Divided Diference Operators
Omar Ortiz
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
[email protected]
January 29, 2012
1
Definition
Fix the following data:
V a C-vector space of finite dimension n with basis v1 , . . . , vn .
V ∗ the dual space of V , with basis X1 , . . . , Xn dual to v1 , . . . , vn .
S = C(X1 , . . . , Xn ) the C-algebra of rational functions on V .
W ⊆ GL(V ) a finite reflection group generated by the reflections s1 , . . . , sr , acting on V ∗
by contragredient:
wX(v) = X(w−1 v)
for w ∈ W, X ∈ V ∗ and v ∈ V ;
and on S by extension:
w(gh) = (wg)(wh)
and w
g
h
=
wg
wh
for w ∈ W and g, h ∈ S.
NW = hf ∈ S; tw for w ∈ W | tu tv = tuv , tw f = (wf )tw i the Nil-Hecke Algebra of W .
For each reflection sα ∈ W , where α denotes the root of the reflection, the divided
difference operator Tα ∈ NW is defined by
1
(1 − tsα ).
α
And for any w ∈ W with reduced expression w = sα1 · · · sαl we define Tw ∈ NW as
Tα =
Tw = Tα1 · · · Tαl .
1
2
Actions
The Nil-Hecke Algebra acts naturally on the algebra of rational functions S by:
f · g = f g (polynomial multiplication), for f ∈ NW and g ∈ S;
tw · g = wg (contragredient action), for tw ∈ NW and g ∈ S.
Therefore, the divided difference operator Tα acts on S by
Tα f =
f − sα f
α
for f ∈ S.
NW also acts on the algebra ZW of global sections of the structure sheaf over the
moment graph of W ,
ZW = {ψ : W −→ S | ψ(u) − ψ(v) ∈ αS if u = sα v},
by
f · ψ(w) = (wf )ψ(w), for f ∈ NW and ψ ∈ ZW ;
tu · ψ(w) = ψ(wu), for tu ∈ NW and ψ ∈ ZW .
Therefore, the divided difference operator Tα acts on ZW by
Tα ψ(w) =
3
ψ(w) − ψ(wsα )
wα
for ψ ∈ ZW .
BGG bases
Recall the definition of the coinvariant algebra SW ,
SW =
C[X1 , . . . , Xn ]
,
hP ∈ S W | P (0) = 0i
where S W denotes the algebra of W -invariant polynomials.
Q
Let Q = {α | sα is a reflection} and w0 be the longest element in W (when expressed
as a reduced word in the generators).
Theorem 3.1 (BGG). {Pw }w∈W is a basis of SW as a C-vector space, where Pw0 =
and Pw = Tw−1 w0 Pw0 .
2
1
|W | Q
Theorem 3.2. {ψ w }w∈W is a basis of ZW where
Q , if w = w0 ;
w0
ψ (w) =
0 , if w 6= w0 .
and ψ w = Tw−1 w0 ψ w0 .
Theorem 3.3. The map
φ : SW
Pw
/ ZW
/ ψw
is and algebra isomorphism.
4
Example
Let V = C − span{v1 , v2 , v3 } and V ∗ = C − span{X1 , X2 , X3 }.




s s s
s s s
0 1 0
1 0 0
T T Let s1 = 1 0 0 = T
, s2 = 0 0 1 =
,
T
T
s
s
s
s
s
T
s
0 0 1
0 1 0
and W = hs1 , s2 i = {c, s1 , s2 , s1 s2 , s2 s1 , s1 s2 s1 } ⊆ GL(V ). Since w0 = s1 s2 s1 is also a
reflection, we will also denote it as s3 ,


s s s
0 0 1
l ,
,


.
s3 = s1 s2 s1 = 0 1 0 = l,
l
s s ls
,
1 0 0
The action of W on V ∗ is given by the inverse transpose of the above matrices. But
since in this case s−t
i = si for i = 1, 2, 3; the action is the same as in V .
Let us denote by αi the root in V ∗ corresponding to si for i = 1, 2, 3. Then
α1 = X1 − X2 ,
α2 = X2 − X3 ,
α3 = X1 − X3 .
The coinvariant algebra is
SW =
C[X1 , X2 , X3 ]
.
hX1 + X2 + X3 , X1 X2 + X1 X3 + X2 X3 , X1 X2 X3 i
As a vector space,
3
SW

1,



X1 , X2 ,
= C − span
X 2, X X ,


 12 1 2
X1 X2 .
The BGG basis theorem gives us
Pw0 = 61 α1 α2 α3 = X12 X2 ,
Ps1 s2 = 31 α2 α3 = X1 X2 ,
Ps2 s1 = 31 α1 α3 = X12 ,
Ps1 = 31 (α1 + α3 ) = X1 ,
Ps2 = 31 (α2 + α3 ) = X1 + X2 ,
Pc = 1,
as a basis for SW , i.e.
SW

1,



X1 , X1 + X2 ,
= C − span
X 2, X X ,


 12 1 2
X1 X2 ;
which clearly agrees with the basis above.
For ZW we get the following basis elements
0
0
0
0
0
0
,
ψ s1 s2 =
,
ψ w0 =
0
0
0
−α1 α3
−α1 α3
α1 α2 α3
0
ψ s1
0
=
α3
α3
ψ s2
α
= 2
α2
ψ s2 s1 =
0
−α2 α3
0
,
α3
−1
ψc =
−1
−1
.
−1
−1
α3
4
0
,
0
−α2 α3
−1
0
α1
,
α1
0