
x
b e t h e correspond-
i s determined up
(B,c)
double c o s e t
K-space
t o t h e one
belongs.
G
I4omC(e , v~:) i s isomorphic a s a
g e n e r a t e d by a l l
(c)
let
G.
x
If
B, C,
G,
If
Bh x Cp
B = C
i s one o f a l g e b r a s .
and
.
h = L!,
,
t h e n t h e isomorphism i n ( b )
24 5
~ o r n ~ ,( V~ f i s t h e number of
P
such t h a t Bh x C $, 0 f o r some,
The dimension o f
(dl
double c o s e t s
(B,c)
hence f o r every
x
D
P
in
D , o r , e q u i v a l e n t l y , such t h a t t h e
r e s t r i c t i o n s of , h and
Proof:
(a)
Assume
KG- module,
Slnce
T
C
X.E B\G/C
VhG
a l l of
to
B n x Cx -1
,
om^ (VhG , VyG ) .
E
determines
TBh
bBh = h ( b ) B h ,
TJ3, =
xy
a r e equal.
This i s c l e a r .
(b )
as a
G
T.
Since
Let
TBh
we g e t by a v e r a g i n g o v e r
c B x C/,.
x h
Thus
T
generates
Bh
C
=
x s G/C
B that
i s r e a l i z e d on
cx x C
Bh,
-
hence o n
( B ]-1 C cxBh x C
by r i g h t m u l t i p l i c a t i o n by
P
Y
Conversely, any such r i g h t m u l t i p l i c a t i o n y i e l d s a homomorphism,
which proves (b ) .
(c)
By d i s c u s s i o n i n ( b ) .
The f i r s t s t a t e m e n t f o l l o w s from ( a ) and ( b ) .
B 1 = B n x C x - 1 and C1 = x - l ~ xT\ C , and lyi ] and
(dl
Let
systems of r e p r e s e n t a t i v e s f o r
t
Bh
= C h
1
and
(y;l)yi
GI,
=
T
'9
B?t.
since
1f
=
T
BhBlhBl,xp xC ,LL
x clx-'
h = x4,
=
Bl
this
s i n c e t h e elements
Remarks:
B ~ \ B and C / C ~ .
-1
C p ( 2 . ) z . . Then
J
J
(a)
.
with
h $, x u
If
x z
on
B1,
I B I ~ B ~ and~
product i s
yibl
-3
(xp)(xcx
?
-
J
=
lzJ ]
Set
il(c)
then
BlhBl,xP
then
Bh x Cfi $, 0
=
0.
are a l l distinct.
T h i s i s a s p e c i a l c a s e o f a theorem o f G. Mackey.
(See, e . g . , F e i t f s n o t e s . )
(b)
The a l g e b r a o f
(c)
i s a l s o c a l l e d t h e commuting
v
a l g e b r a s i n c e it c o n s i s t s o f a l l endomorphisms o f
I
commute w i t h t h e a c t i o n of
~
Theorem 4 7 :
Let
If
(a)
u s u a l way,
every
w
(b)
V
:
h , fl
If
w $ 1
v
(a)
if
(b)
and o n l y i f
h
f o r e x a c t l y one
G
Since
Vh
morphic i f and o n l y i f
and
w
V:
wh
then
w
h = wfi f o r some
~xercise.: (a)
HO~~(V:
= WY
i.e.,
a g r e e on
i.e.
W,
E
W.
E
by ( c ) and ( d l
B
f o r only
f~WBW-',
w
= 1.
a r e i r r e d u c i b l e , they a r e isoG
h
dim
and
dim HomG(Vh,
h o l d s e x a c t l y when
v,)G
=
1, which., a s above,
w
f o r some (hence f o r e x a c t l y one)
, vG ~ =) 1~~ 1
=
(b)
for
.
( S c h u r r s ernm ma).,
o f Lemma 84, i f and o n l y i f
H,
wh $ h
i s i r r e d u c i b l e i f and o n l y i f i t s commuting
a l g e b r a i s one-dimensional
hence on
i n the
B
b o t h s a t i s f y t h e c o n d i t i o n s of ( a ) ,
i s isomorphic t o
Proof:
-
extended t o
1s
i r r e d u c i b l e i f and o n l y i f
G
Vh
such t h a t
W
H
.is a c h a r a c t e r on
then
E
G.
b e a ( p e r h a p s t w i s t e d ) f i n i t e Chevalley group.
G
h
that
0
if
h
=
w,u
f o r some
E
W.
w,
otherwise.
I n Theorem 47 t h e c o n c l u s i o n i n ( b ) h o l d s even i f
t h e condition i n (a) doesn?t.
Here
Wh
i s the stabilizer of
h
particular, t h a t if
1w 1 - dimensional.
Theorem 48':
Let
one-dimensional
=
h
in
W.
We see,, i n
1 t n e n t h e commuting a l g e b r a of
V:
is
But more i s t r u e .
V
b e t h e KG-module induced by t h e t r i v i a l
Wmodule.
Then t h e commuting a l g e b r a
~
n (v)
d ~
KW.
i s isomorphic t o t h e group a l g e b r a
Reformulations :
(a)
The m u l t i p l i c i t i e s o f t h e i r r e d u c i b l e components o f
KW- modules.
a r e j u s t t h e degrees of t h e i r r e d u c i b l e
(b)
(c)
KG
of
--
f : G -3 K
The a l g e b r a of f u n c t i o n s
(f(bxbT)= f(x)
for a l l
b,b
1
E
spanned by t h e
KW.
BwXw i s i s o m o r p h i c t o
d o u b l e c o s e t sums
B
A,
The s u b a l g e b r a , c a l l it
b i i n v a r i a n t under
8) w i t h c o n v o l u t i o n a s m u l t i -
KW.
p l i c a t i o n is isomorphic t o
The t h e o r e m i s e q u i v a l e n t t o ( a ) by S c h u r t s Lemma and t o
Proof:
( b ) by Lemma 84, w h i l e ( b ) and ( c ) a r e c l e a r l y e q u i v a l e n t .
4
s h a l l g i v e a p r o o f o f ( b ) , due t o J. Tits.
average i n
KG
We
w i l l denote t h e
w
of t h e elements o f t h e double c o s e t
The
BwB.
4
A
w
elements
a
V
form a b a s i s of
i s a simple r o o t ,
ca
and
A
1 i s t h e u n i t element.
Ixa I-'
w i l l denote
.
A
[wa
i s g e n e r a t e d as a n a l g e b r a by
(1) A
If
I
a simple]
subject t o the relations
(a)
(P)
Proof:
A
"2
wa
=
A
cal
A A f '
wawbwa
4
(1-ca)wa
.
=
A
4
for all
A
. ..
wbwawb
We o b s e r v e t h a t i f e a c h
ca
,
a.
a s i n Lemma $3 ( a ) .
i s r e p l a c e d by
KW, by Appendix 11.38.
r e l a t i o n s go o v e r i n t o a d e f i n i n g s e t f o r
Since
r,s
E
B U BwaXa
K.
Since
i n v e r s e , we g e t
s =
IBW~X,]
-
i s a g r o u p , we have
Bwa X
r
B .
=
1 then these
(Fwaxa) = r 3 + sBwaXa
with
c o n t a i n s w i t h each o f i t s e h m e n t s i t s
(BW,X,
Thus
1,
(a)
and t h e n from t h e t o t a l c o e f f i c i e n t
holds i n
A,
and s o d o e s
(P)
by
24-8
,
Lemma 2 5 , C o r .
A
wa
which a l s o shows t h a t t h e
generate
A.
A
Conversely, l e t
A1
b e t h e a b s t r a c t a s s o c i a t i v e algebra (with
A
g e n e r a t e d b y symbols
w
E
h
W
wa
(a)
subject t o
w
choose a minimal e x p r e s s i o n
A A
w = wawb
... .
By Lemma 83 ( a )
t h e e x p r e s s i o n chosen.
Thus t h e
A
w
(@I.
For each
wa wb " *
and s e t
( B ) t h i s i s i n d e p e n d e n t of
=
and
( a ) it follows t h a t
By
form a b a s i s f o r
dimension a s
and
1)
which,
A1,
h a v i n g t h e same
i s t h e r e f o r e isomorphic t o it.
A,
( 2 ) There e x i s t s a p o s i t i v e number c =: C ( G )
such t h a t
na
C
= c
w i t h n a a p o s i t i v e i n t e g e r d e p e n d i n g o n l y on t h e t y p e
a
o f G.
The m u l t i p l i c a t i o n t a b l e o f A i n t e r m s o f t h e b a s i s $1
i s g i v e n b y polynomials i n
Proof:
Consider
possibilities for
set
-',
c = q
c
2~ ( q2 )
f o r example.
4
Ixa(
d e p e n d i n g o n l y on t h e
are
q2
and
Here t h e two
q3
If we
by Lemma 6 3 ( c ) .
t h e n t h e corresponding values of
which depend o n l y on t h e t y p e .
type.
na
are
2
For each of t h e o t h e r t y p e s t h e
verification i s similar.
From t h e f i r s t s t a t e m e n t o f ( 2 ) a n d t h e
e q u a t i o n s o f t h e p r o o f of
(1) t h e s e c o n d s t a t e m e n t f o l l o w s .
(3) An a . s s o c i a t i v e a l g e b r a
with multiplication table
AC
g i v e n b y t h e p o l y n o m i a l s o f ( 2 ) e x i s t s f o r e v e r y complex number
In particular
Proof:
and
=
KW
Since t h e t y p e o f t h e group
G
=
A
3,
and
A1
c.
.
c o n t a i n s a n i n f i n i t e number
24 9
o f members, t h e m u l t i p l i c a t i o n t a b l e i s a s s o c i a t i v e f o r an
i n f i n i t e s e t of values o f
( 4 ) AC
c., h e n c e f o r a l l v a l u e s .
i s semisimple f o r
c
=
f o r a l l b u t a f i n i t e number of v a l u e s o f
Ac
Proof:
is f o r
KG-module and f o r
i n both cases.
nonzero a t
c
c =
c
=
C(G)
c.
1 a group a l g e b r a
1 since then
Ac
Ac
Completion o f p r o o f .
is an algebraically closed f i e l d ,
KW, h e n c e s e m i s i m p l e
i s a polynomial i n
c,
i s semisimple, hence nonzero
f o r a l l b u t a f i n i t e number o f v a l u e s o f
(5)
c = 1 , and
t h e commuting a l g e b r a o f a
The d i s c r i m i n a n t o f
=
for
c(G).,
Since
A
c.
A
i s s e m i s i m p l e and
i s a d i r e c t sum o f c o m p l e t e
metric a l g e b r a s , o f c e r t a i n degrees over
K
(see, e.g.,
J a c o b s o n f s S t r u c t u r e o f Rings o r F e i t f s n o t e s ) ,
and s i m i l a r l y
KW. We h a v e t o show t h a t t h e d e g r e e s a r e t h e same i n t h e
for
two c a s e s .
If
a
i s any f i n i t e - d i m e n s i o n a l a s s o c i a t i v e a l g e b r a
which i s s e p a r a b l e , i . e .
which i s s e m i s i m p l e when t h e b a s e f i e l d
i s extended t o i t s a l g e b r a i c c l o s u r e ,
invariants of
we d e f i n e t h e n u m e r i c a l
t o b e t h e degrees of t h e r e s u l t i n g matric alge-
The proof o f Theorem 4 8 w i l l b e c o m p l e t e d by t h e f o l l o w i n g
bras.
lemma.
Lemma 85 .:
Let
R
b e a n i n t e g r a l domain,
q u o t i e n t s , and
f
a homomorphism o f
a
K
be a finite-dimensional
QF
and
$
and
R
F its
f i e l d of
K.
onto a f i e l d
associative algebra over
R,
Let
and
aK t h e r e s u l t i n g a l g e b r a s o v e r F and K. If
aK a r e s e p a r a b l e , t h e n t h e y h a v e t h e same n u m e r i c a l
250
invariants.
I n f a c t , from (3) and t h e lemma w i t h
and f i r s t
that
f
A
c (G)
c->
;
and t h e n
: c
f
d
R = K[c],
+ 1,
,
= A,
it follows
KW have t h e same n u m e r i c a l i n v a r i a n t s , hence
and
t h a t t h e y a r e isomorphic.
Proof of t h e lemma:
(a)
0
Assume t h a t
i s a f i n i t e - d i m e n s i o n a l semisimple
L,
a s s o c i a t i v e a l g e b r a o v e r an a l g e b r a i c a l l y c l o s e d f i e l d
that
.
bl,b2,.
2
bn
a r e independent indeterminates over
that
@/L,
form a b a s i s f o r
L,
that
that
b = E xibi
~ ( t i)s t h e c h a r a c t e r i s t i c polynomial o f
fi L(xl,. . .
t h e l e f t on
w i t h the
Pi
written as
:Xn)
Then:
(al)
The
a r e t h e n u m e r i c a l i n v a r i a n t s of
(a21
pi
pi
dgtPi
f o r each
~ ( t= )T
If
L(xl).
.. )xn)
For
s any
Q ~ ( i~
)
J
TT
~ ~ ( t )
Pi
such t h a t
(all
q
and
=-
J
(a21
dg &
t j
i s i r r e d u c i b l e over
,
qj
f a c t o r i z a t i o n over
f o r each
j,
and t h a t
E. .. I f
then it agrees
a r e t h e numerical i n v a r i a n t s
we may assume t h a t
End L'
b
=
63
is the
Cx. E.
l j
.
13
in
[x. -1, t h e n
13
s o t h a t we have t o show t h a t d e t (tI
terms of t h e matric u n i t s
-
0 .
J
complete m a t r i c a l g e b r a
P ( t )= d e t (tI
a c t i n g from
1.
i
w i t h t h e one above s o t h a t t h e
Proof:
and
d i s t i n c t monic polynomials i r r e d u c i b l e over
.. , x n ) .
(a3
,
b
~ ( t= )
L(xl,.
=
. ,xn
xl,x2,..
L(x. -1.
1J
13
X
=
-
X)
This i s so since specialization t o
t h e s e t of companion m a t r i c e s
PP
y i e l d s t h e general equation
of degree
some
Qk
p.
In
(a31
k
j,
with
i f some
were r e d u c i b l e o r e q u a l t o
J
t h e n any i r r e d u c i b l e f a c t o r Pi of
Q-
Q~
would v i o l a t e
(a2).
I
.I-
(b)
Let
R'"
R
b e t h e i n t e g r a l c l o s u r e of
in
F
( c o n s i s t i n g o f a l l elements s a t i s f y i n g monic polynomial e q u a t i o n s
over
R).,
*
R [xl,..
F (xl,
and
xl,x2,.
. xn ]
..
xn
indeterminates over
R[xl,.
i s t h e i n t e g r a l closure of
-
F.
Then
.. x n ]
in
-:,xn)
Proof : S e e , e. g.
, Bourbaki , Commutative
Algebra, Chapter V.,
Prop. 13.
If
(c)
R*
i s as i n (b)
, then
4-
can b e extended t o one o f
K
into
R-
any homomorphism of
-
into
R
K.
-0.
Proof:
By Z o r n t s lemma t h i s can b e reduced t o t h e c a s e
where it i s almost immediate s i n c e
(dl
Completion o f proof.
UF/~?
hence a l s o f o r
F
and a l s o o v e r
-
K.
f :
a homomorphism
.#.
morphism o f
>k
R"
into
and
ixi]
K
Let
i s a l g e b r a i c a l l y closed.
{ai]
K
be a b a s i s f o r &/R,
independent i n d e t e r m i n a t e s o v e r
The g i v e n homomorphism
-+
a .
R*r = ~ [ a ] , )
f :R>
-
K
defines
By ( c ) i t e x t e n d s t o a homo-
and t h e n n a t u r a l l y t o one o f
.. ,xn] i n t o -K[xl,. . . xn 1. If a = Z x.1 a 1. and
P i i s i t s c h a r a c t e r i s t i c polynomial, f a c t o r e d
t= ) r[ Pi ( t )
R [xl,.
~ (
-
F(xl).
..) x n )
a s b e f o r e , t h e n t h e c o e f f i c i e n t s o f each
Pi
over
are
rn
252
i n t e g r a l p o l y n o m i a l s i n i t s r o o t s , hence i n t e g r a l o v e r t h e
c o e f f i c i e n t s of
P,
hence belong t o
R [xl,.
hence i n t e g r a l over
.fa
1
.
.. ,xn]
by (b 1.
...,xn],
R[xl,x2,
1,
f ( a ) = Pxif ( a
Thus i f
t h e n i t s c h a r a c t e r polynomial has a corresponding f a c t o r i z a t i o n
Pf ( t ) = r
[Pif ( t )P i o v e r -K(xl,
x n ) . By ( a l l t h e pi a r e
...,
Q , and by (a21 t h e y s a t i s f y
t h e numerical i n v a r i a n t s of
pi = dgtPi
= dgtPif,
s o t h a t by ( a ? )
a l s o t h e numerical i n v a r i a n t s of
If
Exercise:
h
u s u a l way, t h e n
with
6=a
a K , which
H
i s a c h a r a c t e r on
they a r e
p r o v e s t h e lemma.
B
extended t o
KWh
~ n d ~ ( ~ ;i s
) isomorphic t o
.
i n the
(Observe t h a t
t h i s r e s u l t i n c l u d e s b o t h Theorem 47 and Theorem 48.1
Remark:
Although
A
i s isomorphic t o
KW t h e r e d o e s n o t seem
t o b e any n a t u r a l isomorphism and no one h a s s u c c e e d e d i n decomp o s i n g t h e module
of Theorem 48 i n t o i t s i r r e d u c i b l e com-
V
We may o b t a i n some
p o n e n t s , e x c e p t f o r some g r o u p s o f low r a n k .
p a r t i a l r e s u l t s , i n terms o f c h a r a c t e r s , by inducing from t h e
parabolic
Lemma 86:
subgroups and u s i n g t h e f o l l o w i n g simple f a c t s .
Let
n be a s e t o f simple r o o t s ,
corresponding subgroups o f
and
G
Vn
G
and
the
W
( s e e Lemma 3 ~ 1 , and V n
t h e c ~ r r e s p o n d i n gt r i v i a l modules i n d u c e d t o
and s i m i l a r l y f o r
(a)
A
,
(b)
Gn
W
and
G;
.
a'
system of r e p r e s e n t a t i v e s i n
d o u b l e c o s e t s becomes
G
W and
W,
in
G
W
for the
a system o f r e p r e s e n t a t i v e s f o r
double cosets.
G
dim H 0 m G ( V , ,
G
(Wn,Wn~)
W
W 1.
V,T ) = dim HomW(Vn, vnt
Proof:
(a)
(b)
By Lemma $&{dl and ( a ) .
C o r o l l a r y 1:
for
Exercise.
W. I f
Let
X
G
G.
Proof:
Let
(X, )G
W, then
(x' ,xW
)
i s irreducible,
-X
+
X
=
and s i m i l a r l y
W=~n&:
= E nnX,
denote t h e average of
( Xn G ,n ~ G =~dim
) G H O ~ ~ ( GV , ,v z r ) ,
so t h a t
X
i s a s e t o f i n t e g e r s such t h a t
In,]
a n i r r e d u c i b l e c h a r a c t e r of
one o f
VnG
denote t h e c h a r a c t e r of
Xv
is,up t o sign,
over
and s i m i l a r l y f o r
1, i t f o l l o w s t h a t
is
G.
W.
We h a v e
since
xW
(xG,xGl G =
1,
i s i r r e d u c i b l e , by t h e o r t h o g o n a l i t y r e l a t i o n s f o r
f i n i t e group
characters.
Remarks : ( a )
I n a b e a u t i f u l paper i n B e r l i n e r Sitzungsberichte,
1900, F r o b e n i u s h a s c o n s t r u c t e d a complete s e t o f i r r e d u c i b l e
S n , 1.e.
c h a r a c t e r s f o r t h e symmetric g r o u p
type
xW
n
An-1
a s a s e t of i n t e g r a l combinations o f t h e c h a r a c t e r s
Using h i s method and t h e p r e c e d i n g c o r o l l a r y one can de-
corgpose t h e c h a r a c t e r o f
An- 1.
t h e Weyl group o f
V
i n Theorem 48 i n c a s e
(See R. S t e i n b e r g T.A.M.S.
(b)
g r o u p o f o r d e r 8.
1951).
i. e. t h e d i h e d r a l
B2
I t h a s f i v e i r r e d u c i b l e modules ( o f dimensions
W
of t y p e
1 7 1 , 1 1 , 2 7 while t h e r e a r e o n l y f o u r X ; , S
Corollary 2 :
i s of type
The s i t u a t i o n of ( a ) does n o t h o l d i n g e n e r a l .
C o n s i d e r , f o r example, t h e g r o u p
(c)
G
t o workwith.
A r e s u l t of a g e n e r a l n a t u r e i s a s f o l l o w s .
I f t h e n o t a t i o n i s a s above and
Lemma 66 ( d ) , t h e n
x
=
Z ( -1 )
n~z
(-1
is as i n
i s an i r r e d u c i b l e character
of
]u(.
and i t s d e g r e e i n
G
Proof:
x
Consider
= C
( -1)n
X,W .
By ( 8 ) on p. 1 4 2 , extended t o
t w i s t e d groups (check t h i s u s i n g t h e h i n t s g i v e n i n t h e proof o f
Lemma 66) ,
?:
det
=
,
an
irreducible character.
i s a l s o one by Cor. 1 above.
We have ~ ~ (= -1 ) ~
i s u n t w i s t e d and t h e b a s e f i e l d h a s
G
Theorem 4'
applied t o
N
Z ( -1)% ( q ) / ~ n ( q ) = q
26.
If
and t o
G
I
=
U
Hence
I,
q
elements,
2
x
G
.
If
G
then by
t h i s can b e c o n t i n u e d
Gn
a s i n ( 4 ) of t h e proof o f Theorem
i s t w i s t e d , t h e proof i s s i m i l a r .
G
We c o n t i n u e w i t h some remarks on t h e a l g e b r a
A
of Theorem
48(b).
Lemma 87:
The homomorphisms o f
- ca
4.
f (wa) = 1 o r
4
f (wa)
Proof:
and
a
i s c o n s t a n t on each
b
simple, l e t
a r e g i v e n by:
K
f o r each s i m p l e r o o t a ,
condition t h a t
For
onto
A
subject t o t h e
W-orbit.
n(a,b)
denote t h e order of
We c l a i m t h a t
(
a and b b e l o n g t o t h e same
wawb i n W.
W-orbit i f and o n l y i f t h e r e e x i s t s a sequence o f s i m p l e r o o t s
a = ao 2 '1
,
,ar
=
b
such t h a t
)"
he e q u a t i o n
show t h a t
wa-
and
a
1 with
n
i s odd f o r every
I.
odd can b e r e w r i t t e n t o
a r e c o n j u g a t e , s o t h a t i f t h e sequence
wai+l
1
e x i s t s then
=
n(ai,ai+l)
and
b
a r e conjugate.
If
a
and
b
a r e conjugate,
and wb , and t h i s remains t r u e when we p r o j e c t
wa
i n t o t h e r e f l e c t i o n group o b t a i n e d b y imposing on Tld t h e a d d i t i o n a l
then so are
relations :
group
wa
(wcwd)
and
wb
2
=
1 whenever
n (c ,d)
i s even.
I n t h i s new
must b e l o n g t o t h e same component, s o t h a t t h e
r e q u i r e d sequence e x i s t s .
4
By
4
e x a c t l y when
f (wa) = f (wb)
e x a c t l y when
f
Theorem 4 8 ) .
Since
(a)
t h e c o n d i t i o n o f t h e lemma h o l d s
(
whenever
n(a,b)
- ca
A
(solve t h e quadratic),
i.e.
(of t h e proof of
e x a c t l y when
f
preserves
we have t h e lemma.
By f i n d i n g t h e a n n i h i l a t o r i n
Remark:
($1
preserves t h e relations
f (wa) = 1 o r
i s odd,
A
of t h e kernel of each of
t h e homomorphisms o f Lemma 87, we g e t a one-dimensional
I
in
A.
This c o r r e s p o n d s t o a n i r r e d u c i b l e submodule o f m u l t i p l i c i t y
1
in
V, r e a l i z e d i n t h e l e f t i d e a l
of
KG1
KG.
ideal
By working o u t t h e
c o r r e s p o n d i n g i d e m p o t e n t , t h e d e g r e e o f t h e submodule can b e found.
Exercise:
with
4,
(a)
=
If
l ~ l , and
t r i v i a l one.
[l,wa ]
I
a,
show t h a t
t h a t t h e corresponding
(Hint:
relative t o
show t h a t
4
f (wa) = 1 f o r a l l
by w r i t i n g
and w r i t i n g
W
(a)
4
I =KC q w
W
KGmodule i s t h e
a s a union o f r i g h t c o s e t s
h
A
i n t h e form (wa-l)(w,+ ca) = 0 ,
i s as i n d i c a t e d . )
(b)
If
A
f(wa)
= -c
a
for a l l
a,
show t h a t
and t h a t i n t h i s c a s e t h e dimension i s 1 U 1.
-1 , show t h a t e2 =me
i s t i l e g i v e n sum and c = qw
I = K 1 ( d e t w)$,
(Hint : i f
with
m
= C
e
c
W
(c)
87,
W
=
JG/BI/I
For
uJ.)
G
=
B (q)
2
work o u t a l l ( f o u r ) c a s e s o f Lemma
hence o b t a i n t h e d e g r e e s of a l l ( f i v e ) i r r e d u c i b l e components
(dl
Remark:
Same f o r
A2
and f o r
G2
.
It c a n b e shown t h a t t h e module o f ( b ) i s i s o m o r p h i c t o
t h e one w i t h c h a r a c t e r
xG a s
i n Lemma 86, Cor.2.
I f we r e d u c e
256
t h e element
mod
C(det w ) I U / x w
lFwG
IBI]u(
(which i s
we s e e from t h e proof o f Theorem 46 ( e l
P,
of
I
t h a t t h e given
h = 1 and
module r e d u c e s t o t h e one o f t h a t theorem f o r which
every
e)
,ua =-1. This l a t t e r module can i t s e l f b e shown t o b e
isomorphic t o t h e one i n Theorem 43 f o r which
f o r every
a.
< h ,a
>
=
q(a)-l
From t h e s e f a c t s and WeylTs formula t h e c h a r a c t e r
o f t h e o r i g i n a l module can b e found up t o s i g n and t h e r e s u l t u s e d
t o prove t h a t t h e number of
of
G
is
1u12
.
p-elements
p)
(of o r d e r a power o f
For t h e d e t a i l s s e e R. S t e i n b e r g , Endomorphisms
o f l i n e a r a l g e b r a i c g r o u p s , t o appear.
F i n a l l y , we s h o u l d mention t h a t t h e a l g e b r a
i n v o l u t i o n g i v e n by
c, ---3 1 and
A
>
-
4
wa >
KW
l
- c a -Wa
4
w
reduces t o
A
admits a n
for all
a (which i n c a s e
-+ ( d e t
w)w).
The p r e c e d i n g d i s c u s s i o n p o i n t s up t h e f o l l o w i n g
Problem:
Develop a r e p r e s e n t a t i o n t h e o r y f o r f i n i t e r e f l e c t i o n
groups and u s e it t o decompose t h e module
V
(or t h e algebra
A)
of Theorem 4 8 .
It i s n a t u r a l t h a t i n s t u d y i n g t h e complex r e p r e s e n t a t i o n s o f
G
we have c o n s i d e r e d f i r s t t h o s e induced by c h a r a c t e r s on
f o r representations of c h a r a c t e r i s t i c
set.
I n characteristic
t h e simplest case
0,
p
this
B
since
l e a d s t o a complete
however, t h i s i s n o t t h e c a s e , a s even
shows. One must d e l v e deeper. There2
f o r e , we s h a l l c o n s i d e r r e p r e s e n t a t i o n s o f G induced by (oneG = SL
dimensional) characters
h
on U
.
We can n o t expect such a
r e p r e s e n t a t i o n ever t o be i r r e d u c i b l e s i n c e i t s degree ~ G / u ( i s
l/ 2
too large (larger than I G ]
, b u t what we s h a l l show i s t h a t
if
h
i s s u f f i c i e n t l y g e n e r a l t h e n a t l e a s t it i s m u l t i p l i c i t y - f r e e .
G
I n o t h e r words, EndG(Vh)
(If
i s A b e l i a n , hence a d i r e c t sum o f f i e l d s .
i s n o t s u f f i c i e n t l y g e n e r a l , we can e x p e c t t h e Weyl g r o u p
h
t o p l a y a r o l e , a s i n Theorem 4 8 . )
Before s t a t i n g t h e t h e o r e m , we prove two Iemn1.a~.
Lemma 88:
Let
b e a f i n i t e f i e l d and
k
h
a nontrivial character
cI.
from t h e a d d i t i v e group of
b e w r i t t e n uniquely
Proof:
The map
k
: t ->h(ct)
h
C
c -+Ac
w
For
If
(a)
W
E
a
Then e v e r y c h a r a c t e r c a n
f o r some
c
i s a homomorphism o f
k
and i t s k e r n e l i s c l e a r l y
Lemma 89:
K
into
E
.
k
into its dual,
0.
t h e following conditions a r e equivalent.
and
art. p o s i t i v e m o t s r;nd one o f thcrr, i s
wa
simple, then so i s t h e other.
If
(b)
a
i s s i m p l e and
is positive, then
wa
wa
is
simple.
(c)
w
u s u a l and
w,
=
w ow n
+
(c)
of s i m p l e r o o t s , w i t h wo
t h e corresponding o b j e c t of
2'
There a r e
Proof:
n
f o r some s e t
w,
Because
maps
.
.
W,
w.
possibilities for
(a)
as
n
-n
onto
n
mutes t h e p o s i t i v e r o o t s w i t h s u p p o r t n o t i n
(+) per-
and
(sane proof a s
f o r Appendix '1.11).
(a)
+
(b)
Obvious.
(b) 9 (c)
hence s i m p l e , by
then
wa
Then
wa
0.
=
wb
w.
Write
+
n
Let
wc.
be t h e s e t of simple r o o t s kept p o s i t i v e ,
We c l a i m :
a
=
Here
b + c
wc
0
>
(
if
with
supp b
a
0
&
and
supp a
n , supp c
by t h e c h o i c e o f
n,
-rc ,
GTTand
n.
258
s u p p wc
wn
not i n
in
n
n
then
Thus wa c 0. If
wb.
ww,a
(x)
< 0 by
and
a
i s a simple root
w h i l e if a i s
(::k),
t h i s h o l d s by t h e d e f i n i t i o n o f
whence
n
.
Thus
ww, -- wo
,
(c).
Theorern 49:
and
2 supp
Let
>-
h :U
b e a f i n i t e , perhaps t w i s t e d , Chevalley group
G
J.
K' a c h a r a c t e r s u c h t h a t
1 if
a
i s simple,
= 1 if
a
is
p o s i t i v e b u t n o t s i m p l e . Then :V
is multiplicity-free.
I n other
G
words, ~ n ( vd ~ )~i s A b e l i a n , o r , e q u i v a l e n t l y , t h e s u b a l g e b r a A
of
spanned by t h e elements
KG
Here
Z h (u-')u
Uh =
UEU
c h o s e n ' a s , i n Lemma Ef3( b )
Remarks :
(a)
a
If
I
t h e case,
( k( =
3.
H
,w
W) i s Abelian.
E
w
and we assume t h a t t h e
are
.
=
Xa
c
u , so
1 i s s u p e r f l u o u s , b u t t h i s i s n o t always
o r F4 w i t h k
= 2 or for
with
2
G2
I n these l a t t e r groups, t h e r e a r e other p o s s i b i l i t i e s ,
e.g.
for
B
which b e c a u s e o f t h e i r s p e c i a l n a t u r e w i l l n o t b e g o n e i n t o h e r e .
The p r o o f t o f o l l o w i s s u g g e s t e d by t h a t o f G e l f a n d
(b)
1
(h E
i s not simple, then usually
t h a t t h e assumption hlxa
I
I
9
Uh hGUh
1
and Graev, Coalady, 1963, who h a v e g i v e n a p r o o f f o r
I
I
I
and
SLn
announced t h e g e n e r a l r e s u l t f o r t h e u n t w i s t e d g r o u p s . T. Yokonuma,
C. Renduss, P a r i s , 1967, has a l s o given a proof f o r t h e s e l a t t e r
g r o u p s , b u t h i s d e t a i l s a r e u n n e c e s s a r i l y com.plicated.
P r o o f o f Theorem. 49: The f a c t t h a t
A
i s A b e l i a n w i l l f o l l o w from
t h e e x i s t e n c e o f an ( i n v o l u t o r y ) antiautomorphism
that
(a)
f U
=
U
.
f
of
G
such
259
(c)
fn = n
have
For s i n c e
UnU
For e a c h d o u b l e c o s e t
f
(here
n
N = X
E
extended t o
antiautomorphism
Uhn Uh
such t h a t
HB).
i s an
A
and ' t h e n r e s t r i c t e d t o
KG
0 , we
and a t t h e same t i m e t h e i d e n t i t y (by
( c ) ) it i s clear t h a t
i s Abelian.
A
(a)? (b)?
The e x i s t e n c e o f
f
will
b e proved i n s e v e r a l s t e p s .
set
w
By Lemma 89 we need o n l y p r o v e t h a t i f
a
UhnUh
0
and
n
E
=
wow,
f o r some
of s i m p l e r o o t s .
n
Proof:
wa
$
HE, then
(1) I f
positive then
wa
i s simple.
i s s i m p l e and
Writing t h e f i r s t
above
Uh
X
component on t h e r i g h t , and t h e second w i t h t h e Xa
wa
component on t h e l e f t , we g e t %a , h "'a , h n-I 0 . S i n c e h i s
n o n t r i v i a l on Xa i t i s a l s o s o on \2,
whence wa i s s i m p l e
with the
by t h e a s s u m p t i o n s on
h,
which p r o v e s (1).
The c o n d i t i o n i n ( 1 ) e s s e n t i a l l y f o r c e s t h e c o r r e c t
,. =
- wo a .
.L
definition of
a
4
1
.
.
f.
We s e t
If
i s simple, so i s
a
I n o.rder t o s i m p l i f y t h e d i s c u s s i o n i n one o r two s p o t s we
assume h e n c e f o r t h t h a t
If
a'
G
is
i t s r o o t s y s t e m ) i s indecomposable.
(i.e.
G
u n t w i s t e d , we s t a r t w i t h t h e g r a p h automorphism correspond-
,
I
.
ing to
'
inversion
( s e e t h e C o r o l l a r y on p. 1561,
x ->x
that the result
on
U.
on
h
-1
,
f
s a t i s f i e s , not only
compose i t w i t h t h e
and f i n a l l y w i t h a d i a g o n a l automorphism s o
fU
=
U
but also
hf = h
T h i s i s p o s s i b l e b e c a u s e o f Lemma 89 aizd t h e assumptions
i n t h e theorem.
If
g r a p h automorphism ( b e c a u s e
t h e e x p l i c i t - isomorphism
is
G
.L
-,
t w i s t e d , t h e n we may omit t h e
i s then t h e i d e n t i t y ) ,
xa/Bxa
2 k
of
(2)
and u s e
o f t h e proof of
Theorem. 36 i n combination w i t h Lemma 89 t o a c h i e v e t h e second
260
condition.
We s e e t h a t
(2)
i s a n i n v o l u t o r y a n t i a u t o m o r p h i s m which s a t i s f i e s
f
t h e required conditions
( c.
also satisfies
(a)
and
(b)
If
a
(5)
If
a
f i x e d by
-,-
Xa
a,
then
=
a,
t h e r e e x l s t s a n o n t r i v i a l element of
.I,
of' t y p e
i s t h e i d e n t i t y on
f
, - t t 8u )
(4).
t h i s f o l l o w s from
A1
p $ 2,
if
and
o f Lemma 63 ( c )
( t, u )
t = 0,
For
u = 1 if
p = 2,
For types
since
a n d 2 ~ 2w e m a y c h o o s e ( 0 , l ) and
C2
i s t h e i d e n t i t y on { ( o , u ) ] a n d { ( o , u , o ) 1.
f
(6)
-
The e l e m e n t s
(6a)
f a - w
(6b)
If
a
wa
G
E
b
and
e r e s i m p l e and
n
E
N
n
as i n
(6b)
f
x1,x2
E
X-a
E
,
Xa
with
since
as i n (51,
and c h o o s e
a n t i a u t o m o r p h i s m and f i x e s
the
Xa
x
E
Xa
(a simple)
Xa
.
w r i t e it a s
wa a c c o r d i n g l y .
X-a
(0,1,0)
,
then
and t h e i n n e r automorphisms
From e a c h o r b i t we s e l e c t a n e l e m e n t
xa
of
i s such t h a t
h ( n x n- l ) = h(x) f o r a l l
Under t h e a c t i o n o f
by e l e m e n t s
Xa
a.
- .
choose
Xa
may b e s o c h o s e n t h a t :
f o r every simple root
and
n Xan--1- Xb
nGan -1 = wb
Proof:
.
(check t h i s , r e f e r r i n g t o t h e c o n s t r u c t i o n o f
f ( t, u ) = ( t
.
xa/fix a
=
2 ~ 2we c h o o s e t h e e l e m e n t
t = 2, u = 2
f)
H.
h E
f.
Proof: For
type
T1.,
We must p r o v e t h a t i t
A s c o n s e q u e n c e s o f t h e c o n s t r u c t i o n we h a v e :
f o r every
(4)
.
,
in
form o r b i t s .
.I<
If
a''
(s) xa
Since
it a l s o f i x e s
=
=
a,
we
x Fx
with
l a 2
f i s an
-
wa
by t h e
261
u n i q u e n e s s o f t h e above form.
arbitrarily.
f a,
a"
If
We t h e n u s e t h e e q u a t i o n s
fGa
=
i
w
=
wb
We must show t h i s can b e done c o n s i s t e n t l y , t h a t we always
r e t u r n t o t h e same v a l u e .
=
aj+l
Let
..
al,a2,.
al - an
simple r o o t s such t h a t
=
an b e a sequence o f
and f o r each j e i t h e r
a
)
a :: o r e l s e t h e r e e x i s t s
j
n . such t h a t t h e assumptions i n
J
i n p l a c e of a , b , n .
Let g d e n o t e
(6b) holds w i t h
a. anJ ' ~ + 1J'
t h e p r o d u c t of t h e c o r r e s p o n d i n g sequence o f
We must show t h a t
and i n f a c t
g
g
fixes
ma
c
and t h e u s u a l formulas f o r
c o r r e s p o n d i n g c o n d i t i o n on
Lemma 89 t h a t
If
X
a
-w,
by
If
&X
=
0,
then
,
whether
(
f 0,
a
then
f
so t h a t
c = 1,
.
We have
xa/65 Xa '
a c t s on
m u l t i p l i c a t i o n by a s c a l a r
f
5;:
and
n a
G t o t h e o r b i t of
of ( 6 a ) and ( 6 b ) t o e x t e n d t h e d e f i n i t i o n o f
a.
$ 1,
xa e Xa,xa
we choose
g
G
.
in 's
J
,
gXa - Xa
identified with
=
X
-a
'
by
k,
a s f o l l o w s from t h e d e f i n i t i o n o f
Since
in.
and each
g
hg
=
i n , it
h
by t h e
f o l l o w s from
x,/&
i s t h e i d e n t i t y on
f i x e s t h e element
g
f t s and
.
X
a
above, hence a l s o
xa
i s a n automorphism o r an antiautomorphism.
i s twisted so t h a t
-1,
a"
=
a.
If
g
i s an
automorphism, t h e n by t h e proof of Theorem 36 from ( 5 ) on i t s
<
restriction to
while i f
g
Xa,X-a
>
is t h e i d e n t i t y so t h a t it f i x e s
-wa .
t h e choice of
If
G
,
If
f
so t h a t it f i x e s
-
wa by
i s u n t w i s t e d , t h e above proof i s q u i t e s i m p l e .
We assume h e n c e f o r t h t h a t t h e
(7)
wa
i s a n antiautomorphism t h e n by t h e same r e s u l t i t s
r e s t r i c t i o n c o i n c i d e s w i t h t h a t of
Remark:
-
-w = w-a w-b ...
wa
are a s i n (6).
a s i n Lemma 8 3 ( b ) and
-1 -1
w : : = w0w wo
,
C
-
-
then
.
f w = w*
Proof:
Since
. ..
wawb
...
i s minimal,
.,*
i s also.
a
i s a n a n t i a u t o m o r p h i s m i t f o l l o w s from
wb:+ w
I
.
'
(check t h i s . )
f
... -wb , w-a ,--
-
that
(6a)
Since
fw =
- ' - Wb:
-
.0-
= W-I.
Wa"
.
"
(8) I f
w
( 9 ) If
Proof:
Then
i s a s i n (1) t h e n
n
=
%.
i s s i m p l e and
f n = n.
w=wow,.
By ( l ) , n e K H w i t h
wa
fw
i n t h i s case (see (7))
w" - w
Proof:
i s as: i n (1) t h e n
h ( n x n-'1
=
, that
w,
ntj,n-'
(
~ ( w )= N (wow,)
= N
=
--
- Wwa
-
Thus
wo
(:;<::)
=
,
(wo) - N (w,)
--
-ww,
i t f o l l o w s t h a t i f we p u t t-
w
and
w,
we w i l l g e t one f o r
b y Lemma 83 ( b ) , a n d s i m i l a r l y
= w
WW,W
since
n
g e t h e r minimal e x p r e s s i o n s f o r
wo
so t h a t
-1
f r o m which we g e t , on p i c k i n g a minimal e x p r e s s i o n
(6b),
for
nGan
,
x e Xa
h(x) f o r a l l
b y t h e i n e q u a l i t y i n t h e p r o o f o f (1). Thus
by
a c x .
Assume
:
;
.
I f now we w r i t e
n
=
L
wo
Gh
,
=
- w
+w
n
then
71
h
w,
commutes w i t h
fn
= f h - fw since
=
Fi0h~i1w b y
4-
= ww,
h
-w,-
= wh
=
Thus
n
f
1
by
f
and
(::)
(
Hence
)
i s a n antliautornorphism
(3)
and
-.
(8)
--
since
wo
=
ww,
since
h
commutes w i t h
w,
.
s a t i s f i e s condition
i s complete.
(c)
and t h e p r o o f o f Theorem 49
,
263
Exercise:
(a)
(l), t h e n
UkwUk
H,
as i n
denotes t h e k e r n e l of t h e s e t
.n t h e n t h e dimension o f
of s i m p l e r o o t s
w
0.
Deduce t h a t i f
(b)
i r r e d u c i b l e components o f
Remark:
{ F ~ ] i s a s i n ( 6 ) and
Prove t h a t i f
VkG
,
A,
i n Theorem
hence t h e number of
49 i s
C
] H ~ I.
The n a t u r a l g r o u p f o r t h e p r e c e d i n g theorem seems t o b e
t h e a d j o i n t g r o u p extended by t h e d i a g o n a l automorphisms, a g r o u p
o f t h e same o r d e r a s t h e u n i v e r s a l g r o u p , b u t w i t h something e x t r a
a t t h e t o p i n s t e a d o f a t t h e bottom.
t h a t t h e dimension above i s j u s t
-
For t h i s g r o u p ,
I I ( 1~~ 1
-
+ 1)
=
n o t a t i o n of t h e e x e r c i s e j u s t b e f o r e Lemma 83.
t h is case
Remark:
VhG
i s i n d e p e n d e n t of
h
prove
GI,
I I q(a)
i n the
Prove a l s o t h a t i n
.
The problem now i s t o decompose t h e a l g e b r a
49 i n t o i t s s i m p l e (one-dimensional) components.
A
of Theorem
If t h i s were done,
i t would b e a m.ajor s t e p t o w a r d s a r e p r e s e n t a t i o n t h e o r y f o r
A s f a r a s we know t h i s h a s been done o n l y f o r t h e g r o u p
For n o t e v e r y i r r e d u c i b l e
i n one i n d u c e d by a c h a r a c t e r o n
(see
It would n o t , however, b e t h e
Gelfand and Graev, Doklady, 1 9 6 2 ) .
complete s t o r y .
A1
G.
U,
c i p r o c i t y , c o n t a i n s a one-dimensional
i . e.
G-module
,
i s contained
by F r o b e n i u s r e -
U-module,
a s t h e following,
o u r f i n a l , example, due t o 1vI. Kneser, shows ( a l t h o u g h i t i s f o r
some t y p e s o f groups s u c h a s
A,).
A s remarked e a r l i e r , r e d u c t i o n mod 3 y i e l d s a n isomorphism
o f t h e subgroup
group
W
W+ o f e l e m e n t s o f d e t e r m i n a n t
of t y p e
E6
o n t o t h e group
G
=
1 o f t h e Weyl
SO5 ( 3 ) ,
the adjoint
264
group of type
B2(3).
If we r e v e r s e t h i s i s o m o r p h i s m a n d e x t e n d
t h e s c a l a r s we o b t a i n a r e p r e s e n t a t i o n o f
V.
The a s s e r t i o n i s t h a t
f i x e s no l i n e o f
,
V.
U,
a
on a complex s p a c e
3-Sylow s u b g r o u p of
G,
Consider t h e f o l l o w i n g diagram.
T h i s i s t h e Dynkin d i a g r a m o f
(a7
i .e .
G
i s t h e unique r o o t i n
-
w i t h t h e lowest r o o t
ad j o i n e d
E6
a7
D ( s e e Appendix I I I . ~ ~ u)n,i q u e
because a l l r o o t s a r e conjugate i n t h e present case. It i s
connectled a s shown b e c a u s e o f symmetry and t h e f a c t t h a t e a c h
p r o p e r s u b d i a g r a m must r e p r e s e n t a f i n i t e r e f l e c t i o n g r o u p . )
choose a s a b a s i s f o r
V
the
1
a s
with
a
3
of t h r e e bases of mutually orthogonal planes.
r o t a t i o n o f 12C0
i n t h e plane
ial,a2>
t h e o t h e r two p l a n e s , and s i m i l a r l y f o r
group
omitted, a
w1w2
We
union
acts as a
and a s t h e i d e n t i t y i n
w4W5
and
W6W7
.
The
W+ a l s o c o n t a i n s a n e l e m e n t p e r m u t i n g t h e t h r e e p l a n e s
c y c l i c a l l y a s shown, b e c a u s e o f t h e c o n j u g a c y o f s i m p l e s y s t e m s
a n d t h e uniqueness o f l o w e s t r o o t s , and t h e f o u r e l e m e n t s g e n e r a t e
a
3-subgroup of
W+
.
It i s now a s i m p l e m a t t e r t o p r o v e t h a t
t h i s s u b g r o u p f i x e s no l i n e o f
V.
APPENDIX ON FINITE REFLECTION GROUPS
The r e s u l t s (and some o f t h e terminology i n what f o l l o w s )
a r e m o t m a t e d by t h e t h e o r y o f semisimple L i e a l g e b r a s , b u t no
knnowledge o f t h i s t h e o r y i s assumed. The main r e s u l t s a r e s t a r r e d .
V
w i l l be a finite-dimensional r e a l o r r a t i o n a l Euclidean
By a r e f l e c t i o n (on
space.
h y p e r p l a n e H.
=
H,
If . a i s a nonzero v e c t o r o r t h o g o n a l t o
the
ca , i s g i v e n by
r e f l e c t i o n , denoted
1)
i s meant a r e f l e c t i o n i n some
V)
( y EV).
p-z(p,a)/(a,a).a
We o b s e r v e t h a t
i s a n automorphism o f
V, o f o r d e r 2.
A useful f a c t is:
(2)
If
w i s a n automorphism o f
then
V,
P
To prove t h i s , a p p l y b o t h s i d e s t o
and t h e i n v a r i a n c e o f
C
(
,
under
-%
-
maw-'
a*
t h e n u s e (I)
V,
w.
w i l l d e n o t e a f i n i t e s e t of nonzero elements o f
V
such
that :
(3) a E Z => - u E Z
(4) a E Z
The e l e m e n t s of
and
ka@C
if
k +
-+ 1 .
crZ = C .
a
C
w i l l b e c a l l e d r o o t s , and
g r o u p g e n e r a t e d by a l l
ra ( a E Z )
( 5 ) Lemma. The r e s t r i c t i o n of
W
W
w i l l denote t h e
.
to
C
i s faithful.
w
F o r , each
of
W
f i x e s p o i n t w i s e t h e o r t h o g o n a l complement
C.
(6) Cor.
W is finite.
( 7 ) Examples.
(a)
i s t h e r o o t system o f a semisimple
C
If
L i e a l g e b r a o v e r t h e complex f i e l d , t h e n
Weyl g r o u p . ( b )
eeg.
i s t h e corresponding
i s any f i n i t e group g e n e r a t e d by r e f l e c t i o n s ,
W
If
W
t h e group o f symmetries o f a r e g u l a r s o l i d ,
C
may b e t a k e n
as t h e s e t of u n i t normals t o t h e h y p e r p l a n e s i n which r e f l e c t i o n s
of
( 8
W
t a k e place.
Definitions,
A s u b s e t of r o o t s i s c a l l e d a p o s i t i v e system
i f it c o n s i s t s of t h e r o o t s which a r e p o s i t i v e r e l a t i v e t o some
ordering of
a subset
V.
V+
of
(Recall that t h i s involves t h e s p e c i f i c a t i o n of
V
which i s c l o s e d under a d d i t i o n and u n d e r
A
m u l t i p l i c a t i o n b y p o s i t i v e sca.l.2-s and s a t i s f i e s t r i c h o t o m y . )
s u b s e t of r o o t s , s a y
,
l i n e a r l y independent s e t ,
i s a s i m p l e system i f
and
(b)
combination o f t h e elements of
(a)
is a
every r o o t i s a l i n e a r
i n which a l l nonzero c o e f f i c i e n t s
a r e e i t h e r a l l p o s i t i v e o r a l l negative.
:k
( 9 ) P r o p o s i t i o n . ( a ) Each s i m p l e system i s c o n t a i n e d i n a u n i q u e
p o s i t i v e system.
(b)
Each p o s i t i v e system c o n t a i n s a unique
s i m p l e system.
If
i s a s i m p l e system, t h e n c l e a r l y t h e
11
a l l positive'1
r o o t s i n ( 8 b ) form t h e unique p o s i t i v e system c o n t a i n i n g
whence ( a ) .
a s u b s e t of
Now l e t
P
P
b e any p o s i t i v e system.
which g e n e r a t e s
P
Let
under p o s i t i v e l i n e a r
,
be
c o m b i n a t i o n s and i s minimal r e l a t i v e t o t h i s p r o p e r t y .
a , fl E r r , a $ . B
(';')
aCI I3
=
'
-
p
ccr
with
rap = I:c Y y
c
=> ( c , ~ <
)- 0 . Assume n o t , s o t h a t
> 0 . Assume g p E P , s o t h a t
TT ,
(y s
cy
>_ 0 )
,
>
-
while if
1
,
rr
of e l e m e n t s of
p o s i t i v e c o m b i n a t i o n of t h e
hence of
Now a l i n e a r r e l a t i o n on
a s a p o s i t i v e combination
,
equally a contradiction.
P
rr
= 0
P
= o- and u s i n g
(
rr
and
and t h e n e v e r y
a~ = 0 because t h e
S i m i l a r l y e v e r y bR = 0
Thus
i n d e p e n d e n t , i s a s i m p l e system.
P
Lemma.
Let
1-r .
(b)
p
E P*
then use
0
<
write
(
Pick
and
e
are
consists
.
c
f 0 =>
= ~ c ~ n E( c~
(c:
5) <_ 9
.
( e ,a) > 0 .
so t h a t
c E
the positive
P
, CI c>_ 0) a s i n ( U b ) , and
e,?) = I:ca( P , c ) .
"(11) Main lemma.
.--
oa o = - a
P
,
c, p E
(a)
there exists
For ( b )
,o- ) <_ 0
is
P
be a s i m p l e system and
system c o n t a i n i n g
= (('
which a r e n o t p o s i t i v e c o r i b i n a t i o n s of
o t h e r s , hence i s u n i q u e l y d e t e r m i n e d by
(10)
.
0
From t h e d e f i n i t i o n of
a simple system any simple system contained i n
of t h o s e e l e m e n t s of
B
>_
G?S
.
a l l positive.
.
(::)
a,,bg
(e,fl
we g e t
)
,
I: aca = I: b p
may be w r i t t e n
w i t h t h e two sums o v e r d i s j o i n t p a r t s of
9iriting t h i s a s
rr
0
- d;lP E P l e a d s t o a c o n t r a d i c t i o n , whence
Similarly
whence ('
1, t h e l a s t e q u a t i o n ,
a c o n t r a d i c t i o n t o t h e m i n i m a l i t y of
it e x p r e s s e s
,
<
I f cs
9 as a
written s u i t a b l y , expresses
o t h e r e l e m e n t s of
Then
Let
rr
,
rcc:(P\c:) = P\
P\c7
P
a
be a s i n ( 1 0 )
and
rr .
c~
Then
.
P = 2 c g ? (cg
>_ 0 ; 0
~ n ) (3)
BY
some
,
B
>
0 (B
a).
Application of
% p EP
Hence
does n o t change t h i s
Ca
'
\ a.
( 1 2 ) Theorem. Any, two s i m p l e ( o r positive) systems a r e c o n j u g a t e
under
W.
P
By (9) we need o n l y c o n s i d e r two p o s i t i v e systems, say
and
P
P = P
t
.
t
.
Assume
relative t o
IG~P n (-P
1)
n-1,
=
1
t
n (-
P
a
whence
cap
a
Then t h e r e i s a r o o t
such t h a t
P
t
> 0.
n
i n d u c t i v e assumption
P.
t
n = \ ~ n ( )-I . ~ If
Weuseinductionon
t
t
simple
By (11),
P).
IP~-c-P
n = 0 ,
I
=
is conjugate t o
n-1.
BY t h e
P ; hence s o i s
.
n ,P
Henceforth
(13) D e f i n i t i o n .
w i l l be a s i n (10)
p EZ , p
If
ca G as i n
a Err
a:ld 3 i r l t t e n h t
If
Lemma.
p E P,
Let
t h e minimum v a l u e o f
pt
.
p
T
{ahla En ] .
Wofm p
on t h e s e t
is
be a minimum p o i n t and assume, i f p o s s i b l e , t h a t
By ( l o b )
there is
a
PT
contradiction t o t h e choice of
(15) Corollary.
Won = E .
then
P nn.
Wo
whence by (1) h t G , - P ' ~h t
(b)
ht
,
.
lo be t h e group g e n e r a t e d by
and i s t a k e n on o n l y on
Let
($b)
= Z
i s c a l l e d t h e h e i g h t of
(&)
and f i x e d .
(a)
i.e.
If
so t h a t
(
P
,a) > 0,
and by (11) r a p >fO ,
p
P EP
Every r o o t
En
t
.
,P
p
a
r,t h e n
ht
I s conjugateunder
P>
Wo
1.
to a
1
simple root.
By ( 1 4 ) , ( a )
-
then
'j
=
f > 0,
i s c l e a r and so i s ( b )
-
whence
If f'
0.
0,
( w E W o , a En), s o t h a t
wa
f=' =
>
i f 'j
(wras)a.
(16)
Theorem. W
P
If
-
WCW0
p
i s a r o o t we have
( l 5 b ) , whence
{ca
la E TT
i s g e n e r a t e d by
1.
,a E
1,
element o f
Wo
wa (w E Wo
=
, an
cy = wbaw -1 ( s e e ( 2 ) )
i n (&I.
I. e. W = Wo
by
. Hence
W = W o .
and
I1
(17)
Definition.
roots
p
The f u n c t i o n
w E W, ~ ( w ) w i l l d e n o t e t h e number of
For
such t h a t
>
f'
N
0
w
and
p
0.
I n o t h e r words,
Prove t h i s .
(19)
Lemma.
(a)
If
If
(at)
(b)
If
1
(b )
Let
whence
get
(b)
w E W,
Assume
If
w-Ia
>
w-lo
wa> 0,
a!
N(CT-W)
0,
N(caw)
then
N(ma)
then
and
To g e t
(at)
(bt )
replace
=
N(wra)
~ ( w )= P ~ w - ' ( - P ) .
(a).
.
0, t h e n
then
wu. < 0 ,
E-j-
w
by
=
+ 1.
N ( w ) - 1.
N(w)+l.
=
~ ( w -) 1.
S(craw) = w - l a v S ( W ) ,
Then
replace
= N(W)
w
w-l
by
caw
and u s e
in
(a),
(18).
and t o
(20)
Problem.
1
N(w) + bI(wt) mod 2.
(b)
(21)
w
Lemma
1
N(ww) (_N(w) + N ( w )
(a)
Assume
=
det w
wlw2..
=
.
llN(~)
(-
1
and
=
E w).
N(ww )
w w
t
.wn (wi= cra . ' a i E T I ) .
If
1
~ ( wq) n , t h e n f o r some I
( a ) ai W ~ + ~ W ~ +* w
~ j.a .j + l
(b)
Wi+1Wi+2.'.Wj+l
(c)
W = W
1 2
t
* * *
* *
-
< n-1.
Since
wi+l' ''Wjaj+l > O
by
(111,
..
N(w)
By (19b) and
j
ajcl
.W
>
n7
n,
q
.
we g e t ( b ) ,
y
with
w.
and
J +l
whence
i &j ,
missing.
f o r some
wi ( w ~ + ~.wjaj+=
*.
)< 0
we have
Using ( 2 )
we h a v e :
1 ,
W ~ W ~ . : . W ~ ~ ~ +0 ~
0,
f o r some
which i s ( a ) .
a = aj+l 9
-1 i - j - -
= W1
.W.
...W-J
1+1
t
w
, (
with
w
W~+~...WCZ-
and
=a-
j ~ + 11
=
w
~
+
~ and
*'Wj
and t h e n r e p l a c i n g t h e l e f t s i d e of ( b )
by t h e r i g h t s i d e i n t h e p r o d u c t f o r
w and u s i n g
w:
= 1, we
get (c).
Problem.
N(W)
(22)
q
Prove, conversely, t h a t ( a ) , (b) o r ( c ) implies
n.
Cor.
w E W,
If
then
w
minimal e x p r e s s i o n o f
~ ( w ) i s t h e number of t e r m s i n a
a s a p r o d u c t of r e f l e c t i o n s correspond-
i n g t o simple r o o t s .
*
w
...
w w
w
1 2
n
((a) or (b)), ~ ( w 5
) n,
Let
=
(23)
Theorem. For
then
w = 1.
w
E W,
b e a minimal e x p r e s s i o n .
and by
if
By (19)
> n.
(2 lc) , N ( ~ )
wP = P
or
w TI =
The t h r e e a s s u m p t i o n s a r e c l e a r l y e q u i v a l e n t .
or
Now
N (w) = 0 ,
~ ( w =) 0
2.7%
w
i m p l i e s t h a t t h e minimal e x p r e s s i o n o f
w
whence
*
(24)
i n ( 2 2 ) i s empty,
1.
=
Theorem.
i s simply t r a n s i t i v e on t h e p o s i t i v e systems,
W
and a l s o on t h e s i m p l e systems.
By ( 1 2 ) and ( 2 3 ) .
( 2 5 ) Problem.
choose a minimal e x p r e s s i o n a s
W,
...
=
pi
and s e t
w
For
ww
wn (wi =
1 2
= wlw2...wilai
w
i n (221,
(a)
.
~ n ( s)o t h a t
rai, ai
Prove t h a t
complete l i s t of a l l r o o t s f' such t h a t
(b)
-P
Since
( n = N(%)
=
9.
(1- i ( n )
1
P > 0 and w-l
such t h a t
IPI)
woP = -P.
as above.
Write
Prove t h a t
(26)
Definition.
Thus
D
D
w i l l denote t h e region
i s a c l o s e d convex cone, and i f
a s i m p l i c i a 1 cone w i t h v e r t e x a t
f a c t t o some
P EV
Every
p
t
in
D
S
= wlw2
1i
c 0.
*
wn
-n
is
..
W.
{ f E ~ ] ,( ae) > O , a E n ] ,
TT spans
i t i s even
V
0.
i s c o n j u g a t e t o some
p -1
such t h a t
P
t
E D,
in
i s a nonnegative
.
combination o f t h e elements o f
Let
wo
(
I11 A fundamental domain f o r
Lemma.
P
(Hint :, (19) , (21) )
a complete l i s t o f a l l p o s i t i v e r o o t s .
(27)
is a
i s a p o s i t i v e system, t h e r e e x i s t s by ( 2 4 ) a
wo E W
unique
n = N(w)),
b e t h i s s e t of nonnegative combinations ( i n o t h e r
words, t h e d u a l cone of
by t h e d e f i n i t i o n
t h e conjugates
5
D)
.
2/ d T
p t - of p
We i n t r o d u c e a p a r t i a l o r d e r i n
i f and o n l y i f
under
W
3-
such t h a t
E S.
p1 ,
V
Among
we
p i c k one which i s maximal r e l a t i v e t o t h i s p a r t i a l o r d e r .
Then
a
E
=j,
and
(27)
pT E D
( 2 8 ) Theorem.
Y
+ y y ==> (y
>0
,a)
(by ( I ) ) , whence
follows.
w E W, YE V ,
Assume
- p.
wp
r o o t , and
7
ca b,
Then
w
=
1.
p
n o t o r t h o g o n a l t o any
$
[Restatement: w
l j w F =P=>,p
i s o r t h o g o n a l t o some r o o t .
P
By ( 2 7 ) we may assume
E D. For
-1
= (a,w
p ) = ( a , ? ) > O . Hence w a E P ,
s o t h a t wP = P .
(29)
w
Then
meI f p
v
c o n j u g a t e s under
W
=
E P,
(wa,? )
for a l l
a E P ,
a
1 by ( 2 3 ) .
i s n o t o r t h o g o n a l t o any r o o t , i t s
are a l l distinct.
(And c o n v e r s e l y , o f
course. )
(30)
Core The o n l y r e f l e c t i o n s i n W a r e t h o s e i n h y p e r p l a n e s
orthogonal t o r o o t s , i . e .
Let
u
be
-r
u $1,u p
p
-.>
to
(32)
some
W.
a
-Lemma.
EX).
H not
The r o o t s b e i n g f i n i t e i n number,
p
.CW,
H,
n o t o r t h o g o n a l t o any r o o t .
Then
by ( 2 8 ) .
S be a s e t of r o o t s such t h a t
( 3 1 ) Problem. Let
generates
cp(f
any r e f l e c t i o n i n a h y p e r p l a n e
o r t h o g o n a l t o any r o o t .
there exists
t h o s e of t h e form
{cr, 1 a E 5 ]
Prove t h a t e v e r y r o o t i s c o n j u g a t e , under
S, and e v e r y r e f l e c t i o n i n
Assume
p ,FED,
wEW, w p
W,
W t o some ca( a E s ) .
=
c . Then
(a)
w
i s a product of simple r e f l e c t i o n s ( i . e r e l a t i v e t o simple r o o t s )
fixing
f .
(b)
P
=c.
For ( a ) we u s e i n d u c t i o n on
N(w)
.
If
N(w)
=
0,
then
w
= 1 by
w a c 0.
and
by
(23).
Then
N(w) > O .
Assume
(0-,wa)
O >
Pick
=(p: a ) > O ,
a
so t h a t
( ? , a ) = O
whence
f a y = P a Since (
= o - , and
7
(19b ) , t h e i n d u c t i v e a s s u m p t i o n a p p l i e d
= ~ ( ~ ) - l
N(WG,)
to
w,
yields (a).
Clearly (a) implies (b).
:
(33) Theorem.
D
i s a f u n d a m e n t a l domain f o r
words, e a c h element o f
of
V
W
on
V.
In other
i s c o n j u g a t e t o e x a c t l y one element
D.
By
(27)
and
(32b)-
(34) Problem. If
D
and
w E W,
show t h a t
?-w?
is
a n o n n e g a t i v e combination o f p o s i t i v e r o o t s .
(35) R e s t a t e m e n t . The r e f l e c t i n g h y p e r p l a n e s ( t h o s e o r t h o g o n a l
t o roots) partition
V
fundamental domain f o r
i n t o c l o s e d chambers, each o f which i s a
W.
For a g i v e n chamber, t h e r o o t s normal
t o t h e w a l l s and i n w a r d l y d i r e c t e d form a s i m p l e s y s t e m , and each
s i m p l e s y s t e m i s o b t a i n e d i n t h i s way.
Prove t h e s e a s s e r t i o n s
and a l s o t h a t t h e a n g l e between two w a l l s o f a chamber i s always
11
1
1
t
2%
n
(36) Theorem. If
S i s any s u b s e t o f
which f i x e s
w
every
1
I
.
a submultnple of
V , t h e subgroup o f
S p o i n t w i s e 1s a r e f l e c t i o n group.
W which f i x e s
W
I n o t h e r words,
S pointwise i s a product of r e f l e c t i o n s
which a l s o do.
I
Remark.
(36) i s a n e x t e n s i o n of
(28).
Verify t h i s .
I
For t h e p r o o f o f
(36) we may assume t h a t
S
i s indepedent,
1
1
i
hence f i n i t e , and u s i n g i n d u c t i o n , r e d u c e t o t h e c a s e where
S
I
P,
h a s a s i n g l e element, say
(27).
Then ( 3 2 a )
(37) Prouern. For each s u b s e t
fl
'
]
1a
{
group g e n e r a t e d by
of
W(n' )
n let
(38)
Theorem.
For
a,$
,
E
let
n(a,$)
denote t h e
W(n'n n")
Prove t h a t
ITJ Generators and r e l a t i o n s f o r
*
by
f y i e l d s our r e s u l t .
CT =
with
D
which may b e t a k e n ' i n
=
W.
denote t h e order
i n W.
(So n ( a , a ) = 1, w h i l e n ( a , $ ) > 1 i f a $ $.)
a B
Then t h e group W i s d e f i n e d by t h e g e n e r a t o r s [6a
1 a En
of
0-0-
subject t o the relations
{Ao-
n(a,b)
=
aCTB
o t h e r words, t h e g i v e n elements g e n e r a t e
r e l a t i o n s imply a l l o t h e r s i n
1
. .wr
=
a , $ ~ 1n.
In
and t h e g i v e n
W
W.
By (16) t h e g i v e n elements g e n e r a t e
r e l a t i o n (*) wlw2.
I
1 (wi =
Suppose t h e
W.
CJ-
a.
ai
l7 ) h o l d s i n W.
1
We w i l l show i t i s a f o r m a l consequence o f t h e g i v e n r e l a t i o n s ,
by i n d u c t i o n on
r = 2s.
If
r.
s = 0
By (20b) o r by (19)
r
i s even,
t h e r e i s n o t h i n g t o show.
suppose
say
s
>
0.
We s t a r t w i t h t h e o b s e r v a t i o n :
(0)
(3)
Case 1.
N (w1w2.
i s equivalent t o
al $ w2w3
Suppose
..ws +l)
=
N( w ~ ~
wi+lwi+2
...wr w1 w2 ...wi = l ( l-< -i cr ) .
...wsas + l . We have
w ~- w~s +-2) ~ - s + l , by
Hence by ( 2 1 ) we have ( 2 1 a ) and (21b)
- -
1 < i <j c
- s.
Since
i , j = 1,s
f o r some
(19a).
i ,j
such t h a t
i s excluded i n t h e p r e s e n t c a s e ,
< s.
b o t h s i d e s o f (21b) have l e n g t h
By o u r i n d u c t i v e
a s s u m p t i o n we may r e p l a c e t h e l e f t s i d e o f ( 2 1 b ) by t h e r i g h t
side i n
(c).
w:
If
i s t h e n r e p l a c e d by
1, t h e i n d u c t i v e
assumption can b e a p p l i e d t o t h e r e s u l t i n g r e l a t i o n t o complete
t h e proof, i n t h e present case.
al $ a3.
Case 2 . Suppose
(If
s
By t h e f i r s t c a s e we may assume
by
(0)
also
a2
=
w w
al
1, t h i s c a s e d o e s n r t o c c u r . )
=
...wsa s + l
w w
3
2
and t h e n
. ~ ~ + , ~ whence
a ~ + ~
•
(*:?I
W2w3"'W s + l = W3w4" ' Ws+2
( ) is substituted into (
If
=
by
(2).
we c a n s h o r t e n
Thus we a r e r e d u c e d t o showing t h a t
(
a s above.
)
i s a consequence o f t h e
(2k:c;c)
o r i g i n a l r e l a t i o n s i n (38) , i. e. t h a t
W W W
3,2 ~ " ' W s + l w s + ~ w s + l * *=
*W
1~ i s .
Since
a3
al = w w
w6. " .s + l , w e a y e back i n Case 1. 2 3
+-
...
Because of
al -
Case 3 . Suppose
Then
(
Here
s
so t h a t
(39)
above t h e o n l y c a s e t h a t remains is:
(0)
-
a3
h a s t h e form
-
a5
(o.0- I S
B
=
must b e a m u l t i p l e o f
(*)
(a)
@J
=
a c t s on a n n - d i m e n s i o n a l
Sn.
1 with
-
=G"4=a6=...
2
CL
=
P
a
1'
n ( a , ~ ) , the order of
=
CT
a
2
c
a B
'.
(cc r )n ( a , ~ )= 1.
a B
The symmetric g r o u p o f d e g r e e
n
s p a c e by permuting t h e c o o r d i n a t e s
r e l a t i v e t o an orthomormal b a s i s
- -
(1c i c n ) .
&-i
The t r a n s -
corresponds t o t h e r e f l e c t i o n i n t h e hyperplane
(ij)
orthogonal t o
=
a
and
i s a consequence of t h e r e l a t i o n
Examples.
position
...
E
-
E
-j
.
We may t a k e
r e l a t i v e t o a lexicographic ordering
C =
,
=
[ E ~ - E- -
-
J
,'
{ E ~ -E
/
i
~
$
j],
11
+ 5~ i
and
-
n-11
t
276
i
i
S1; i s g e n e r a t e d by t h e t r a n s p o s i t i o n s
(1i
- i c n - 1) s u b j e c t t o t h e r e l a t i o n s wi2
wi
Thus
-
= ( i i+l)
1, ( w ~ w ~3 + =~ 1)
)
=
.
0,
= 1 i'f li- jl > 1.
( b ) W = Octn
The
J
o c t a h e d r a l g r o u p i n c l u d e s s i g n changes a s w e l l as p e r m u t a t i o n s
and
(w.w.)
1
!
o f t h e c o o r d i n a t e s , so has o r d e r
X
+
= {
+ E - ~ Ii
with
( a ),
E -
So, comparing
J
J]
)
and
n
Here we may t a k e
= ! E . ~ - & ~ + E~ ,
- -
)l<i<n-l].
we have one more g e n e r a t o r . wn and n
wn2 = 1, (wnm1 w )4 = 1, ( wI- wn l 2 = 1 i f i ~ n - 2 .
-
more r e l a t i o n s
W
e observe that
Sn
and
Octn
a r e t h e g r o u p s o f symmetries o f
t h e r e g u l a r s i m p l e x and t h e r e g u l a r cube.
( 4 0 ) Problem. Prove t h a t
W i s d e f i n e d by t h e g e n e r a t o r s
{c
31 a E Z ] s u b j e c t t o t h e r e l a t i o n s
(Hint.
Using (15b) and ( 1 6 ) show t h a t t h e g r o u p s o d e f i n e d i s
g e n e r a t e d by
{ca( a
E
]
as a consequence of
Using ( 2 1 ) show t h a t any n o n t r i v i a l r e l a t i o n
(y = q;
,
I
(B), and t h e n
w1w2.
..wn
=
1
E T ) which h o l d s i n W can b e s h o r t e n e d a s a
consequence of (A) and (B)
ai
We c o n s i d e r some r e f i n e m e n t s i n o u r r e s u l t s which o c c u r
i n t h e c r y s t a l o g r a p h i c c a s e , when
for a l l
a,B E c ,
2 ( a , ~ /)
(a,@)
which we h e n c e f o r t h assume.
i s an integer
(This case
o c c u r s when we have r o o t s y s t e ~ . :o f 'Lie a l g e b r a s . )
(41) Refinement of .(8), I n t h e p r e s e n t c a s e , a l l c o e f f i c i e n t s
277
i n (8b) a r e i n t e g e r s .
Prove t h i s , by i n d u c t i o n on
(42)
*.
(43)
Problem.
that
(2
that
f'
p
ht
Under t h e assumptions of
- tv p
(44) Problem.
rap
or
cpa
P
( s e e (13) 1.
C)
(341,
i s a n i n t e g e r f o r every
assume a l s o
a E
n. Show
i s a n o n n e g a t i v e i n t e g r a l combination o f t h e
e l e m e n t s o f (7
a
(
i s always a n i n t e g e r .
p, a)/ ( a , a )
prove t h a t
ht
.
If
+ p
.
o!
and
B are roots,
is a root.
a $ - $
( H i n t : prove t h a t
a
and
( a , ~ ) 0,
+ 8
equals