A proof
of a con.lecture
T.J.
Enright and
§ I..Introductio.n
Let
or
SO(n,2)
Lie g r o u p
where
a conjecture
irreducible
tensor
is
connected.
highest
for
of Kashi~Bra
and
produc~
highest
so(n,2)
In this
weight
Our results
unitary
SU(p,q)
is the l i n e a r
Lie algebra
irreducible
unitarizaDle.
preliminaries.
SO(n,2)
is simply
apd Verglae
R. P a r t h a s a r a t h y
be one of the groups
whose
fication
the
G
apd
of K a s h l w a r a
Mp(n,~)
real s e m i s l m p l e
and whose c o m p l e x i -
paper we describe
modules
of
SU(p,q)
Vergne,
weight
or
G
and
which are
Mp(n,]R)
namely,
modules
ef the S e g a l - S h a l e - W e i l
all
settle
that a l l the
occur in ~he
representation.
There has been other research related to the conjecture.
In
[4], F. Williams proves the conjecture for the case of SU(2,2).
In
the more general case of SU(p,q), a proof of the conjectrue has been
given by H. Jacobsen.
Let
go
go = ko + Pc
is simple
be a semisimple
a Caftan
and that
complexifications
that
Zo=
action
IRZ t
on
p
Let
h°
Let
r
where
+i.
the suffix
the eigenvalues
and
P
of
over
~
go" A s s u m e
centre
and
that
o. It is k n o w n
Z
f o r the a d J o i n t
E p ; [Z,X] = + i X ]
be the set
of roots
be the positive
of
of
of
system
go
g
contained
containing
g
go
z o. Denote
Let
be a Borel s u b a l g e b r ~
A
of
has a nontrivial
be a Cartan s u b a l g e b r a
p+. Let
let
decomposition
by d r o p p i n g
are
p+ = { X
k°
Lie algebr~
in
h
with respect
of roots
k o.
and
to
h
corresponding
* This research has been supported by the Alfred P. Sloan Foundation
and the National Science Foundation grant # MCS 78-02896
75
to
r. Let ~ k
(resp. A n ) denote the set of compact
(resp. noncompact) roots. Thus
Pk = P ~ ~ k
In this paper
P
~ = ~kU
and
~m"
Imt
Pn = P ~ ~ n
will be referred to as the
special positive s2st~n.
Let W(resp. Wk) denote the Weyl group of
(resp. k) relative to
(~,H)
for
g
h. For
~ s h
= Hom(h,C)
let
denote the irreducible highest weight module
with highest weight
~
with respect to
P.
(See [i] for definitions). We wish to know when
is unitarizable with respect to
If
real on
a
(~,H~)
go"
is unitarizable, one knows that
iho, k-integral and
spin module for the Lie algebra
orthogonal group
~
is
Pk-dominant. We will now
describe another necessary condition. Let
p
g
so(p)
L
be the
of the special
S0(p) (the symmetric bilinear form on
beimg the restriction of the Killing form). By compos-
ing with the adjoint representation of
obtain the spin module
Let
6n -- ½ ~ a P
6 = ~Z~E
n
a.
L
for
k
on
p, we
k.
P ~, 6k = ~ S P k
a
and
76
(1.1)
Let
Propositigp.
~
be the
su~module
Suppose
Pk-highest
of
~
weight
(
Killing
Proof
,
)
form to
is the dual of the restriction
being that in
here we are considering,
highest
weight modules
(1.2)
Corollary.
tarizable
k
let
of
unitarizable,
: Let
k
-Pn,
whereas
the notation
H . If
of
[lJ,
P.
(~ ,Hp)
of an
is ~ -
CP + 6, p + 6).
~ = ~ + 6 n.
(q.e.d)
is k-integral
the irreducible
k
and Pk-dominant,
module with
Pk
k .
proposition.
irreducible
Proof
submodule
denote
PkU
be the highest weight
~ ~ h . If ~
highest weight
(1.3)
~
: Apply 1.1 taking
Vh
to
with respect to
Let
2.6J
L2J, highest weight
following
(~ + 6, ~ • 6) ~
I~t
of the
[2, Proposition
in
were defined with respect
irreducible
k
h.
the only difference
Proof
of an irreducible
~k ~ >i (~ + 6 , p + 6)
: This is the assertion
modules
is ttuitarizable.
~ L. Then
(~.+ 6 ~ , ~ +
where
(~ ,H )
Let
submodule
then
~
of
be the highest weight
Vp ~ p_. If
(~ + 6, ~ + 6) ~
W~ = U(g) @ U(k+p+) V
(~p,Hp)
of an
is
(p + 6, ~ + 6).
, where
U(
)
denotes
77
the enveloping algeora.
know
that
H
~ k @ i>io Si~P-) ~ V . We
l
w
is the unique irreducible
quotient of
W .~It can De s h o w n that all irreducible subquotients
of
W
then
are highest weight modules
either
weight
V
occurs in
H
of
or
g. If
~
of an irreducible subquotient
V~ ~_ ¥
~ p_,
is the highest
of
W . This last
f a c t can be deduced using the property %hat in any
positive
root %he coefficient
of the noncompact simple
root is zero or one. In the first case 1.2 ~
the second
Hence
case, ~ w E W g
such that
1.3.
In
9 + 6 = w(~÷8).
(9 + 6, ~ + 6) = (~ + 6, ~ + 6)
which implies
1.3.
§ 2.
G = SU(p,q)
Let
oasis
of
el,e2,...,ep+ q
be the standard orthonormal
IR p+q. The special positive
system
P
has
roots {ei - ej, 1 ~ i ~ J ~ P + q ~
Pk = { e l -
ej, l~< i ~ j ~< p
P n =~ei
~
-L eJ ' 1
or
p + 1 ~< i < J ~ p + q ~ .
i ~ p, p + 1 ~ j ~ p + q~.j
We can normalize the Killing f o r m on
for each root
a
~. Any linear form
p ÷ q - tuple
the c o r r e s p o n d e n c e
so that
(a,g) = 2
,
# ~ h c~n be written as
(ml,m 2 ..... ,mp+q)
~
= m i - mj
g
s u c h that ~ m i = 0
if
with
~ = e i - ej. More-
over
= m]2 + .... + (mp+q) 2 .
(2.l)
Also
is k-integral
and d o m i n a n t with respect to
Pk
if
78
and only if
or
p + 16
m i - mj E ~ +
i < J~
Now let
whenever
i % i < J ~ p
p + q.
p = (ml,...,mp+q)
and suppose that
the irreducible
highest weight m o d u l a r for
highest
weight
p
(2.2)
Lemm~.
Let
~ = (nl,n 2 ..... ,np+q)
~
is k-integral
Suppose
that the irreducible
weight
~
~q
i=l
Proof
with
is unitarizable.
that ~ n i = O. Suppose
zable
g
occurs in
V
k
and assume
and Pk-dominant.
module w i t h highest
$ p_. If
(~,Hp)
is ~ I t a r i -
then
P+q
p+q_2i+l) 2
~ i=l (mi +
2
(ni+ P+q~2i+l) 2
: Since we have
6 = (p+~-l~
P+q-3
p+q-2i+l
n-~_1,z_~.~_=)
L
the assertion
2.2 follows from 1.3 and 2.1.
We will now apply
(2.3)
Lemma.
Suppose
2.2 to a particular
m I = m 2 .....
m I W ml+ 1
mp+q = rap+q_ 1 = ... = mp+q_j+ 1 ~ mp+q_j.
- (e i - ep+q_j+l).
Proof
since
Thus
: Since
~
m i ~ mi+ 1
Then
V~
occurs
is k-integral
m i >I l+mi+l;
~ - (ei-ep+q_j+ l)
have the same length,
is
Pk
and
Take
and
~ =
in
p_ ~ Vp.
Pk
dominant and
similarly
dominsnt.
e i - ep+q_j+ 1
~.
mp+q_j >ll+mp+q_j+ 1.
Since all roots
is an extreme weight
79
of
p_. The a s s e r t i o n
ing o b s e r v a t i o n
cibles.
Let
k
be an extreme
Then
W
2.3 now follows from the f o l l o w -
: Suppose
V
and
V
weight of
~
VA+ B ~
Vk+~ _= V ~
V
W. Assume ~ + ~
V
of
~
VA+ ~ @
W
and let
is dominant.
V . But
contains an invariant
W.
Now a p p l y i n g the inequality
lemma
are two irredu-
be the highest weight of
is the PRV component
W G VA+TI ~
W
2.2 taking
~
as in
2.3 we obtain
m p + q _ j + 1 >i mi+p+q-i- j
But s i n c e
m I = m 2 = .. = m i
we obtain the c o n a i t i o n
and
mp+q ....
mp+q_j+ 1
m p + q ~ ml+p+q-i- j. Thus we have
established
(2.4)
Theorem.
m i = 0, m l >~
Let
~ = (ml,m2,..,mp+ q)
... >~ mp, m p + I >i mp+ 2 >j ... >i mp+q
m i - mj c ZZ whenever
Let
P
l~< i ~ J _< p
Choose
or
p+l ~ i <
and
J ~
p+q.
be the positive system of roots ~ e i - e j, i ~< i ~ j ~< p+q].
S u p p o s e that the irreducible
SU(p,q)
where
with P-highest
i,
1 ~< i ~< p
m I = m 2 = ... = m i
m p + q >I m l + p + q - i - j.
and
highest weight module f o r
weight
and
j ,
~
1%
is unitarizable.
J ~< q
maximal such that
mp+q ...... m p + q _ j + 1. Then
80
We will now see using C1, part III] that the
condition in Theorem 2.4 is also sufficient
SU(p~,q)
irreducible
arlzable.
hi+ 1 ~ 0
Since
highest weight module to be ~=It-
In fact, define
ni = mi - mI
and let
and
for the
~ = (nl,n 2 ..... ,np+q)
k = np+q. ~ote that
nl=m2=..=ni=O,
np+q = Zp+q_ 1 ..... np+q_j+ 1 = k ~ np+q_j°
k >i (p-i) + (q-J), we conclude
6.3, 7.2] that the
highest weight
U(p,q)
from El, Theorem
highest weight module with
(nl,n2, .... ,np+q)
occurs im the k-fold
tensor product of the harmonic representation
Ll~. Its restriction to
highest weight module
all the irreducible
SU(p,q)
is precisely
(m I .... ,mp+q)
unitary
~or
U(p,q),
the
SU(p,q).
highest weight modules
Thus
of
of
SU(p, q).
a = Mp(n,m).
Let
of
SU(p,q)
of
occur in some tensor product of the h~rmonic
representation
§ 3.
where
el,... ,en
be the standar~ orthonormal
IRn. The special positive system
P
basis
has roots
(e i + ej, 1 ~< i - j ~< n~ U ( 2 e i, 1 ~< i ~ n ]
Pk = ~ e i - e j ,
Pn = { e i
w~ have
l~
+ ej, i ~
i ~ J ~ n~
i ~ j ~ n]U
1 2 e i ~ i ~< i~< n~.
(e i - ej,e i - ej) = (e i + ej,e i + ej) = 2
(2ei,2e i) = 4.
and
81
Let
~
be an n-tuple,
~ = (ml,m2,...,m n)
and suppose that the irreducible
for ' g
with highest weight
(3.1)
We note that
and dominant
kI ~ k 2 9
6
=
with respect to
.... ~ k n
p_ ~
+ (kn+l)
~ = (k I .... ,kn)
k
module
V . If
H
n
Pk
n
~ Z
~
occurs
then
2
(mi+n-i+l)
.
i=l
: This is ~m~ediate
from 1.3 and 3.1
specialize 3.2 for a
We mow/carefully chosen
that
and
and suppose that the
is unitarizable
i=l
Suppose
be k-integral
with highest weight
2
(ki+n-i+l)
P~opf
if and only if
2
(kn_l+2)
Let
is k-integral
k i - kj E ~ . Also, we have
2
and JJ~+ 6 II = (kl+n) 2
with respect to
irreducible
In
Pk
2
+..+
Lemma.
dominant
is 1~n~tarizable.
and
2
+ (k2+n-1)
weight module
~ = (kl,k2,...,kn)
(n,n-i ..... ,2,1)
(3.2)
~
highest
~
such that
(q.e.d.)
V~ _~ V~ ~ p_.
m I = m 2 = ... = m i ~ mi+ 1. We distinguish
two cases.
Case I. mi+ l~< m i
Case
We first
-
II.mi+ l = m i - 1
conslder
Case I.
2.
and
mi+ l = m i + 2 ..... ~ + j ~ m i + j + 1
82
(3.3)
~emma.
mi+ 1 ~
p_'~
Suppose
m i - 2. Take
~ ~ ~ - 2e i. T h e n
mi
and
V~
occturs
V.
Proof.
Since
clearly
of the
can
m I >i m 2 ~
$
highest
take
(3.4)
root
and
of
VT
occurs
Proof
: Obviously
the
proof
in
m I = m 2 ......
p_ @ V
~
of 2.3
m i, m i + 1 = mi-1,
is
V
~ = ~ - e i - el+ j.
.
Pk
we w i l l
of
t~k-orbit
(q.e.d.)
m i + j ~ m i + j + 1. Take
p_ ~ V
character
is in t h e
..
II.
mi+ 1 = mi+ 2 .....
in
p_. Hence
Suppose
Then
-2e i
~ mi+ 1
3.3 follows.
up Case
Lemma.
occurs
.. ~ mi_ 1 >i m i - 2
is P k - d o m i n a n t .
be a p p l i e d
We n o w
The
m I = m 2 .....
dominant.
use Weyl's
equals
To show that
character
~
~(s)
saW k
V~
formula.
eS(~+6k)/~
where
s 6k
=~s
c W k E(s)
p n e-~.
E
e
. The
~ence,
~(s)
character
the c h a r a c t e r
e
s (~-~+6 k)
of
of
p_
equals
p_ @ V
. We w i l l
prove
is
the lemma,
sEW k , ~EP n
by s h o w i n g
that the
to
~ = e i + el+ j
s = l,
term.
That
s = 1
term
is, we w i l l
and
in the n u m e r a t o r
does
show
not c a n c e l
that
corresponding
with any o t h e r
s ( ~ - ~ + 6 k) = ~ - e i - e i + j + 6 k
~ = e i + el+ j. Since
s6 n = 6n, it is
83
equivalent
and
to s h o w i n g
s(~-~+6)
= ~-ei-ei+j+6
Let
Bince
~ + 6 =
m l~
(al,a2,...,an).
=
i+J
bi = ai - 1
and
Then
m 2 >~ ... ~> ran, a k > a~ if
~-ei-ej+6
(bl,b2, .... b n).
Then
and
ak=mk+n-k+l
k < £ . Let
bk = ak
if
a i _ 1 = a± + l, a i = ai+ 1 + 2,
= ai+ j + 1
a i + j >~ a i + j + 1 + 2. Hence
nor
ai+ j
and
is an
element
~ow for some
(3.G)
of the set
~ c Pn
s(~-~+6)
~-~+~
=
Cl,C2,...,c n
particular
nor
(Cl,C2,...,Cn).
k = i
cf
set
- e i - el+ j + 6
the proof
cr = ar
ready
to
that n e i t h e r
if
= i + J, that
to
Hence
of 3.4.
We are now
In
{ Cl,C2,...,Cnj
provej
. Writing
r ~ k
or
is,
s(~-ei-ei+j+6)
~ - e i - ei+ j
is r e g u l a r .
that
b l , b 2 , . . . , b n.
and
3.6 r e d u c e s
- e i - ei+ j + 6. Since
ai
I bl'b2' .... 'bn~"
i n 3.~ i m p l y
b e l o n g to the
= e i + el+ j. Thus
neither
Then 3.6 implies
is a p e r m u t a t i o n
we c o n c l u d e
ai+j_ 1
suppose
= e k + ez, 1 .< k .< ~.< n, n o t e
Hence,
of
= ~ - e i - ei+ j + 6.
our o b s e r v a t i o n s
ai+ j
k ~ i or
bi+ j = ai+ j - 1. B e c a u s e
our a s s u m p t i o n s
ai
s = 1
~ = e i + el+ j.
(3.15)
Let
~
=
is P k - d o m i n a n ~
s = 1. This
completes
84
(3.7)
is
Theorem.
k
(i.e:
Let
integral
(i.e.,
m I >~ m2>~
i) If
~ =
(ml,m2,...,mn).
m i - m j ~ Z~)
... >~ mn).
and
Suppose
then
±i) If
-m
I
mi+j+ 1 ~
Proof
: In case
that
3.2 b e c o m e s
domlm~nt
is n n ~ t a r i z a b l e .
and
m i + 1 ~< ml-2,
~ n-i
mI = m 2 =
and
Pk
=~
m I = m 2 = ... = m i
Suppose
... = m i, m i + 1 = ... = m i + j = m l - 1
m l - 2 , then
i, a p p l y
-m I ~ n - i - ~
3.2 t a k i n g
~ = ~ - 2e i. N o t e
(mi_2+n_i+l) 2 >~ (mi+n_i+l) 2
which
reduces
to
In case
We find
-m i >~ n-i.
ii, a p p l y
3.2
Since
m I = mi,
taking
-m 1 ~ n-i
~ = ~ - e i - el+ j.
3.2 be comes
(mi-l+n-i+l) 2 + ( m i + j _ l + n _ i _ j + l )
2
(mi+n-i+l) 2 + ( m i + j + n - i - j + l ) 2
This
reduces
to
and
m i + j = ml-1,
Theorem
unitarizability
covering
prove
the
group
-m i - m i + j >~ 2 ( n - i ) - j + l .
we o b t a i n
3.7 gives
of
following
-m I >~ n-i -j.
a necessary
of h i g h e s t
weight
Sp(n,IR).
result.
Since
In [i],
(q.e.d.)
condition
modules
mi = m1
for the
of the u n i v e r s a l
Kashiwara
and Vergne
85
(3.8)
Let
Theorem.
(Kashiwara-Vergne
~ = (ml,m2,...,mn) ; Suppose
I
m I >i m 2 ~ ..... > m m. Assume
the s i t u a t i o n
m i - mj ¢ ~
and
m i ¢ 2Z/2. (This is precisely
in which the nighest weight module becomes
a representation
g r o u p of
[i, Theorem 8.8]).
of
Mp(n,~)
the two sheeted covering
Sp(n,~R)). I~
ml-m 2
- m I >i m i n ( n - 1 , - - ~ + n - 2 ,
(ml-m2) + (ml-m~) +n-3,
,
2
....
(ml-m~+ .... +(ml-mn) )
2
t h e n the highest weight module
~act
~
~
occurs as an irreducible
is unitarizable.
submodule
In
of
W @ W ®
... ® W, tensor product of f i n i t e l y many copies
of the
Segal-Shale-~eil
representation
They then conjecture
weight module for
2p(n,IR)
of
Mp(n,l~).
that any unitarizable
highest
is obtained as above.
We will
n o w prove this by showing that our necessary condition
3.7
already implies the condition in the theorem 3.8.
(3.9)
of
Theorem.
Mp(n,~)
Proof
and
: Let
highest weight module
occurs as an irreducible
tensor product
Shale-Weil
Every tmitarizable
submodule
of finitely m a n y copies of the
of the
Segal-
representation.
~ = (ml,m2, .... ,ran) when
m i - mj ~ ~ .
Let
~
mi ~ m 2 ~
... ~
mn
De the highest weight module
86
of
sp(n,IR)
w i t h highest weight
unitarizable.
We assume that
is a' r e p r e s e n t a t i o n
such that
of
I t Either
i = n
Case
If. i ~ n
and
Let
dk =
Since
and suppose ~
m i E -~-
so that
~
M p ( n , ~ ) . Choose a maximal
m I = m 2 = ....
Case
~
or
m i. We aistinguish
i ~ n
and
i
two cases.
mit I ~ mi-2
mi+ 1 = mi-1
(ml-m2) + ( m l - ~ ) + ' ' ' + ( m l - m k ) + n - k .
2
~
is unitarizable,
in case I by t h e o r e m
3.7, i, -m I >i n-i. Also, d i = n-i. Thus, the condition
in T h e o r e m 3.8, namely,
satisfied.
Hence
~
-m l ~
min~dl,d2,...,dn)
occurs as an irreducible
is
submodule
of the t e n s o r product of finitely many copies of the
Segal-Shale-Weil
representation.
In Case II, choose a maximal
j
such that
mi+ 1 = . .. = mi+ j. By theorem 3.7, ii, -m I ~ n-i-2~. Also,
di+ j = n-i-2~. Thus, the condition of theorem 3.8, namely,
-m I >i m i n ( d l , d 2 , . . . , d n )
3.8, ~
product
is salisfied.
occurs as a n irreducible
Hence by theorem
submodule of the t e n s o r
of finitely many copies of the Segal-Shale-Weil
representation.
~'his proves t h e o r e m 3.9.
87
4.
go = so(m,2)
Let
and
go : gl or g2' where
g2 = so(2n-2,2).
standard
Let
orthonormal
positive system
P
gl = so(2n-l,2)
el,e2, .... ,en
basis of
for gl
]Rn. The special
has roots
Pk = l e i
+ ej , 2 ~< i < J ~ n~
Pm = ( e l
+ e j ~ 2 ~< J ~< m J
The special
positive system
Pk = l e i
P
for
for
g2<~
gl<~
gl' whereas
Whenever
~
Pk
or
as usual,
~ +
necessary
is
is
Pk'
. As usual,
6 = ( ~ r ~ , 2 ~ - ~ , . . . , l)
... ,1,0)
then
and sufficient
When
m2
for
g2"
=
~
m 2 = 0.
=
..
[3]
conditions for
go = so(2n-l,2)
(~,~) = 0
of this, in the statements
will exclude
~. ~
p
(~,a) = 0, V ~ ~ Pk' Nolan Wallach
V ~ E Pk<~
dominant,
We have
6 = (n-l, n-2,
to be tmitarizable.
(~,~) = 0
.
m 2 ~> m 3~> .... ~> mn_ 1 >i I mml
has determined
has roots
m 2 >~ m3>~ ... >i m n >I 0, while
JJ~I/ 2 = m]2 + m 2 +...+ ~ .
for
g2
~ = (ml,m2, .... ,ran)
k integral~=~ {m2, .... ,mnl_c2Z
for
n~
+- ej ~ 2 % i < j ~< n ~
Writing
dominant
~ e i, 2~< i ~
i ell"
Pn = ( el + ej ~ 2 ~< j ~< n ~
(4.1)
be the
• = mm
=
0.
or so(2n-2,2),
If
~
is
V ~ E Pk~=~ m 2 = 0. In view
of the following results, we
88
(4.2)
Theorem.
Let
Suppose
that
is g - i n t e g r a l
or
~Z~ + ~
Choose
and
i
Proof
weight
clearly
highest
~
suppose
Since
weight
of
can
to the
following
representation
is in the
occurs
i = n.
V~
fact
and
then
V
in
in
that
g
and s i n c e
integral
-m I ~
n-~.
This
go = so (2n-2,2).
-m I ~
n-1.
Pk
of the
1.3
2n-i-1
m 2 = m i-
argument
V
~
Let
V
we c o n c l u d e
~ = ~ - e I . It
p_
(this c o r r e s p o n d s
if
T.
~
is t h e s p i n
representation
then
L --> I
that
1.3 a n d 4.1 to
But since
m 2 = m n = 6,
completes
orbit
-ml-m i ~
since
is the n a t u r a l
Applying
is
~ p_. A p p l y i n g
get
so(2n-1)
L
we c o n c l u d e
V
m n = ~.
Clifford multiplication
L @ V).
in
the a b o v e
: for
with
m n >i l, t h e n t a k i n g
dualizing
occurs
g
- m l - m 2 ~/ 2n-i-1.
Weyl group
4.1 we
If
occurs
if
of
m i > m i + 1 ~ O,
- m l - m 2 >i 2 n - i - 1
mn ~ 1
be s h o w n t h a t
then
e ~
If the
~ = ~ - e I - el, ~
and u s i n g
If
... = m i.
,H )
mj
m 2 ~ O.
setting
and u s i n g
Next suppose
= ~ - eI - en
(~
i.e.
Since
p_, V
is the s a m e as
- m l - m 2 >I n-1.
dominant
m2 = m3 =
module
(ml,m2,...,mn).
i ~< n-1.
-el-e i
~ = ~ - e I - ei
which
Pk
is u n i t a r i z a b l e
m i >I 1. H e n c e
dominant.
to
highest weight
: Firs¢
and
I..l =
Let
m 2 ~ m 3 >i ... ~/ m n ~/ 0. A s s u m e
maximal such that
irreducible
highest
~
go = s o ( 2 n - l , 2 ) .
~
~ = ~ - e1
is a s s u m e d
we c o n c l u d e
Zhe p r o o f o f 4.2.
we s e e
to be
that
Next,
we c o n s i d e r
89
(4.3)
Theorem.
Suppose
or
~
~ + ~
Choose
the
is
m2 ~
Proof
~
: First
mj
occurs
e ~
in
or
and
V
Next suppose
that
ing
If
~
1.3
Pk
to
in
V
- m l + m n ~ n-2,
that
is g - i n t e g r a l
in f a c t
that
~
inequality
and
2n-i-2.
m 2 = J mnJ,
and note
V
~p_.
Apply-
4.1 we get
mn <
is
in
and
Fk
0, we
choose
dominant
and
V~
1.3 and 4.1, we c o n c l u d e
go = s o ( 2 n - l , 2 )
and
-ml-m 2 ~
m2 ~ 0
~
and
~ = ~-el-e i
Thus
4.3 is proved.
p a p e r we will a c t u a l l y
: Let
m i > J mi+lJ ~ 0,
dominant
i.e.,
If
- m l - m 2 ~ n-2.
converse
then
to
occurs
- m l - m 2 ~ n-2.
i.e.,
Pk
and using
with
m i ~ 1. Set
is
V
g
If
- m l - m 2 ~ 2n-i-2.
Since
1.3
m 2 ~ 0.
of
~ = ~ - eI - en
and
x p_. A p p l y i n g
the a p p r o p r i a t e
~
2n-i-2,
choose
and n o t e
In a l a t e r
~
+ ~, c l e a r l y
dominant
i.e.,
= ~ - e I + en
occurs
~
~ = W - e I - en
- m l - m n > n-2
then
c
mi_ 1 = ~mil.
(~,H~)
i & n-1.
that
i.e., mj
Imml . A s s u m e
module
i = n. ~ I n c e
m n > 0, we
is
dominant,
that
-ml-m i ~
~ = (ml,m2,...,mn).
m 2 = m 3 = ....
weight
note
Let
... ~ m n _ 1 ~
~ p_. A p p l y i n g
4.1, we g e t
m n 40.
Pk
is u n i t a r i z a b l e ,
suppose
= ~ - eI - ei
using
~
highest
weight
and since
V
~
and
maximal such that
irreduciOle
highest
go = s o ( 2 n - 2 , 2 ) .
g-integral
and
i
Let
Pk
or
is u n i t a r y .
so(2n-2,2)
dominant.
in T h e o r e m
prove the f o l l o w i n g
If
and s u p p o s e
-ml-m 2 satisfies
4.3 or in T h e o r e m
4.4,
90
This is proved in the same way as [2, Theorem B]
is proved.
C o m b i n i n g with the result
thisldescribes
all the irreduciole
modules
(n,2),
of
SO
complex-ification
algebra
of Wallach in [3J
unitary highest weight
the real semisimple Lie group whose
is simply connected and which has Lie
so(n,2).
BiblioATaph~
1.
q~. F~ashiwara and M. Vergne,
representations
Math.
2.
44, 1 -
R. Parthasar~thy,
On the B e g S - S h a l e - W e l l
and harmonic
polynomials,
47 (1978).
Criteria
for the u n i t a r i z a b i l i t y
of some highest weight modules,
Acad.
3.
Inv.
Proc.
Indian
Sci. 89, 1 - 24 (1980).
N o l a n R. Wallach,
The analy%ic
Discrete series,
c o n t i n u a t i o n of the
If.
h. Floyd Williams, Unitarizable highest weight modules of
the conformal group, preprint (1978).
~athematic s Department
UCSD
La Jolla, Ca. 92093
U.B.A
School of F~thematics
Tata Institute of Fundamental
Homi B h a b h a Road
Bombay 400 005 (India)
Research