Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra

17 Pages·0.26 MB·English

by  Thomas J. Haines

Formulae relating the Bernstein and Iwahori-Matsumoto
presentations of an affine Hecke algebra
Thomas J. Haines, Alexandra Pettet
2
0
0
2 Abstract
n Weconsiderthe“antidominant”variantsΘ−λ oftheelementsΘλ occurringintheBernsteinpresenta-
a tion of an affine Heckealgebra H. Wefind explicit formulae for Θ− in terms of the Iwahori-Matsumoto
λ
J generators Tw (w ranging over the extended affine Weyl group of the root system R), in the case (i) R
8 is arbitrary and λ is a minuscule coweight, or (ii) R is attached to GLn and λ = mek, where ek is a
1 standard basis vectorand m≥1.
Intheabovecases, certain minimalexpressions forΘ−λ playacrucialrole. Suchminimal expressions
T] exist in fact for any coweight λ for GLn. We give a sheaf-theoretic interpretation of the existence of a
R minimalexpression forΘ−λ: thecorrespondingperversesheafontheaffineSchubertvarietyX(tλ)isthe
push-forwardofanexplicitperversesheafontheDemazureresolutionm:X˜(tλ)→X(tλ). Thisapproach
h. yields, for a minuscule coweight λ of any R, or for an arbitrary coweight λ of GLn, a conceptual albeit
at less explicit expression for thecoefficient Θ−λ(w) of the basis element Tw, in terms of thecohomology of
m a fiberof the Demazureresolution.
AMS Subject classification: 20C08, 14M15.
[

1
1 Introduction
v
6
7 Let be the affine Hecke algebra associated to a root system. There are two well-known presentations
H
1 of this algebra by generators and relations, the first discovered by Iwahori-Matsumoto [7] and the second
1
by Bernstein [12], [14]; cf. 2.2.1 below. The Iwahori-Matsumoto presentation reflects the structure of the
0
Iwahori-Hecke algebra C∞(I G/I) of the split p-adic group G attached to the root system: the generators
2 c \
0 Tw correspondtothecharacteristicfunctionsofIwahoridoublecosetsIwI,wherewrangesovertheextended
/ affineWeylgroup. TheBernsteinpresentationreflectsthedescriptionoftheHeckealgebraasanequivariant
h
K-theoryofthe associatedSteinbergvariety,whichplaysa roleinthe classificationofthe representationsof
t
a , see [9], [2]. The Bernstein presentation has the advantage that one can construct a basis for the center
m H
of by summing the generators Θ over Weyl-orbits of coweights λ; the resulting functions are known as
λ
H
: Bernstein functions.
v
i It is of interest to give an explicit relation between the generators in these two presentations. More
X
precisely, one would like to write each Θ as an explicit linear combination of the Iwahori-Matsumoto basis
λ
r elements T . A direct consequence would be the explicit description of the Bernstein functions (and thus
a w
the center of ) in terms of the Iwahori-Matsumoto basis. This problem was considered earlier by the first
H
author[4],[5]becauseofcertainapplicationstothe studyofShimuravarieties,andwascompletely answered
there for the case where λ is a minuscule coweight. More recently, O. Schiffmann [16] has given explicit
formulae for all elements in a certainbasis for the center Z( ) of anaffine Hecke algebra of type A; from
H H
this one can derive a formula for the Bernstein function z , where µ is any dominant coweightof a groupof
µ
type A.
In this paper we consider the “antidominant” variants Θ− of the elements Θ . The support of these
λ λ
functions is somewhatmore regularthan the originalfunctions Θ , cf. Lemma 2.1. In section 3 we consider
λ
the case where λ is a minuscule coweight, and we prove the following explicit formula for Θ−.
λ
Theorem 1.1 Let λ X be minuscule. Then
∗
∈
Θ− = R˜ (Q)T˜ .
λ x,tλ x
{x:λX(x)=λ}
1
Here λ(x) is the translation part of x “on the left” defined by the decomposition x = t w (w W ),
λ(x) 0
∈
T˜ is a renormalization of the usual Iwahori-Matsumoto generator T , Q = q−1/2 q1/2, and R˜ (Q) is a
x x x,y
−
variant of the usual R-polynomial of Kazhdan-Lusztig [8].
The formula above is analogous to the expression for Θ found by the first author in Proposition 4.4 of
λ
[5]:
Θ = R˜ (Q)T˜ .
λ x,tλ x
{x:t(x)=λ}
X
Here t(x) is the translation part of x “on the right” defined by the decomposition x = wt (w W ).
t(x) 0
∈
However our proof is simpler and more direct than that of loc.cit., and the same arguments appearing here
also give a short proof of the formula for Θ . In fact one can derive the formula for Θ− from that for Θ ,
λ λ −λ
athnednvιi(cΘe-ver)s=a.ΘIn−deaendd,ιifiιnt:eHrch→anHgesdetnhoetfeosrmthuelaaentfio-rinΘvolutaionnddΘe−te.rmWienerdembayrqk1t/h2a7→t Tqh1/e2oraenmd1T.x17→remTax−in1s,
−λ λ −λ λ
valid for Hecke algebras with arbitrary parameters.
InthefourthsectionwestudycoweightsofGL oftheformλ=me ,wheree isthek-thstandardbasis
n k k
vector and m Z . The case m = 1, studied in [4] and [5], has relevance to a certain family of Shimura
+
∈
varieties with bad reduction, known as the Drinfeld case. The general case is referred to as multiples of the
Drinfeld case. We prove the following formula for Θ− .
mek
Theorem 1.2 Let 1 k n and m 1. Then
≤ ≤ ≥
Θ− = R˜ (Q)T˜ .
mek x,tmek x
{x:λ(Xx)(cid:22)mek}
Here denotes the usual partial order on the lattice X .
∗
(cid:22)
Theorems 1.1 and 1.2 yield explicit expressions for the Bernstein functions z (µ minuscule) and z ,
µ me1
respectively; see Corollary 3.6 and 4.2. The expressions in these special cases seem much simpler than the
corresponding ones given by Schiffmann [16].
Theorems 1.1 and 1.2 rely on the existence of certain minimal expressions for Θ−: these are expressions
λ
of the form
Θ− =T˜ǫ1 T˜ǫrT˜ ,
λ t1 ··· tr τ
where t = t t τ (t S , τ Ω) is a reduced expression and ǫ 1, 1 for every 1 i r. In
λ 1 r i a i
the final two se·c·t·ions, we∈discuss∈how one can approach a general form∈ul{a fo−r Θ}− when λ is≤an a≤rbitrary
λ
coweightof GL , through minimal expressions (which always exist in this setting, cf. section 5). The result
n
is much less explicit than Theorems 1.1 and 1.2, and involves the geometry of the Demazure resolution
X(t ) X(t )ofthe affineSchubertvarietyX(t ). We defineaperversesheafΞ− onthe affineflagvariety
whoλse→correspλonding function in the Hecke algebλra is ε Θ−. It turns out that Ξλ− is supported on X(t ).
λ λ λ λ
Wee see thatthe existenceofa minimalexpressionforΘ− is analogousto the existence ofa certainexplicitly
λ
determined perverse sheaf on X(t ) whose push-forward to X(t ) is Ξ−. More precisely, we conclude the
λ λ λ
paper with the following result (cf. Theorem 6.7, Corollary6.8 for a completely precise statement).
e
Theorem 1.3 Let λ be a minuscule coweight of a root system, or any coweight for GL . Choose a minimal
n
expression for Θ− and let m : X(t ) X(t ) denote the corresponding Demazure resolution for X(t ).
λ λ → λ λ
Then there exists an explicit perverse sheaf on X(t ) (determined by the choice of minimal expression for
λ
Θ−) such that e D
λ
Rme( )=Ξ−.
∗ D λ
Consequently, if we denote the coefficient of T in the expression for Θ− by Θ−(x), then we have
x λ λ
Θ−(x)=ε Tr(Fr ,H•(m−1(x), )),
λ λ q D
for any x t in the Bruhat order. Here the right hand side denotes the alternating trace of Frobenius on
λ
≤
the ´etale cohomology of the fiber over x X(t ) with coefficients in the sheaf .
λ
∈ D
Given a coweight λ for an arbitrary root system, let λ denote the dominant coweight in its Weyl-orbit.
d
WeremarkthatthereisasimilarformulaforΘ−,provided thatλ isasumofminusculedominant coweights.
λ d
2
2 Preliminaries
2.1 Affine Weyl group
Let (X∗,X ,R,Rˇ,Π) be a rootsystem, where Π is the set ofsimple roots. The Weyl groupW is generated
∗ 0
by the set of simple reflections s :α Π .
α
{ ∈ }
We define a partial order on X (resp. X∗) by setting λ µ whenever µ λ is a linear combination
∗
(cid:22) (cid:22) −
with nonnegative integer coefficients of elements of αˇ :α Π (resp. α:α Π ). We let Π denote the
m
{ ∈ } { ∈ }
set of roots β R such that β is a minimal element of R X∗ with respect to .
∈ ⊂ (cid:22)
In section 4 we will use the following description of the relation for coweights of GL : (λ ,...,λ )
n 1 n
(cid:22) (cid:22)
(µ ,...µ ) if and only if λ + +λ µ + +µ for 1 i n 1, and λ + +λ =µ + +µ .
1 n 1 i 1 i 1 n 1 n
··· ≤ ··· ≤ ≤ − ··· ···
Let W be the semidirect product X ⋊W = t w : w W ,x X , with multiplication given by
∗ 0 x 0 ∗
{ ∈ ∈ }
txwtx′w′ = tx+w(x′)ww′. For any x W, there exists a unique expression tλ(x)w, where w W0 and
∈ ∈
λ(x) Xf.
∗
∈
Let f
S = s :α Π t s :α Π W.
a α −αˇ α m
{ ∈ }∪{ ∈ }⊂
Define length l:W Z by
→ f
f l(txw)= α,x 1 + α,x .
|h i− | |h i|
α∈R+:wX−1(α)∈R− α∈R+:wX−1(α)∈R+
LetQˇ be the subgroupofX generatedbyRˇ. The subgroupW =Qˇ⋊W ofW is a CoxetergroupwithS
∗ a 0 a
the set of simple reflections. The subgroup is normal and admits a complement Ω= w W :l(w)=0 .
{ ∈ }
For w W denote ε =( 1)l(w) and q =ql(w) (for q any parameter). f
w w
∈ −
The Coxeter group (W ,S ) comes equipped with the Bruhat order . We extend it fto W as follows:
a a
≤
we say wτ fw′τ′ (w,w′ W , τ,τ′ Ω) if w w′ and τ =τ′.
a
≤ ∈ ∈ ≤
Let µ X be dominant. Following Kottwitz-Rapoport [10], we say x W is µ-admissiblefif x t
∗ w(µ)
∈ ∈ ≤
for some w W . We denote the set of µ-admissible elements by Adm(µ).
0
∈
f
2.2 Hecke algebra
2.2.1 Presentations
The braid group of W is the group generated by T (w W) with relations
w
∈
f TwTw′ =Tww′ whenever l(fww′)=l(w)+l(w′).
The Hecke algebra is defined to be the quotient of the group algebra (over Z[q1/2,q−1/2]) of the braid
H
group of W, by the two-sided ideal generated by the elements
(T +1)(T q),
f s s
−
for s S . The image of T in is again denoted by T . It is known that the elements T (w W) form
a w w w
∈ H ∈
a Z[q1/2,q−1/2]-basisfor . The presentationof using the generatorsT andthe aboverelationsis called
w
H H
the Iwahori-Matsumoto presentation. f
For any T , define a renormalization T˜ = q−l(w)/2T . Define an indeterminate Q= q−1/2 q1/2. The
w w w
elements T˜ form a basis for , and the usual relations can be written as −
w
H
T˜ , if l(sw)=l(w)+1,
T˜ T˜ = sw
s w QT˜ +T˜ , if l(sw)=l(w) 1,
(cid:26) − w sw −
for w W and s S . There is also a right-handed version of this relation. Note that T˜−1 =T˜ +Q.
We∈will denote∈T˜a (λ X ) simply by T˜ . s s
tλ ∈ ∗ λ
For λf X , define
∗
∈ Θ =T˜ T˜−1
λ λ1 λ2
3
where λ = λ λ , and λ ,λ are dominant. The elements Θ generate a commutative subalgebra of .
1 2 1 2 λ
− H
It is known that the elements Θ T (λ X , w W ) form a Z[q1/2,q−1/2]-basis for . These generators
λ w ∗ 0
∈ ∈ H
satisfywell-knownrelations(seeProp. 3.6,[14]);incasetherootsystemissimplyconnected,thesearegiven
by the formula
Θ Θ
λ s(λ)
Θ T T Θ =(q 1) − ,
λ s− s s(λ) − 1 Θ
−αˇ
−
where s=s and α Π. The presentationof with generatorsΘ T and the above relations is called the
α λ w
∈ H
Bernstein presentation.
We also define
Θ−λ =T˜λ′1T˜λ−′21
where λ=λ′ λ′, and λ′,λ′ are antidominant.
1− 2 1 2
The involution a a of Z[q1/2,q−1/2] determined by q q−1 extends to an involution h h, given by
→ 7→ →
a T = a¯ T−1 .
w w w w−1
X X
ItisimmediatethatΘ =Θ−. ClearlytheBernsteinpresentationgivesrisetoananalogouspresentation
λ λ
using the generators Θ−T in place of Θ T .
λ w λ w
2.2.2 Bernstein functions
For each W -orbit M in X , define the Bernstein function z attached to M by
0 ∗ M
z = Θ .
M λ
λ∈M
X
When the W -orbit M contains the dominant element µ, this function is denoted by z .
0 µ
From Corollary 8.8 of Lusztig [12], we have z =z . Consequently,
µ µ
z = Θ−.
µ λ
λ∈XW0(µ)
2.2.3 A support property
The preceding formula implies that when one studies Bernstein functions there is no harm in working with
the functions Θ− instead of the functions Θ . We do so in this paper because their supports enjoy a nice
λ λ
regularity property, given by the following lemma.
Lemma 2.1 For λ X , we have
∗
∈
supp(Θ−) x : λ(x) λ .
λ ⊂{ (cid:22) }
Proof. Write
Θ− = a (Q)T˜ ,
λ y y
yX≤tλ
where a (Q) Z [Q] (see Lemma 5.1 and Corollary 5.7 of [4]).
y +
Choose a∈dominant coweight µ′ such that µ′ +λ(x) is also dominant for any x in the support of Θ−.
λ
Thus we have
T˜−1 =Θ− =Θ−Θ− = a (Q)T˜−1T˜ .
−(µ′+λ) µ′+λ µ′ λ y −µ′ y
y
X
Let y ∈ supp(Θ−λ). We claim that tµ′y belongs to the support of T˜−−µ1′T˜y. Indeed, under the specialization
map H→Z[W] determined by q1/2 7→1, the element T˜−−µ1′T˜y maps to tµ′y. Since no cancellation occurs on
the right hanfd side above, we see from this that tµ′y ∈ supp(T˜−−(1µ′+λ)), and thus
tµ′+λ(y)wy =tµ′y tµ′+λ,
≤
4
where y = t w . Since µ′ + λ(y) and µ′ + λ are both dominant, it is well-known that this implies
λ(y) y
µ′+λ(y) µ′+λ. The lemma follows.
(cid:22)
In the case where λ is minuscule, this statement can be considerably sharpened; see Corollary 3.5.
We remark that Lemma 2.1 plays a key role in the proof of Theorem 4.1.
2.2.4 R-polynomials
For any y W, let y = s s τ (s S ,τ Ω) be a reduced expression for y. Then for any x, we can
1 r i a
∈ ··· ∈ ∈
write
f T˜−1 = R (Q)T˜ .
y−1 x,y x
xX∈W
e
where Rx,y(Q) Z[q1/2,q−1/2]. These coefficients Rfx,y(Q) can be thought of as polynomial expressions in
∈
Q (as the notation suggests) because of the identity
e e
T˜−1 =(T˜ +Q) (T˜ +Q)T˜ .
y−1 s1 ··· sr τ
3 The minuscule case
We sayλ X is minuscule if α,λ 0, 1 ,for everyrootα R. Suchcoweightsare the concernof this
∗
∈ h i∈{ ± } ∈
section.
The purpose of this section is to present an analogue of Proposition 4.4 from [5] using Θ− instead of
λ
Θ . For simplicity, the theorem is given here for affine Hecke algebras with trivial parameter systems. The
λ
generalization to arbitrary parameter systems is straightforward (see [5] for notation and details). Similar
arguments to those appearing here apply to Θ , giving a short proof of Proposition 4.4 from [5].
λ
Theorem 3.1 Let µ− be minuscule and antidominant, and λ W (µ−). Then
0
∈
Θ− = R (Q)T˜ .
λ x,tλ x
x :Xλ(x)=λ
e
We begin with some lemmas. For a proof of the first lemma, refer to Proposition 3.4 of [5], where a
similar result is given (see also the proof of Corollary 6.6).
Lemma 3.2 Let µ− be an antidominant and minuscule coweight, and let τ Ω be the unique element such
∈
that tµ− ∈ Waτ. Let λ ∈ W0(µ−). Suppose that λ−µ− is a sum of p simple coroots (0 ≤ p ≤ l(tµ−) = r).
Then there exists a sequence of simple roots α ,... ,α such that the following hold (setting s =s ):
1 p i αi
(1) α ,s s (µ−) = 1, 1 i p;
i i−1 1
h ··· i − ∀ ≤ ≤
(2) there is a reduced expression for tµ− of the form tµ− =s1···spt1···tr−pτ;
(3) there is a reduced expression for t of the form t =t t (τs ) (τs )τ;
λ λ 1 r−p 1 p
··· ···
(4) Θ− =T˜ T˜ T˜−1 T˜−1T˜ ,
λ t1··· tr−p τs1 ··· τsp τ
where t S , j 1,... ,r p .
j a
∈ ∀ ∈{ − }
Lemma 3.3 Let x W, and suppose that xs >x for all α Π. Then l(xw)=l(x)+l(w) for all w W .
α 0
∈ ∈ ∈
Since µ− is antidominfant, this lemma applies to the expressiontµ− =s1···spt1···tr−pτ. It then applies to
the expression t t τ as well. It follows that we can think of the formula in Lemma 3.2(4) as
1 r−p
···
Θ− =T˜ T˜−1 T˜−1 =T˜ T˜−1
λ wλ s1 ··· sp wλ w−1
wheret =wλw,withw W andwλtheminimallengthrepresentativeforthecosett W . Thisobservation
λ 0 λ 0
∈
is helpful towards proving the main result of this section.
5
Lemma 3.4 For λ, s ,... ,s , and t ,... ,t as in Lemma 3.2, the mapping
1 p 1 r−p
y : y s s x : λ(x)=λ and x t
1 p λ
{ ≤ ··· } −→ { ≤ }
defined by
y t t τy =wλy
1 r−p
7→ ···
is bijective.
Proof of Theorem 3.1. Let w = s1···sp, and wλ = t1···tr−pτ, so that tµ− = wwλ and tλ = wλw. We
have
T˜−1 T˜−1 = R (Q)T˜
s1 ··· sp y,w y
y : y≤w
X
e
The expression for Θ− of Lemma 3.2, together with the fact that T˜ T˜ = T˜ for all y W (since
λ wλ y wλy ∈ 0
l(wλy)=l(wλ)+l(y)), implies
Θ− = R (Q)T˜ .
λ y,w wλy
y : y≤w
X
e
Using the recursion formula of Lemma 2.5 (1) from [5], we obtain R (Q) = R (Q). In view of the
y,w wλy,tλ
bijection given in Lemma 3.4, we have
e e
R (Q)T˜ = R (Q)T˜ ,
wλy,tλ wλy x,tλ x
y X: y≤w x :Xλ(x)=λ
e e
which completes the proof.
For the minuscule case, Theorem 3.1 yields the following improvement on Lemma 2.1.
Corollary 3.5 Let λ X be minuscule. Then
∗
∈
supp(Θ−)= x : λ(x)=λ and x t .
λ { ≤ λ}
Here we have used Lemma 2.5 (5) of [5], which asserts that R˜ (Q)=0 if and only if x y.
x,y
6 ≤
The Bernstein function z has a very simple form when µ is minuscule (cf. Theorem 4.3 of [5]):
µ
Corollary 3.6 If µ is dominant and minuscule, then
z = R˜ (Q)T˜ .
µ x,tλ(x) x
x∈AXdm(µ)
4 Multiples of the Drinfeld case
Fix positive integers n and m, and an integer 1 k n. In this section, we establish in Theorem 4.1
a formula for the Θ− functions of GL when λ =≤me≤(where e denotes the coweight of GL with kth
λ n k k n
coordinate equal to 1, and all other coordinates equal to 0).
In this section, we adopt the following notation: for 1 i n 1, let α = αˇ = e e , and let
i i i i+1
≤ ≤ − −
s =s . We single out the element τ Ω given by τ =t s s .
i αi ∈ (1,0,...,0) 1··· n−1
Theorem 4.1 For the coweight me of GL , we have
k n
Θ− = R (Q)T˜ .
mek x,tmek x
x : λX(x)(cid:22)mek
e
6
Consequently, we have a result for me analogous to that of Corollary 3.5 for λ minuscule, that is,
k
supp(Θ− )= x : λ(x) me and x t .
mek { (cid:22) k ≤ mek}
We also get the following explicit formula for the Bernstein function z , analogous to Corollary 3.6:
me1
Corollary 4.2
z = R˜ (Q) T˜ ,
me1  x,tλ  x
x∈AXdm(µ) λ(x)(cid:22)Xλ,x≤tλ
 
where the inner sum ranges over λ W (me ) such that λ(x) λ and x t .
0 1 λ
∈ (cid:22) ≤
Werequirethreelemmasbeforetheproofofthetheorem. Inthefollowingargumentsweusethenotation
to denote products even though we are working in a non-commutative ring. We will use the following
convention: n a will denote the product a a a (in that order).
Q i=1 i 1 2··· n
Lemma 4.3QFor the coweight me of GL , we have
k n
Θ− =(T˜ T˜ T˜ T˜−1 T˜−1)m.
mek sk−1··· s1 τ sn−1··· sk
Proof. From Lemma 3.2, we have
Θ− =T˜ T˜ T˜ T˜−1 T˜−1.
ek sk−1··· s1 τ sn−1··· sk
Then the formula for Θ− follows from the fact that Θ− =Θ−Θ− for all ν ,ν X .
mek ν1+ν2 ν1 ν2 1 2 ∈ ∗
Lemma 4.4 Let w,y W˜. Then
∈
T˜ T˜ = a T˜
w y x x
x=wy
X
where w and y range over certain subexpressions of w and y, respectively.
ee
Proof. This is an easy induction on the length of y.
e e
Lemma 4.5 Let x be a subexpression of
t =(s s τs s )m
mek k−1··· 1 n−1··· k
such that for some 1 i k 1, at least one s is deleted. Then λ(x)(cid:14)me .
i k
≤ ≤ −
Proof. We can write
x=u τv u τv
1 1 m m
···
for suitable subexpressions u ,... ,u and v ,... ,v of s s and s s , respectively. Suppose
1 m 1 m k−1 1 n−1 k
··· ···
that p is the least index such that u =s s . Then
p k−1 1
6 ···
p−1 m
x= t s s v (t u s s v ) u τv ,
ek k··· n−1 i ej p 1··· n−1 p i i
(cid:18)i=1 (cid:19) (cid:18)i=p+1 (cid:19)
Y Y
for some j with 1 j <k. Since s s v(e )=e for any subexpression v of s s ,
k n−1 j j n−1 k
≤ ··· ···
p−1 p−1 −1
t s s v (t ) t s s v =t .
ek k··· n−1 i ej ek k··· n−1 i ej
(cid:18)i=1 (cid:19) (cid:18)i=1 (cid:19)
Y Y
7
It follows that the translation part λ(x) is the sum of e and a non-negative integral linear combination of
j
vectors e (i=1,...,n). Indeed, the translation part of
i
p−1 m
t s s v (u s s v ) u τv
ek k··· n−1 i p 1··· n−1 p i i
(cid:18)i=1 (cid:19) (cid:18)i=p+1 (cid:19)
Y Y
is necessarily a vector (b ,...,b ) where b Z for every i.
1 n i +
∈
We thus see that one of the first k 1 coordinates of λ(x) is positive (namely the j-th coordinate is),
−
and this implies that λ(x)(cid:14)me .
k
Proof of Theorem 4.1. LetE =( 0,1 n−1)m ( 1 k−1 0,1 n−k)m. UsingT˜−1 =T˜ +Qandexpanding
{ } − { } ×{ } s s
the left hand side, we can write
m
T˜−1 =Θ− + Qm(n−1)−σ(ǫ) T˜ǫik−1 T˜ǫi1T˜ T˜ǫin−1 T˜ǫik. (1)
−mek mek sk−1 ··· s1 τ sn−1 ··· sk
ǫ∈E i=1
X Y
Here ǫi 0,1 and we denote ǫ = (ǫ1,... ,ǫm), and ǫi = (ǫi,... ,ǫi ), i = 1,... ,m, and σ(ǫ) =
j ∈ { } 1 n−1
m n−1ǫi.
i=1 j=1 j
From Lemma 2.1, we know that supp(Θ− ) x : λ(x) me . Thus we need only prove that if T˜
P P mek ⊂{ (cid:22) k} x
is in the support of the second term on the right hand side, then λ(x) (cid:14) me . Indeed, then the first and
k
second terms on the right hand side of (1) have disjoint supports, and so the coefficients of like terms will
be equal in T˜−1 and Θ− .
−mek mek
Let ǫ=(ǫ1,... ,ǫm) E, and consider
∈
m
T˜ǫik−1 T˜ǫi1T˜ T˜ǫin−1 T˜ǫik.
sk−1 ··· s1 τ sn−1 ··· sk
i=1
Y
By Lemma 4.4, if x is in the support of this product, then x is a subexpression of
m
sǫik−1 sǫi1τsǫin−1 sǫik.
k−1 ··· 1 n−1 ··· k
i=1
Y
SinceE excludestheelementsof( 1 k−1 0,1 n−k)m,weknowthatforsome1 i mand1 j k 1,
{ } ×{ } ≤ ≤ ≤ ≤ −
that ǫi = 0. But this is equivalent to the deletion of some s (1 j k 1) from the expression
j j ≤ ≤ −
t = (s s τs s )m. By Lemma 4.5, we have λ(x)(cid:14)me , and the proof is complete.
mek k−1··· 1 n−1··· k k
5 Minimal expressions
We say Θ− has a minimal expression if it can be written in the form
λ
Θ− =T˜ǫ1 T˜ǫrT˜ ,
λ t1 ··· tr τ
where t = t t τ (t S , τ Ω) is a reduced expression and ǫ 1, 1 for every 1 i r. Such
λ 1 r i a i
··· ∈ ∈ ∈ { − } ≤ ≤
expressions played a key role in Theorems 3.1 and 4.1.
Lemma 3.2 asserts that Θ− has a minimal expression whenever λ is minuscule. If λ is any coweight for
λ
GL , then we may write
n
λ=λ + +λ
1 k
···
where each λ is minuscule and
j
l(t )=l(t )+ +l(t ).
λ λ1 ··· λk
8
It follows that for any coweight λ of GLn, there is a minimal expression for Θ−λ. Letting wλi denote the
minimal representative for the coset tλiW0 and writing tλi = wλiwi (wi ∈ W0), we may recover a minimal
expression for
Θ− =T˜ T˜−1 T˜ T˜−1
λ wλ1 w−1··· wλk w−1
1 k
by choosing reduced expressions for every wλi and wi.
ClearlyasimilarresultwouldfollowforanyrootsystemwiththepropertythatΘ− hasaminimalexpres-
ω
sion for every Weyl conjugate ω of every fundamental coweight. It seems to be an interesting combinatorial
problem to determine the root systems (besides that for GL ) which satisfy this property.
n
In principle, a minimal expression for Θ− allows one to write it as an explicit linear combination of the
λ
Iwahori-Matsumoto generators T , simply by using the formula T˜−1 =T˜ +Q and expanding the product.
w s s
The result is a linear combination of certain products
T T T
s1··· sg σ
(s S , σ Ω), where s s σ ranges over certain subexpressions of t (which subexpressions occur is
i a 1 g λ
∈ ∈ ···
governedbythesignsǫ intheminimalexpression). Thesemayinturnbesimplifiedbyusingthewell-known
j
formula
T T = N(s,w,q)T ,
s1··· sg w
w
X
(cf. [11],Lemma3.7). Heres=(s ,...,s )andN(s,w,q)isthenumberofF -rationalpointsonthevariety
1 g q
Z(s,w)consistingofallsequences(I ,...,I )wherethe I areIwahorisubgroupsofGsc(F¯ ((t))) (hereGsc is
1 g i q
the simply connected groupassociatedto the givenroot system) suchthat the relativepositions of adjacent
subgroups satisfy
inv(I ,I )=s
i−1 i i
for all 1 i g, and I =wI w−1, where I is a fixed “standard” Iwahori subgroup.
g 0 0
Wefo≤rgo≤thecumbersometaskofdescribingmorecompletelytheresultingexpressionsforΘ− intermsof
λ
thegeneratorsT . ThecombinatoricsarebestdescribedinthegeometricframeworkofDemazureresolutions.
w
We explain this in the following section.
Remark. Let λ be a coweightfor GL , and write λ=m e + +m e . One finds a similar expression for
n 1 1 n n
Θ− starting from ···
λ
Θ− =Θ− Θ− ,
λ m1e1··· mnen
and making use of Theorem 4.1.
6 Sheaf-theoretic meaning of minimal expressions
The goal of this section is to describe a sheaf-theoretic interpretation of a minimal expression for Θ−: the
λ
corresponding perverse sheaf on the affine flag variety is the push-forward of an explicit perverse sheaf on a
Demazure resolution of the Schubert variety X(t ). We proceed to illustrate this statement in more detail.
λ
6.1 Affine flag variety
Let k =F denote the finite field with q elements, and let k¯ denote an algebraic closure of k. Let G be the
q
split connected reductive group over k whose root system is (X∗,X ,R,Rˇ,Π). Choose a split torus T and
∗
a k-rational Borel subgroup B containing T, which give rise to R and Π.
Denote by l the affine flag variety for G. This is an ind-scheme over k whose k-points are given by
F
l(k)=G(k((t)))/I ,
k
F
where I =I G(k[[t]]) is the Iwahori subgroup whose reduction modulo t is B.
k
Fix a prim⊂e ℓ=char(k), and make a fixed choice for √q Q¯ℓ (for Tate twists).
LetDb( l)den6 otethecategoryDb( l,Q¯ ). Bydefinition∈Db( l,Q¯ )istheinductive2-limitofcategories
Db(X,Q¯ )wFhereX lrangesovercalFlprojℓectivek-schemeswchiFchareℓclosedsubfunctorsoftheind-scheme
c ℓ ⊂F
9
l. The category Db(X,Q ) is the “derived” category of Deligne [3]: Q the projective 2-limit of the
cFategories Db (X,Z/cℓnZ). ℓFor any finite extension E of Q contained in Q¯ℓ⊗, the definition of Db(X,E) is
ctf ℓ ℓ c
similar, and by definition Db(X,Q¯ ) is the inductive 2-limit of the categories Db(X,E).
c ℓ c
For f : X Y a morphism of finite-type k-schemes, we have the four “derived” functors f ,f :
∗ !
Db(X,Q¯ ) D→b(Y,Q¯ ) andf∗,f! :Db(Y,Q¯ ) Db(X,Q¯ ). This notationshouldcause no confusion,since
c ℓ → c ℓ c ℓ → c ℓ
we never use the non-derived versions of the pull-back and push-forward functors in this paper.
We define the category P ( l): it is the full subcategory of Db( l) whose objects are I-equivariant
I
F F
perverse sheaves for the middle perversity (by definition the latter have finite dimensional support).
The I-orbits on l correspond to W. Given w W, we denote by Y(w) = IwI/I the corresponding
Bruhat cell, and weFdenote its closure by X(w) = Y∈(w). Further, let Q¯ denote the constant sheaf on
ℓ,w
Y(w), and define =Q¯ [l(w)](l(w)f/2). This is a seflf-dual perverse sheaf on Y(w).
w ℓ,w
A
Let j : Y(w) ֒ X(w) denote the open immersion. We define J = j and J = j . These
w w∗ w∗ w w! w! w
→ A A
are perverse sheaves in P ( l) satisfying D(J )=J . (Here D denotes Verdier duality.)
I w∗ w!
F
Given P ( l) we may define the corresponding function [ ] on l(k), which we may identify with
I
G ∈ F G F
an element in :
H
[ ](x)=Tr(Fr , ),
q x
G G
where Frq denotes the Frobenius morphism on Flk¯ (raising coordinates to power q). We have
[J ]=ε q−1/2T
w! w w w
[J ]=ε q1/2T−1 .
w∗ w w w−1
6.2 Convolution of sheaves
Following Lusztig [15], one can define a convolution product ⋆ :P ( l) P ( l) Db( l). We formulate
I I
F × F → F
this in a way similar to [6]. Given P ( l), i = 1,2, we can choose X(w ) such that the support of
i I i i
G ∈ F G
is contained in X(w ), for i = 1,2. We may identify l with the space of all “affine flags” for G(k((t)));
i
F L
there is a base point whose stabilizer in G(k((t))) is the “standard” Iwahori subgroup I. Then X(w) is
0
L
identified with the space of all affine flags such that the relative position between the base point and
0
L L
satisfies
L
inv( , ) w
0
L L ≤
in the Bruhat order on W. The “twisted” product X(w )˜X(w ) is the space of pairs ( , ′) l l
1 2
× L L ∈ F ×F
such that
f
inv( , ) w
0 1
L L ≤
inv( , ′) w .
2
L L ≤
Wecanfindafinite-dimensionalprojectivesubvarietyX lwiththepropertythat( , ′) X(w )˜X(w )
1 2
⊂F L L ∈ × ⇒
′ X. The “multiplication” map m:X(w )˜X(w ) X given by ( , ′) ′ is proper.
1 2
L ∈Now (i = 1,2) determine a well-defined×pervers→e sheaf ⊠˜ oLnLX(w7→)L˜X(w ) (see e.g. [6]). We
i 1 2 1 2
G G G ×
define
⋆ =m ( ⊠˜ ).
1 2 ∗ 1 2
G G G G
The convolution ⋆ Db( l) is independent of the choice of X(w ) and X.
1 2 i
G G ∈ F
The object ⋆ is I-equivariantin a suitable sense, so that we can regardits function Tr(Fr , ⋆ )
1 2 q 1 2
G G G G
as an element of the Hecke algebra .
H
It is well-known that this product is compatible with the function-sheaf dictionary:
[ ⋆ ]=[ ]⋆[ ].
1 2 1 2
G G G G
Here ⋆ on the right hand side is just the usual product in .
H
Later we shall use the following fact, referred to in the sequel simply as associativity: if (i = 1,2,3)
i
G
areobjectsofP ( l)suchthat ⋆ P ( l)and ⋆ P ( l),thenthereisacanonicalisomorphism
I 1 2 I 2 3 I
F G G ∈ F G G ∈ F
⋆( ⋆ )˜( ⋆ )⋆ (the “associativity constraint”). This is provedby identifying eachcanonically
1 2 3 1 2 3
G G G → G G G
with the “triple product” ⋆ ⋆ , whose construction is similar (see section 6.3).
1 2 3
G G G
10