A remark on some bases in the Hecke algebra

0.11 MB·English

by  Rahbar Virk

A REMARK ON SOME BASES IN THE HECKE ALGEBRA
R.VIRK
1
Contents
1
0
1. Introduction 1
2
2. The Hecke algebra 2

n
3. The Ext form 4
a
4. Dualities between the bases 5
J
References 6
8

]
O
C
h. 1. Introduction
t
a InthisbriefnoteweconsidersomebasesintheHeckealgebraandexhibitcertain
m dualities between them. Our results and arguments are completely combinatorial.
[ However,toexplainthe motivationandputthecontentsofthisnoteinperspective

we need to discuss some results from geometric representation theory.
2
LetGbeareductivegroup,B ⊆GaBorelsubgroupandletBbetheflagvariety
v
4 of G. Then the Iwahori-Hecke algebra of G realizes the Grothendieck group of B-
2 equivariantperversesheavesonB.Multiplication inthe Heckealgebracorresponds
9 to convolution of sheaves.
1
In his streamlined treatment of the Hecke algebra, Soergel [So1] considers sev-
.
2 eral pairwise commuting automorphisms (see [So1, Thm. 2.7]. The geometric sig-
1 nificance of some of these is well known. For instance, the automorphism d (2.3.1)
0
corresponds to Verdier duality on sheaves. This note was motivated by trying to
1
: understand the geometric significance of the automorphism b (3.4.1). I will now
v
outline the contents of this note from this perspective.
i
X A cornerstone of Kazhdan-Lusztig theory is the identification of IC-complexes
r (=simple objects) on B with the Kazhdan-Lusztig basis of the Hecke algebra. In
a Thm. 3.7 we identify/construct the basis corresponding to projective objects. In
Thm.4.2weexplainhowbconnects the projectivebasiswith the Kazhdan-Lusztig
basis. Actually, Thm. 4.2 implies that b switches the Kazhdan-Lusztig basis with
the basis corresponding to tilting objects (see [Virk, §9] for the precise statement).
Now,Soergel’s‘tiltingduality’[So2]switchestiltingobjectswithprojectiveobjects
and the Koszul duality of [BGS] switches projective objects with simple objects.
It follows that b is the composition of these two dualities. Another way to see this
is to use [So3, Thm. 4.4] to directly deduce that b switches the Kazhdan-Lusztig
basis with the tilting basis. The aforementioned Koszul duality does not commute
with convolution (it doesn’t preserve the monoidal unit). In particular, it does not
descendtoaringautomorphismofthe Heckealgebra.However,Thm.4.2isacom-
binatorial shadow of the statement that the tilting duality intertwines convolution
and Koszul duality (cf. [BG, Conjecture 5.18, Thm. 5.24, Thm. 6.10]).
1
2 R.VIRK
Acknowledgments I am grateful to W. Soergel for some very helpful correspon-
dence and to Arun Ram for his comments on a preliminary version of this note.
This work is supported by the NSF grant DMS-0652641.
2. The Hecke algebra
2.1. Let(W,S) be aCoxetersystem, ℓ: W →Z the correspondinglength func-
≥0
tion and let ≤ denote the Bruhat order on W. In particular, x < y means x ≤ y
and x6=y. The identity element in W is denoted by e.
2.2. The Hecke algebra H is the free Z[v,v−1]-module Z[v,v−1]T with
x∈W x
Z[v,v−1]-algebra structure given by
L
T T =T if ℓ(xy)=ℓ(x)+ℓ(y), (2.2.1)
x y xy
(T +1)(T −v−2)=0 for all s∈S. (2.2.2)
s s
Set H =vℓ(x)T for all x∈W. Then (2.2.2) implies that each H is invertible. In
x x x
particular, if s∈S, then H2 =1−(v−v−1)H . Hence,
s s
H−1 =H +(v−v−1). (2.2.3)
s s
Note that T =1.
e
2.3. Define a ring involution d: H→H by
d(v)=v−1, d(H )=H−1 . (2.3.1)
x x−1
We will often write H instead of d(H). We call C ∈H self-dual if C =C. For each
s∈S set
C =H +v. (2.3.2)
s s
Then, by (2.2.3), each C is self-dual. In the sequel we will need the following well
s
known formulae:
2.4. Lemma. Let s∈S and let x∈W be arbitrary. Then
H +vH if sx>x;
sx x
C H =
s x (Hsx+v−1Hx if sx<x.
Proof. This is a direct computation and is left to the reader. (cid:3)
The following classical result is due to Kazhdan and Lusztig [KL], the proof
presented here is stolen from [So1].
2.5. Lemma. For each x ∈ W there exists a self dual element C ∈ H such that
x
C ∈H + vZ[v]H .
x x y<x y
Proof. ProcPeed by induction on the Bruhat order. Certainly we can start our in-
duction with C = H = 1. Now let x ∈ W be given and suppose we know the
e e
existence of C for all y < x. If x 6= e find s ∈ S such that sx < x. Then by
y
the induction hypothesis C C = H + h H for some h ∈ Z[v]. Set
s sx x y<x y y y
C = C C − h (0)C . By construction C satisfies the required condi-
tioxns. s sx y<x y y P x (cid:3)
P
2.6. Lemma. Suppose C ∈ vZ[v]H is self dual. Then C =0.
x x
Proof. Write C = xhxHxPfor hx ∈vZ[v,v−1]. Using (2.2.3) one sees that
P Hy ∈Hy+ Z[v,v−1]Hz
z<y
X
forally ∈W.Picky maximalsuchthath 6=0.ThenC =C impliesthath =h .
y y y
Thus, h =0, contradicting our assumption. So C =0. (cid:3)
y
A REMARK ON SOME BASES IN THE HECKE ALGEBRA 3
2.7. Theorem ([KL]). For each x ∈ W there exists a unique self-dual element
C ∈H such that C ∈H + vZ[v]H .
x x x y y
Proof. The existenceis givenbPyLemma 2.5,the uniquenessis providedbyLemma
2.6. (cid:3)
2.8. For each x,y ∈W define polynomials h ∈Z[v] by
y,x
C = h H .
x y,x y
y
X
The following result is well known. It will be key in the sequel.
2.9. Lemma ([KL]). Let s∈S and let x∈W. If sx<x then
C =C C − µ(y,sx)C
x s sx y
y<sx,
sXy<y
where µ(y,sx) is the coefficient of v in h .
y,sx
Proof. We have
C C =C (H + h H
s sx s sx y,sx y
y<sx
X
=C H + h H + h H
s sx y,sx y y,sx y
y<sx, y<sx,
(cid:0) sXy<y sXy>y (cid:1)
=H +vH + h (H +v−1H )+ h (H +vH ).
x sx y,sx sy y y,sx sy y
y<sx, y<sx,
sXy<y sXy>y
We need to look slightly more carefully at the construction of C in Lemma 2.5.
x
Note that for y <sx each h ∈vZ[v]. So, by the proof of Lemma 2.5,
y,sx
C =C C − µ(y,sx)C . (cid:3)
x s sx y
y<sx,
sXy<y
2.10. Proposition. Let s ∈ S and let x ∈ W. For each y ∈ W let µ(y,x) denote
the coefficient of v in h , then
y,x
Csx+ y<x,µ(y,x)Cy if sx>x;
C C = sy<y
s x
((v+v−P1)Cx if sx<x.
Proof. Suppose sx<x, then by Lemma 2.9
C =C C − µ(y,sx)C .
x s sx y
y<sx,
sXy<y
Further, C2 = (v +v−1)C by (2.2.3) and so the result follows by the induction
s s
hypothesis. If sx>x, then once again by Lemma 2.9
C =C C − µ(y,x)C ,
sx s x y
y<x,
sXy<y
whence the result. (cid:3)
The following is originally due to Kazhdan-Lusztig [KL], I learnt the proof pre-
sented here from [So1].
4 R.VIRK
2.11.Proposition([KL]). SupposeW is finite.Letw ∈W bethelongestelement.
0
Then
C = vℓ(w0)−ℓ(x)H .
w0 x
x∈W
X
Proof. LetH = h H ,h ∈Z[v,v−1],beanelementofHsuchthatb(C )H =0
y y y y s
for all s∈S. Then, by Lemma 2.4
P
h (H −v−1H )+ h (H −vH )=0
y sy y y sy y
sy>y sy<y
X X
for all s ∈ S. Consequently, h = v−1h for all s ∈ S. Proceeding by induction we
s e
see that H = h vℓ(y)H . By Prop. 2.10 C C = (v+v−1)C for all s ∈ S.
e y y s w0 w0
The result follows. (cid:3)
P
3. The Ext form
3.1. Define a symmetric Z[v,v−1]-bilinear form h·,·i: H⊗H→Z[v,v−1] by
hH ,H i=δ . (3.1.1)
x y x,y
3.2. Remark. This form correspondsto the form consideredin the proofof [BGS,
Thm. 3.11.4]. Namely, it corresponds to
(−1)idim Exti(M,DNhni)v−n,
i,n
X
where M,N are perverse sheaves on the flag variety, D is Verdier duality, hni is
shift + (half) Tate twist in the derived category and v is a formal variable.
3.3. Proposition. If s∈S, then
hC H,H′i=hH,C H′i
s s
for all H,H′ ∈H.
Proof. It suffices to prove the assertion for H = H and H′ = H with x,y ∈ W.
x y
By Lemma 2.4 we have
δ +vδ if sx>x;
sx,y x,y
hC H ,H i=
s x y (δsx,y+v−1δx,y if sx<x;
and
δ +vδ if sy >y;
x,sy x,y
hH ,C H i=
x s y (δx,sy+v−1δx,y if sy <y.
So there are four cases to consider: if sx > x and sy > y, then the assertion is
evident; if sx > x and sy < y, then certainly x 6= y and so the assertion holds; if
sx<xandsy >y,thenonceagainx6=y andthe assertionholds;finally,ifsx<x
and sy <y, then the assertion is evident. (cid:3)
3.4. Define a ring involution b: H→H by
b(v)=−v−1, b(H )=H . (3.4.1)
x x
Then b commutes with d. Furthermore:
3.5. Proposition. If s∈S, then
hb(C )H,H′i=hH,b(C )H′i
s s
for all H,H′ ∈H.
Proof. Since b(C )=C −(v+v−1), the result follows from Prop. 3.3. (cid:3)
s s
A REMARK ON SOME BASES IN THE HECKE ALGEBRA 5
3.6. Lemma. Assume W is finite. Then for each x ∈ W there exists a P ∈
x
H + vZ[v]H such that hP ,C i=δ for all z ∈W.
x y>x y x z x,z
Proof.PProceed by induction. Let w be the longest element in W. We start our
0
induction with P = H . Now let x ∈ W be given and suppose we know the
w0 w0
existence of P for all y > x. If x 6= w find s ∈ S such that sx > x. Then by the
y 0
induction hypothesis
b(C )P ∈H + h H
s sx x y y
y>x
X
for some h ∈Z[v]. For each y >x let p =hb(C )P ,C i. Set
y y s sx y
P =b(C )P − p P .
x s sx y y
y>x
X
Let’s show that P ∈ H + vZ[v]H . To do this we must demonstrate that
x x y>x y
(a) p ∈ Z[v] and (b) p (0) = h (0), for each y > x. Both of these follow from
y y y
P
the fact that each h ∈ Z[v] and each C ∈ H + vZ[v]H . By construction
y y y z z
hP ,C i=δ for all z ∈W. (cid:3)
x z x,z
P
3.7. Theorem. Assume W is finite. Then for each x ∈ W there exists a unique
P ∈H such that hP ,C i=δ for all y ∈W.
x x y x,y
Proof. Existence was established in the previous Lemma. Uniqueness follows from
the evident fact that the form h·,·i is non-degenerate. (cid:3)
4. Dualities between the bases
4.1. Lemma. Assume W is finite. Let w be the longest element in W. Then for
0
all x∈W, P =CxH for some self-dual Cx ∈H.
x w0
Proof. Proceed by induction. We start our induction with P = H . Now let
w0 w0
x ∈ W be given and suppose we know that P = CyH , with Cy = Cy, for all
y w0
y > x. If x 6= w find s ∈ S such that sx > x. Then, by the proof of Lemma 3.6
0
and the induction hypothesis,
P =b(C )P − p P =b(C )CsxH − p CyH ,
x s sx y y s w0 y w0
y>x y>x
X X
where p = hb(C )P ,C i. To complete the proof it suffices to show that p ∈
y s sx y y
Z[v+v−1] for each y. By Prop. 3.5
hb(C )P ,C i=hP ,b(C )C i.
s sx y sx s y
If sy < y, then b(C )C = (C −(v+v−1))C = 0 by Prop. 2.10. If sy > y, then
s y s y
once again using Prop. 2.10,
b(C )C =(C −(v+v−1))C =C −(v+v−1)C + m C
s y s y sy y z z
z
X
for appropriate integers m ∈Z. The result follows. (cid:3)
z
4.2. Theorem. Assume W is finite. Let w be the longest element in W. Then
0
b(C )H =P
x w0 xw0
for all x∈W.
Proof. By Lemma 4.1 P H−1 is self-dual. On the other hand, by Lemma 3.6,
xw0 w0
P H−1 ∈H H−1+ vZ[v]H H−1.
xw0 w0 xw0 w0 y w0
y>Xxw0
6 R.VIRK
Using the fact that H H−1 =H H−1H−1 =H−1 and applying d we obtain
z w0 z z w0z−1 w0z−1
P H−1 ∈H + v−1Z[v−1]H .
xw0 w0 x yw0
y>Xxw0
By the uniqueness of C (Thm. 2.7) we must have that
xw0
P H−1 =b(C ). (cid:3)
xw0 w0 x
References
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(1999), 1-31(electronic).
[BGS] A.Beilinson,V.Ginzburg,W.Soergel,KoszuldualitypatternsinRepresentationThe-
ory,JournalofA.M.S.9(1996), 473-527.
[KL] D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent.
Math.53(1979), 165-184.
[So1] W. Soergel,Kazhdan-Lusztig polynomials and a combinatoric for tilting modules, Repre-
sentationTheory1(1997), 37-68(electronic).
[So2] W. Soergel, Character formulas for tilting modules over Kac-Moody algebras, Represen-
tationTheory2(1998), 432-448(electronic).
[So3] W.Soergel,Andersenfiltrationand hard Lefschetz,Geom.Funct.Anal.17(2008),no.6,
2066-2089.
[Virk] R.Virk,A note on Hecke patterns incategory O,arXiv:1012.2416v1.
DepartmentofMathematics,University of California,Davis, CA95616
E-mail address: [email protected]