Characters of Reductive Groups over a Finite Field. (AM-107), Volume 107 9781400881772

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Annals of Mathematics Studies Number 107

CHARACTERS OF REDUCTIVE GROUPS OVER A FINITE FIELD BY

GEORGE LUSZTIG

PR IN C E TO N U N IV E R S IT Y PRESS

PRINC ETO N, N E W JERSEY 1984

C o p y rig h t ©

1984 b y P rin ce to n U n iv e rs ity Press

ALL RIGHTS RESERVED

T h e A n n a ls o f M a th em a tics Studies are ed ited b y W illia m B r o w d e r, R o b e rt P . L a n g la n d s, John M iln o r , and E lias M . Stein C o rresp o n d in g editors: S tefan H ild eb ra n d t, H . B la in e L a w s o n , L o u is N ire n b e rg , and D a v id V o g a n

C lo th b ou n d ed itio n s o f Prin ceto n U n iv e rs ity Press b o ok s are printed on a cid -free paper, and bin din g m aterials are ch osen fo r strength and d u ra bility. P a ­ perbacks, w h ile satisfa cto ry fo r personal co lle c tio n s , are n ot u su ally suitable fo r library reb in d in g

IS B N 0 -6 9 1-0835 0-9 (c lo th ) IS B N 0-69 1-0835 1-7 (p a p er)

Prin ted in the U n ited States o f A m e r ic a b y P rin ce to n U n iv e rs ity Press, 41 W illia m Street P rin ce to n , N e w Jersey

☆

L ib r a r y o f C o n gre ss C a ta lo g in g in P u b lica tio n data w ill be fo u n d on the last printed p a g e o f this b o o k

T o Miki, Irene and Tamar

T A B L E OF CO N TE N TS

ix

IN TRO D U CTIO N 1. 2. 3.

C O M PU TA TIO N OF L O C A L IN TE R S E C TIO N COHOMOLOGY OF C E R T A IN L IN E BUNDLES OVER A SCHUBERT V A R IE T Y L O C A L IN TE R S E C TIO N COHOMOLOGY WITH TWISTED C O E F F IC IE N TS OF TH E CLOSURES OF THE V A R IE T IE S

Xw

3 30

G LO B A L IN TE R S E C TIO N COHOMOLOGY WITH TWISTED C O E F F IC IE N TS OF TH E V A R IE T Y X,„ w

58

4.

R E P R E S E N T A T IO N S OF W EYL GROUPS

76

5.

C E L L S IN W E YL GROUPS

134

6.

AN IN T E G R A L IT Y THEOREM AND A DISJOINTNESS THEOREM

180

SOME E X C E P T IO N A L GROUPS

217

8.

DECOMPOSITION OF INDUCED R E P R E S E N T A T IO N S

251

9.

C LA S S IC A L GROUPS

269

10.

C O M PLE TIO N OF TH E PRO O F OF THEOREM 4.23

296

11.

E IG E N V A LU E S OF FROBENIUS

313

12.

ON THE STR U C TU R E OF L E F T C E L L S

324

7.

13. R E LA T IO N S WITH CONJUGACY CLASSES

342

14. CONCLUDING REMARKS

351

A P P E N D IX

358

REFERE N C E S

377

SUBJECT INDEX

382

N O TA TIO N INDEX

383

IN TRO D U C TIO N

One of the aims of this book is to present a cla ssifica tio n of the irreducible representations of the finite group G (F q ), where G is a connected, reductive algebraic group with connected center defined over a fin ite field

. The representations of G(Fq) w ill be always taken

over Qg, an algebraic closure o f the field of £-adic numbers ( I fixed prime not dividing q ).

is a

Th is is because the representations of

G(Fq) are studied (fo llow in g [ D L j ] ) using the f-adic cohomology with compact support of certain lo ca lly closed

G(Fq)-stable subvarieties

Xw

of the fla g manifold of G , with co efficien ts in certain loca lly constant £-adic sheaves

5^ of rank 1 . (Here w

is a Weyl group element, X w is

the set of Borel subgroups which are in relative position w with their transform under Frobenius and 6 is a character of the group of rational points of a maximal torus T

over

F^ , corresponding to w .)

The main object of [D L j ] was the study of the virtual representation A Rip of G(Fq) obtained by taking the alternating sum over i of the G(Fq)-modules Hj,(Xw ,5 ^ ). The main result of the present work is an exp licit formula for the m u ltiplicities with which the various irreducible A representations of G(F^) appear in R ^ . It is shown in [ D L j ] that the set of pairs

(T , 0) can be partitioned in

equivalence cla sses with the follow in g properties:

the equ ivalence

cla sses are in 1-1 correspondence with the semisimple conjugacy cla sses in G *(F ) , HI

where G *

is a reductive group over F , dual to G ; HI

moreover, for any irreducible representation one equivalence class of pairs m ultiplicity in R ^

p of G(F^) there is exactly

(T , 6) such that p appears with nonzero

for some (T , 0) in that equivalence cla ss.

equivalence class corresponds to the semisimple class ix

If this

(s ) in G *(F ^ ),

X

INTRODUCTION

we shall write p e &(Sy

This gives a partition of the set of irreducible

representations of G(Fq) into disjoint pieces simple class

(s )

in G * (F q ). The piece

&(Sy

one for each sem i­

is the set of “ unipotent

representations55 of G(F_ ) ; the corresponding equivalence cla ss of pairs (T , 6) consists of those (T , 0) for which extreme, we have pieces of form

6 is trivial.

where (s )

At the other

is a regular semisimple

class; each of these pieces consists of a sin gle representation of G (F ^ ). For a general (s ) , we show that

is hi 1-1 correspondence

with the set of unipotent representations of a smaller group H (F ^ ), where H is the dual of the centralizer in G * of an element s e ( s ) . Moreover, the m u ltiplicities of the various p e

in the virtual representations

R^, with (T , 0) in the corresponding equivalence class, are the same (up to a sign

= ±1 , depending only on (s ) ) as the m ultiplicities of the

unipotent representations of H (F ) (T 'C H ),

in the virtual representations

defined with respect to H ; the degree of p e

R^/ ,

is obtained

by multiplying the degree of the corresponding unipotent representation of H(Fq) by the part prime to q of the integer

e^|G(Fq)| * |H(F )1_1 . This

result is entirely analogous to the Jordan decomposition of an element in an algebraic group as a product of a sem isimple and a unipotent element. T o state our results on unipotent representations we shall need the concept of left c e lls in a Weyl group.

This concept has its origin in work

of Robinson and Shensted, who described a 1-1 correspondence between the symmetric group @n and the set of pairs (r, O tableaux of the same shape, of s iz e

n.

of standard Young

In this picture, a left c e ll of @n

appears as the subset of @ n corresponding to the set of pairs (r, r ') with t ' fixed and r of the same shape as r ' ; one could also define a two-sided c e ll of @ n as the subset of @n corresponding to the set of pairs

( r , r ') with r, r ' of fixed shape.

Thus, @ n is decomposed into

two-sided c e lls (one for each partition of n ) and each two-sided c e ll is a disjoint union of a number of left c e lls equal to the dimension of the corresponding irreducible representation of © a picture which applies to any Coxeter group.

. Th is is the prototype of

INTRODUCTION

xi

In his study of primitive ideals in the enveloping algebra of a complex semisimple L ie algebra, Joseph has defined the concept of left c e lls in the Weyl group W , as follow s.

For w e W , let

Iw be the annihilator of

the irreducible module of the enveloping algebra with highest weight - w p - p , where p is half the sum of positive roots.

Then w, w ' are

said to be in the same left c e ll precisely when Iw = Iw ^. Joseph shows that each left c e ll can be regarded in a natural way as a basis of a repre­ sentation of W ; this representation is not n ecessarily irreducible, but Joseph [J] shows that it has a canonical “ lo w e s t” irreducible component, which appears with m ultiplicity one and is characterized by the property that its first occurrence in the space of harmonic polynomials on the Cartan subalgebra is earlier than for any other irreducible component. Joseph’s definition of left c e lls and the corresponding Weyl group representations involved some unknown quantities:

the m ultiplicities in

the Jordan-Holder series of the Verma modules with highest weight - w p T p . In [K L 1], Kazhdan and the author have proposed an algorithm for these m ultiplicities and in [K L 2L [BB ], [B K ] it has been shown that this algorithm does indeed give the desired m u ltiplicities. consequences.

This had several

It has allow ed Barbasch and Vogan [B V j], [B V 2] to deter­

mine ex p licitly which irreducible representations of the Weyl group can appear as “ lo w e s t” components o f a left c e ll, (they turn out to be pre­ c is e ly the “ s p e c ia l” representations defined in [ L 6], [ L y ] ) , and to deter­ mine ex p licitly the equivalence relation

E ~

E ' on irreducible repre-

LR

sentations of W generated by the relation: “ E, E ' appear as components of the same left c e l l . ”

(It turns out that this equivalence relation is pre­

c is e ly that of [L y ].) In this book we shall adopt the definition of left c e lls given in [ K L 1] (s e e Chapter 5).

Th is has the advantage that it is elementary, makes

sense for arbitrary Coxeter groups and that it gives rise not only to repre­ sentations of the Coxeter group, but also of the corresponding Hecke algebra.

(T h is last fact is crucial for applications to our group G (F q ).)

* The H eck e a lgeb ra corresponding to a W eyl group has been first studied by N. Iwahori (O n the structure of the H eck e ring of a C h ev alley group over a fin ite field , J. F a c. Sci. U n iv. Tokyo, sect 1A, 1 0 (1 9 6 4) 215-236. I think it would be most appropriate to c a ll it the Iw ahori algeb ra , but the name H ecke ring (or a lg e b r a ) given by Iw ahori h im self has been in u se for alm ost 20 y ears an d ’it is probably too late to change it now.

x ii

INTRODUCTION

On the other hand, as a consequence of [K L 2], [BB ], [B K ] and [V o], this definition of left c e lls coincides with that of Joseph, in the case of Weyl groups. We shall also need the concept of two-sided c e lls in W . In the language of primitive ideals, a two-sided c e ll could be defined as the union of a ll left c e lls with a prescribed “ lo w e st” component.

(It turns

out that the number of left c e lls which make up a two-sided c e ll is equal to the dimension of that lowest component.) Again, we shall adopt the definition of two-sided ce lls given in [ K L 1]; the definition (see Chapter 5) applies to any Coxeter group and gives rise to both left and right module structures over the Hecke algebra or the Weyl group on the vector space with basis given by the elements of the two-sided c e ll. relation

E ~

The equivalence

E ' defined earlier could be also defined by the requirement

LR

that E, E ' appear in the same two-sided c e ll, (for either the left or the right W-module structure). There are properties of left c e lls which are needed in the representa­ tion theory of C(F^) for which no proof is known which is independent of the theory of primitive ideals. We shall now given an example of such a property. irreducible representation of W and let

Let

E be an

E(u) be the “ corresponding”

irreducible representation of the Hecke algebra (s e e 3.1) with parameter u. (Already at this stage, if one wants an e x p licit construction of E(u) starting from E , instead of an abstract existen ce statement, one must make use of the isomorphism [L g ] between the Hecke algebra and the group algebra of W over Q(u1//2) , which depends on the bridge with the theory of primitive id ea ls.) L et Hecke algebra.

(Tx) xeW be the standard basis of the

L e t a E be the sm allest integer > 0 such that the

expression l(X ) , a E ( ~ l ) £(x> u

2

2 T r(T x ,E (u ))

is a polynomial in u1^2 , for a ll x e W , and let cx E < Z

be the constant

INTRODUCTION

term of this polynomial.

x iii

(We shall encounter c x E later in this introduc­

tion .) The property we need is:

“ If E , E ' are irreducible representa­

tions of W appearing in the same left c e ll then a E = a E ' . ”

The only

known proof of this statement is by using the bridge with the theory of primitive ideals.

(We deduce it from [B V j], [B V 2I )

C onversely, our results on representations of G(F^) have some interesting applications to the structure of left c e lls in W . We show, for example (Chapter 12) that, if W is of c la s s ic a l type, then the number of involutions contained in a left c e ll is a power of 2 and this number is the same for a ll left c e lls contained in a fixed two-sided c e ll.

For general

W, we show that in any left c e ll T , there is a unique element x Q such that c v

rp is equal to the m ultiplicity of E

in the W-module F , for

0

any irreducible W-module E . (One can show that x Q = x Q , and is lik ely that x Q is what in theory of primitive ideals one ca lls the Duflo involution in T . )

L et

F*

be the set of elements

c x £ ^ 0 for some irreducible representation E is necessarily a component of T . )

x in F

such that

of W . (It follow s that E

Th ese elements play an essen tial

role in our work. We conjecture that F * = T n F -1 ; it w ill be shown elsew here that this is true for cla s s ic a l groups, in which case

F H T "” 1

consists of the involutions in T . With each two-sided c e ll

c in W we a ssocia te a fin ite group § c .

In the case where W is irreducible,

§c

is either an elementary abelian

2-group or one of the symmetric groups @ 3, © 4,

, and is therefore

determined by its order which is equal to the denominator of the “ formal dim ension” (see 3.3) of the sp ecia l Weyl group representation correspond­ ing to c . Another definition (s e e 13.1) describes

§ c as an ex p licit

quotient of the group of components of the centralizer of a unipotent e le ­ ment u in a complex reductive group with Weyl group W , where u is associated with c by a procedure in volving Springer’s representations of W.

The group § c has also the follow in g interpretation, (s e e 7.1 (ii)).

There exists a left c e ll T

contained in c , such that the matrix (c x E) ,

INTRODUCTION

xiv

(where x runs over T *

and E runs over the irreducible components of

r ), coincides with the character table of § c . Here the indices correspond to the irreducible characters of § c and the indices correspond to the conjugacy In the case where W B n or C n , then § c

is

x E

cla sses of § c . is of type A n , § cis trivial.

If W is of type

an elementary abelian 2-group of order 2^

where d2 + d < n . If W

is of type Dn , then § c

2-group or order 2^ with d 2 < n .

is an elementary

If W is of type E g (resp.

F4 ),

there is a unique 2-sided c e ll c in W such that § c = © 5 (resp.

@ 4 ).

It would be interesting to find a definition of § c in the framework of primitive ideals. L et Wv(c ) be the equivalence class corresponding to c , for the equivalence relation

E ~

E ' on the set of irreducible representations

L R

of W . In Chapter 4, we define (ca se by c a se) a bijection

E

►x E

between Wv(c ) and a subset of the set ® ( § c ) consisting of a ll pairs (x, d) where x is an element of § c (taken up to conjugacy) and a is an irreducible representation of the centralizer of x in §

. (Th is gen­

eralizes the parametrization given in [ L g] of representations of Weyl groups of cla s s ic a l types in terms of sym bols.) If § c

= ® 3,@ 4,or @ 5 ,

then JH(§ ) has respectively 8, 21, 39 elements. In the ca se where § c

is abelian, we have 5K(§C) = § c x § *

§ * is the Pontrjagin dual of

§ c , hence 3H(§C) is a vector space of

even dimension 2d over the fie ld

F2 . It has a natural pairing

I

,

5R(§C) X JH(§C) -> Qj? defined by

(

,)

is the sym plectic form

a (y )r (x ), §c

m = (x ,o ),

where

{m,m'! = 2~c* ( - l / m,m ^ where

Jll(§c ) x 5K(§C) -» F2 defined by

m '= ( y , 0 * The pairing {

,

(- l ) ^ m,m ) =

! makes sense even if

is non-abelian (see Chapter 4, (4.14.3)) and using

it one

can

the Fourier transform of a function on 5H(§C) , just as

in the

abelian case

(see 12.1). Now let r

be a left c e ll contained in c . As a W-module we have

P = 2(E : r )E , where E runs over the irreducible components of F , E

define

XV

INTRODUCTION

and (E : T )

is the m ultiplicity of E in T . Consider the function

on 3K(§C) defined by = 0 for a11 w ' < w -

CHARACTERS OF REDUCTIVE GROUPS OVER A FINITE FIELD

12

We have

w < w =

D

Here /3 -> /3 is the ring involution of

,

\ w \ w

W ,w ,0-

Z [A ] defined by a -» a^ 1 for a e A ,

(See (1 .4 .3).) Comparing the co efficien ts of

(1.7.3)

/ V w ,= ^

2

W

p . w ,w,V>

we get

r. > ,-i£ ,-l W-Wi-'A w x,w,i/f

w '< W j< W

for a ll w /< w . L e t us fix

w '< w . We may assume by induction on

£(w) - 2(w0 that /3 i = 0 whenever w '< w* < w . Then (1.7.3) W j >W,ljf becomes /3 ' j, = [ p ] ^ w >“^ w ^ * ,V^

( 1.7 .4)

[p r ^ w - ^ c w '))*

v

LPJ

'

,

Pvi,VI,lfl

^

Hence

W

= [p]V4 p1//2 ). Hence

The lemma is proved.

(x n£, y 1l ] ) , for the first action (£, £ j) -» (x ny I1£,y~1£1)', for the second action. Th ese actions g iv e rise to two categories

, £^

U x k * x k*-equivariant constructible sheaves with

(s e e 1.4) of

^-structure on M.

is clear that a * defines an equivalence of categories that /3* defines an equivalence of categories other hand, a sheaf in change of variables

£^. ^ ^

(x ,y ) -» (xy, y )

K((2

, .

(L ) ,n

).

On the

in k* x k* . (It is the same sheaf,

CJJ. Combining these three equ ivalences, we get an equivalence £

and

can be regarded as a sheaf in CJJ, via the

with a different action .) This gives an equivalence of categories

categories

It

£^ of

. -> £ , . , hence a canonical isomorphism K (£ . ) -> L ,n (L ) ,n L ,n It is clear that this isomorphism makes the elements

CHARACTERS OF REDUCTIVE GROUPS OVER A FINITE FIELD

16

0

correspond to each other.

0.), ments

It also makes the sheaves

0 ) correspond to each other, hence it makes the ele0 , 3lw 0

correspond to each other.

Thus, the statement made

at the beginning of this section follow s. 1.14.

L et

L e J) and the pull back L / of L ' to 0 (s ) under

: 0 (s )

® . Th ese are line bundles over 0 (s )

and we denote by

. '"W.

( L ') , L

the spaces obtained from them by removing the zero sections.

The group H acts naturally on both these spaces.

The isomorphism y

considered in (1.3.1) defines an H-equivariant isomorphism y ; ( L ')

.

Let

77j = ( L ')

.

/V

-> ( L ') , 772 : L

L et 3” b e a n object of C

.

.

-> L .

L ,n

-» L .

be the natural projections. . Then y*

77*

^

J

is an H-equivariant

sheaf with ^-structure on ( L ') . L et

TgC?) = ^ - l / R ^ C y * ^ )

f

T ' ( ? ) = 2 ( - l ) iR V 1

e K (C

Th ese extend uniquely to

*

) .

v-*-' )

Z [A ]-lin ear maps Tg ,T g : K (C l «

) -»

n)

(Th ey are independent of the choice of y above.) 1.15. LEMMA. Proof.

We have D T

s

= [ p ] _1r D b

as maps K (C

According to [V e], we have Dt71 ,( ) = 771;tcD( ) ,

. )->K(C ) L ,n (L ) ,n D772(

) =

(t7*D ( )) 0 Q g(l) , with a shift of two degrees (in the derived category), since

77

2 is smooth, of relative dimension 1. A lso ,

D commutes with

y * . The lemma follow s. 1.16.

LEMMA.

If i/rm ^ 1 on fin , (m = as ( L ) ) , we have

17

1. LINE BUNDLES OVER A SCHUBERT VARIETY

ws

!

L

[p]9C,W S ,lff Proof.

T o compute Tg (X ^

the restriction SQ of

if

ws < w .

with co efficien ts in

to n ^ ( t ) , L et y € W be such that

lie s over a point in

77^(1') is empty.

ws > w

we must compute for any t e ( L ') , the

cohomology group with compact support of

I'

if

. If y A w , ws , or if y 4 ws > w , then

If y = ws > w , then

is a sin gle point, so that

^ y ' \jj aPP^a fs with co effic ien t 1 in Tg ( ! f ^ ^ ) . If y = ws < w , then n ^ \ t ) is an affin e line and SQ is the constant sheaf Qg over it, hence ^

nnpfl r? w ith o n^ f f i n ien i ^ n tt 0/ aappears with co effic

y = w > ws

f[p] nl

is the most interesting.

in in

Tg(f£^ ^ ) . The remaining case

In this case,

is isomorphic

to k* and Sn « ? m where ? m is the loca lly constant sheaf over 0 i/fm i/fm _ associated with the homomorphism ij/m : (in -> Q | . (We regard fxn as a

k*

quotient of the tame fundamental group of k* .) By assumption, y/m 4 1 on nn , hence the groups H *(k *, SQ) vanish. co efficien t

0 in Tg(f£w

appears with

.

It remains to show that Tg( ^ w back of L

~~L>/

Thus !£w ^

to 0 (s ) C $ x ®

0.) = T g( 9 ^

.

L et L

underpr2 : 0 (s ) -» $ and let

back of L ' to O (s) under prj : O (s)

-» ® . L e t

from L , L ' by removing the zero sections.

L et

L , ( L ')

be the pull­ L ' be the pull­ be obtained

) ' he inclusion of (L O

as an open subset of ( L ') . L et

? 1 : (L/)

-» ( L ')

be the natural projection.

We have

= n ^° j '.

18

CHARACTERS OF REDUCTIVE GROUPS OVER A FINITE FIELD

Now

7 7 is a proper map (a P 1-bundle).

Hence R 1771

= R V j ,( ) .

Hence, in order to prove that Tg (!T^ 0.) = T g (!^^ 0.) it is enough to show that (1.16.2)

R-ij ; ( S ') = R ^ C S ')

where S' = y*n\C^ Let

•

be the line bundle over 0 (s ) associated to the smooth

divisor A = j ( B ' B " ) e 0 (s)| B /= B " }

in 0 (s ) . Then

way a G-equivariant line bundle, since

A

is G-stable in 0 ( s ) . There is

a unique G-equivariant isomorphism of line bundles over The line bundle (up to a scalar) over L | 0 (s)

is in a natural

0 (s ) : L '

—

has a unique G-equivariant trivialization

0 (s ) - A . This gives rise to an isomorphism

S (L ® L ^ ~ m^ )| 0 (s), hence to an embedding j : L

-» ( L ® L ^ ~ m^)

of line bundles with zero section removed, compatible with the imbedding O (s) -> 0 (s ). Then the diagram

v

(L O

L ■

(LO

J

is commutative (up to multiplication by an element in k* ), and we see that (1.16.2) is equivalent to (1.16.3)

R % (S ) = R ^ y S ),

where S = * * ( 3 ^ ) .

We now consider the open subset ll = \B' e ®|B' opposed to B j i C $ where B x is a fixed Borel subgroup.

Then

L

is a trivial line bundle

over 11, hence there exists an isomorphism 8 : L |ll -> U x k *

which is

compatible with the projections on U and is k*-equivariant (with k* acting by scalar multiplication on L |U and by multiplication on the second factor, on U x k * ).

The restriction of the sheaf

to

L|U

19

1. LINE BUNDLES OVER A SCHUBERT VARIETY

is isomorphic to p r ^ (§ ) 0 prj^CS^) over l l x k* , where § structible £-adic sheaf over U .

is a con­

L e t li = S(B/’, B " ) e 0 (s )| B /e 11, B " e 11}.

Th is is an open affin e subset of O ( s ) . We define an isomorphism f

k x l l ^ t l

x :k

as fo llow s.

C

and let

G be an isomorphism of the additive group of k onto the root sub­

group corresponding to s.

Choose a maximal torus

C

and to minus the simple root attached to

Then is defined by c£(a, B ') = (x (a )B 'x (a )"” 1, B ' ) . When B^ varies

the open sets

Ii cover 0 ( s ) , hence, to check (1.16.3) it is enough to

check that (1.16.3) is true when j

jf i : L

is replaced by its restriction

11

|U-A

is replaced by its restriction S '

to L [ l l - A . Using the description of

^

as an external tensor

product over L |ll = t l x k * , ar and of 11 as a product k x K , we get a commutative diagram

JeU

(L®L®(_m))*|li 8

Jl

k x l xk* Id

k*xllxk*

8

k x l l x k*

where j 1(z, u, z ) = (z , u, z mz ' ) , j 2(z , u, z ' ) = (z , u, z ' ) , a 2( z , u , z ') = ( z , u , z ~ mz ' )

and we have:

ScQ = a i(P r^ (§ )ia p r3 (3 ^ r)) = a*a2(Pri(^ m ) i a p r * ( § ) 0 p r 3 ( J ^ ) )

where pr1 , pr3 are the first and third projection of k* x U x k* It is enough to show that R i ( j 2) 5t; (pr*(?^fm )(Slpr^(§)l3pr*(3:^ ) )

when restricted to \0\ x l l x k* . But this higher direct image is (1.16.4)

(R X0 3) ^

m)S p r 3 (§ )® p r * ( 3 ^ )

onto k*

is zero

CHARACTERS OF REDUCTIVE GROUPS OVER A FINITE FIELD

20

where j 3 is the natural inclusion k* -> k . Since if/m stalk of R ^ j ^ S ^ m

at 0 is

0.

on pn , the

Thus, (1.16.4) is zero when restricted

to { 0 l x ll x k * . The lemma is proved. 1.17.

LEMMA.

Assume that ws < w and that iffm

on pn , (m = a g ( L ) ) .

Then T oy L S

Proof.

WSfl/f -

evL W,lA •

We have always s • WL n • s_1 = Wg ^

and s maps R L n to

R s L n * Our assumption on iff implies that m is not d ivis ib le by n . It follow s that s i WL n , so that s maps R ^ n into R + (sin ce the simple root corresponding to s is not in R ^ onto

R+

. It fo llo w s that co n ju gatio n by

) hence s maps R ^ n s

d e fin e s a b ije c tio n of

L ,n

ST

onto S

hence it is an isomorphism of Coxeter groups L ,n

(WT n ,S jT- . i i ) « l~j . i l 9 7

(Wo_,r

j

^ , S_ Ot

JL,n

) . In particular, it is compatible with the L, n

partial order, length function and P-polynomials of these two Coxeter groups. Note also that, for our s , the element w 1 e W has minimal length in w n WT

if and only if w .s

has minimal length in (w 1s)W_,

’

L ,n

1

(T h is follow s from the characterization 1.9 ( i ) . ) Now the result follow s immediately from the previous remarks together with Lemma 1.16. 1.18.

Assume now that L £ £

ag (L ) = 0.

and s g of the

>L

21

1. LINE BUNDLES OVER A SCHUBERT VARIETY

We can now state 1.19.

LEMMA.

(i)

Assum e that SL = L . Then

D$s = [ p ] _1 #s D as endomorphisms of K (C ^ i , < A + ^ws,.A

.

).

• if w < w s

i w X w V 5; ^ ) . « * WS .

If ws < w , then

f l a ^ S ^ W S , ^ = ‘V ^

^ +

Z

, y2(£ (w )-£ (w v ) ) a /Xv x > v s )[p ] I v w Vv lft

,

V16WT

1

L ,n

V1 s gu^1 .

Both X „ T, X. have canonical F -structures (inherited from (G ,F ) ); w w P we shall denote the corresponding Frobenius maps again by F . For w f W , let F ^ : T -> T

be defined by F ^ (t ) = w (F '(t )) and let Tw =

it gt w (g

g B ) . Note

also that G

action of G

the action of Tw and induces on X w

the

given again by left multiplication.

Now let A : T -» k* be a character and let element of £

/T_r — > X TTT, v v w w

acts on X ^ byleft multiplication. This

action of G Fcommutes with TT /

, in such a way that X

(s e e (1.3.2)).

L

be the corresponding

We shall also denote by A its extension to

a character of B , trivia l on U .

L et

n be an integer as in (1.4.1).

We

a ssocia te to A and n a subset Z T „ of W as follow s. i^,n Z T „ consists of the elements w eW for which there exists a characL,n ter Aj : T -» k* such that (2.1.1)

A (F X O )

(N ote that

= A(w- 1( t ) ) A j ( t ) , for

a ll t e T .

, if it exists, is uniquely determined by w .) We shall fix

A, L , n such that Z L n is nonempty until the end of §3. Assume now that w ,

are as in (2.1.1).

The restriction of A j

to Tw is a

homomorphism (2 .1 .2)

V T w -^n .

Consider the variety

(X ^ x /zn)/Tw , where t e Tw acts by

t : ( g , £ ) -» (g t- 1, ^ x ( t ) £ ) ' ( f f fJ-n) ■

CHARACTERS OF REDUCTIVE GROUPS OVER A FINITE FIELD

32

Th is is a variety with a free jxn-action

Fn ? £ 0 : ( g , £ ) -* and its quotient by the /xn-action may be identified with X w , via gB . L et A j

be the inverse image of the principal /xn-bundle

(g, f ) - »

(X . x/xn)/Tw

-> Xxi7 under the map f : G x k* -> G/B , f(g', z ) = g'B . Then A 1 is a W

J-

principal /z -bundle over f~ * (X

(2.1.3)

).

We have

A j = f ( g , g ' , z , f ) e G x G x k * x ^ n |g- 1F X g ) £ wU, gB = g 'B l/(U n wUw- 1)T w

where Uj f U H wUw-1 acts by Uj :(g , g , z , f ) -> ( g u ^ . g ', z , £ ) and t f Tw acts by t :(g , g", z, f )

If (g, g( z , £ )

-» (g t ~ \ g ', z , A 1( t ) f ) .

represents a point in A j , we define u e U , b f B by

g_ 1F '( g ) = w u ,

g'=gb.

Th is gives an isomorphism of A j

with

A 2 = { ( g 'b >U,z ,£ ) e GxBxUxk*x/in |g/_1F /(gO = b_ 1w u F '(b )l/ (U n wUw- 1)T^ where Uj e U fl wUw 1 acts by ux : ( g ' b,u, z , f )

-> ( g ) u1b ,(w ~ 1u1w )u F '(u )“ 1, z , f )

and t e Tw acts by t :(g ',b , u , z , £ ) -> ( g , tb, F X O u F 'C t r 1, z, A1(t)^ r) . Now let (2.1.4)

z) e G x B x U xk* xk*|g' = ( z / z ' ) n \/U H wUw-1

A 3 = {(g ' b, u, z ' A(b)

* F '( g ') = uwb

1,

2. LO CAL INTERSECTION COHOMOLOGY

where

33

t U (1 wUw 1 acts by Uj :(g ', b , u , z ', z ) -* (g', bw_ 1U j1w, uu^1, z', z ) .

For b ( B , we define tb e T

and ub e U by b = tb ub . The map

( g ' b , u, z , f ) -* ( g /',F /( b r 1u~1(w “ 1tbw),U|31, ^ “ 1A 1(tb) z , z ) defines an isomorphism

A 3.

Consider the free action of B x k *

B x k* ?

on A 1 given by

: ( g , g ' , z , f ) -> (g, g ' £ ~ \ z £ , f ) .

Under the isomorphism A 1 — > A 3 constructed above (composition of A 1 ~^> A 2

A 3 ) this becomes the free action of B x k *

on A 3 given

by B x k * ? (/ 3 , £ ) : ( g ', b , u , z ', z ) -> (g'/3_ 1, F X ^ K w -1 ^ 1* ) , ■ t/3u/SutJ01’ The space of orbits of A 3 by this (via the isomorphism A ^ Bxk*-action, i.e. to 2.2.

LEMMA.

A 3/ (B x k * )

^

■

Bxk*-action must then be isomorphic

A 3 ) to the space of orbits of A 1 by its

(X . x u J / T 1 U'

W

. Thus we have

There is a unique isomorphism

be the map given by V (g , z ) = (g, z ) , (w ith the description (1.5.1) of L and let f ' : ( B w B ) x k * -> L |$w , f^ : (B w B )x k * tions.

Let i " : G xk*

G xk*

L |®w

)

be its re s tric ­

be the map given by f " ( g , z ) = (g - 1F '(g ),z n

and let i".1 : f _ 1( XW) (B w B )x k * , f 'lZi : f _ 1(X w „ r) -» (B w B )x k * be its restriction s. Then there is a canonical G F -equivariant isomorphism of lo ca lly constant Qg-sheaves over f “ 1(X w ) = f //“ 1( f /_1( L l®w)) (com p a ti­ ble with the F i-structures) P (2.5.1) Proof.

'* < ? * ,„ > —

■

In 2.1, we have constructed an isomorphism A ^

an isomorphism of principal /^-bundles over f _ 1(X w) . loca lly constant

A^.

Th is is

Taking the

Qg-sheaves corresponding to 4* '■ll n ^ Qg and to these

principal ^ -b u n d les, we find the isomorphism (2 .5.1).

2. L O C A L IN T E R SE C T IO N COHOMOLOGY

2.6.

P ro o f of Theorem 2.4.

smooth fibers

( «

37

Since f is a lo ca lly trivia l fibration with

B x k * ) , the sheaf f^(H *(X a

w

^w fn

be canonically identified with H *(f _ 1(X W) , f * ( 5 ^ fl))

)) on f ~ 1(X

) can

w

or, equivalently,

by (2.5.1), with HiCf _ 1(X W) , Since V f"

is a lo ca lly trivial fibration with smooth fibers

( % B ) and

is an etale su rjective map, the last sheaf is canonically isomorphic to

f 2* f 2* ^ ( L (2 .6 .1)

l»w, ^

^ )*

Using now Theorem 1.24, we see that:

T h e restriction of the s h e a f

is zero unless w ' e Z

f ? (K * ( X

z

w

and i is even; if w ' e Z

w,n

))

to

f _1(X

w

,) (w '< w)

and i is even, it is

lo ca lly constant and admits a filtration by lo ca lly constant, G F -stable Qp-subsheaves defined over F _j each of whose su ccessive quotients is P

isomorphic (as a

G F -e qu iv aria nt,

Qp-sheaf defin ed over

F

) to

j PQ

where w ' f Z tion is

n ^,

represents

w '; the number of nonzero steps in this filtra ­

. (H ere f 3, f 3, f 3 are defined just as f ^ f ^ f ^

in 2.5, with

w replaced by w '. ) In other words, we get the statements of Theorem 2.4, “ lifte d ” under f*

to G x k * . The statements of 2.4 with w 7 Z

or i odd follow .

To

get the statements of 2.4 with w 'e Z and i even, before liftin g under f * , we use the follow in g remark.

Since the lo ca lly trivia l fibration

f : G x k * -> G/B has connected fibers, a constructible

Qp-sheaf ?

on

G/B can be reconstructed from its inverse image f * ? : we have J = R ° f #( f * 5 :) .

Since f is a lo ca lly trivia l fibration, the fact that f X(K*(X

J

is lo ca lly constant on f “^ (X ^ ,) constant on X

J =

3“ .

))

W,n

^) is loca lly

,. W

L et ?

implies that H*(XW,

L

i

be the restriction of f^ K ^ X , ? . )) to f -1 (X , ) . L et 2 w w,n w_ 'D J 1 3 •** 7 ^ N = 0 , (N = n* , ) , be the filtration of ? whose P P p w ,w

38

C H A R A C TE R S OF R E D U C T IV E GROUPS OVER A F IN IT E F IE L D

existen ce is asserted in (2 .6 .1), and let 3^ = R 0( f 3) H,(3:J) be the corre­ sponding sheaves on X

w

/. The exact sequence of sheaves

0 _ p

f j+1

f j +1/ f j - 0

/

can be split in a G

-equivariant manner (but not necessarily in a way

compatible with the F j-structures).

This follow s from the similar

* L property of the filtration of H1( 7r~1(fBw ) ? ^) 1.24.

restricted to L

^

It follow s that we have an exact sequence of sheaves

0

3 j+1

(R ° f3) ;t;( f j+1/ ? j ) -» o

over X

Thus, the 3^ form a filtration of the restriction of Wl / _ K 2(X w , ? ^ ^ ) to X ^ , by lo ca lly constant, G F -equivariant Qp-sheaves defined over F j , such that P s V J i+1 = (R ” f 3) , ( # i+ 1/ J i )

for j = 0,1, ••*, N - l , as required. 2.7.

The

Qp-vector spaces

»

® Q e (- i))

This completes the proof of the theorem.

H*(Xw , 3 ^ ^ ),

( w,w as in 2.4), are

(G F ,F^)-modules in a natural way (using the fact that 3r^ is w,n F/ G -equivariant and defined over F ^ (see 2.3)); an element g ( G P acts as

(g -1 )*

and F^

F/

acts as ( F ^ ) * . Using the spectral sequence

connecting hypercohomology with co efficien ts in a complex, with the ordinary cohomology with co efficien ts in the cohomology sheaves of the complex, we get an equality of virtual (G F ,F^)-modules

(2.7.1) X t - i m V 3*,,) 1

ij

2. L O C A L IN T E R SE C T IO N COHOMOLOGY

Using the decomposition

X

=

hand side of (2 .7.1) as

2

X

w

,, we can rewrite the right-

w _w

2

w < w

U / i xW, ) . W

i, j

Using now Theorem 2.4, we get the follow in g 2.8.

In the setting of Theorem 2.4, we have an equality

COROLLARY.

of (G f ,F^)-m odules

( 2.8.1)

S

( - 1>iHi(Xw 3w1',n)

♦

i

+

i

7

7

w < w w

7 i

( - l A L

fZ

*(x

by

f:(G / U ) F

~*Qg-

e 5 l(T )F

(r

1

, let

f)(g U ) =

9^ to be

It is a G F

( g l O ( g U ) = fC g ^ g U ),

(2.9.1)

L\

2 /

-module:

g ,g 1 e G F , f r ydr .

: 9^r -» 9 ^

,

thevector space of

2

be the endomorphism

rirf (g ' U ) ’ (f e 3dr) •

g U € (G /U )

T? r

g 1 g ^ Un ^U

pdr The endomorphisms rn End

(n x e N (T )

d r ^ d P * For ea ch character 0 : T F g f

and F^

.)

Let r be an integer > 1 . We define

For each

)® Q g (-t)-

has triv ia l action of G F

acts on it as m u ltip lica tion

a ll functions

w w ,n

even

( w / are chosen as in 2.4;

2.9.

h

w ,w c

jr

) form a (Jg-basis of ^ Qg we denote

40

C H A R A C T E R S OF R E D U C T IV E GROUPS O VER A F IN IT E F IE L D

( 2 .9 .2 )

f d = I f

be the linear map defined by ( F 'f ) ( g U ) = f ( F '“ 1(g )U ) •

defined by

41

2. L O C A L IN T E R SE C T IO N COHOM OLOGY

It is clear that (2.9 .8)

F 'r ^

= V ( V F ' : 5>dr -> ?d r , (n ,

$F '

f ^ ) .

Q

Hence F / maps 9 0 into 9 0 and, therefore, the composition 0 F ' t . maps 9 ® into itself. It is, however, not n ecessarily commuting w with the action of G

on j

We now consider the follow in g bijection , due essen tia lly to Shintani, (s e e also [K a 2], [A s 2]):

1Gf (2.9.9)

mod. equiv. rein, x ^ x ^ x F ^ ^ x Q), V x QeG F

\

» (

{G

F^r

To x £G

f

'

mod. equiv. rein,

, there corresponds

-

, -

-

x ~ x Q xF ( x Q), V x QeG -

x AW x w

° )T r (S F V . 9 w

0

9

0

) ,

x pdr

sum over a set of representatives pdr

G

_

,_

1x5 for the orbits of G

by x -» x Q x F ( xq ) .

We may replace this sum by a sum over a ll x

pdr in G

acting on

/ _

* , provided that we replace the factor n~

by

pdr | |G | . It

fo llow s that (2.11.4) is equal to 0 T r ((F V .)® (F V . , ) : ( ?

(2.11.5) o Now, 9 th e

0 x 9

9 0

°® 9

0 ^ pdr 0 )G tD ) .

i 0 > Qp

0 is in a natural way the space dual to

9

can be obtained by associating to f e 9

0 . (A pairing °,

V e 9

0

su m

2

(f-n(gu)^Qe-) rr, dr

g U e (G / U )

Under this duality, the operator F 'r . , on 9 w contragredient of the operator (2.11.5)

Q—1 0 corresponds to the q

_ 1( F /) “ 1 on 9 ° . It fo llow s that (w ') 1 is equal to (2.11.2) and the corollary is proved. r

45

2. L O C A L IN TE R SE C TIO N COHOM OLOGY

2.12.

We shall recall some known properties of the algebra End

*7

It is w ell known that for n j,n 2 f J l(T )

(2 .12.1)

\

F dr

’ n2 ~

dr^dr^*

° F

we have

^ 4 ,

whenever ECn-^n^) = ^ (n j) + £(n2) . We shall denote the roots of G which respect to T a, j8 •••.

They are characters of T .

by Greek letters

For each root a , let

(2.12.2) na be an element of N ( T ) - T

such that na is in the subgroup

pdr of G

generated by the root subgroups (over F j ) corresponding to P

a, - a .

If a is a simple root (with respect to B ) then, as it is w ell known, we have

9^r are invertible. Hence a

the operators rn : 9 ^ -> 9 ^ r are invertible for any We now define

n„ l

(2.12.2)

and l ^ a n 1 2

When

n1 e T l(T )F

Jig to be the subset of ? I(T ) f

elements of the form n

•••n 2

L

consisting of a ll

where a ., •••,

for a ll f e 9 ^ . The statement (iv ) is proved in [K i, (4.10)].

A com pletely sim ilar

proof shows that, with the assumptions of (iv ), we have

47

2. L O C A L IN T E R SE C T IO N COHOM OLOGY

.

(2.13.1)

(f)

f e 9 6 .

for a l l

2.14.

-1 /2(2(n)+£(n„)-2(nn ))dr , ,({) = p a ^ r~ l na (nna ) .

r"i n

0

If the center of G is connected, the sta b ilizer of

cide with

in W coin­

(s e e [D L p 5 .1 3 ]) hence from (2.9.4) we can deduce that the

elements T :

->

, (y

, form a Qj?-basis of End g f

2.15.

Let w , w ^ Z L n

be two elements such that w • WL n = w

let Wj be the element of minimal length in w • WL n , let Z

n'

be a

coherent liftin g (s e e 1.23) of Z = Wj • WL n to ? I(T )F and let WpW, w ' be the elements of Z

corresponding to Wp w, w '.

We have w = w

w '= w1z / (z ,z ' e WL n) and w = WjZ , w ' = W jz' ( z , z ' e

,

n) .

We define an automorphism y of WL n (depending on Z ) by

(2.15.1)

y (y ) = FXwjyw J1), (y fWL>n) .

We have y(WL n) C WL

, since w1 e Z L n . Using the fact that

wl(R p n) C R + and that F '(B ) = B , we see that y(SL n) = SL n , hence

(2.15.2) y is an automorphism of the Coxeter group (WL n, SL n) . We can state the follow in g. 2.16.

r

LEMMA.

In the setting of 2.15, we have, for any y e WL n = W^

( F T 1 -T (w ') " 1 y

-F't =

1 , the expression is the value at prc* of a polynomial with integral co efficien ts

(these co efficien ts being independent of r ).

It fo llow s that:

53

2. L O C A L IN T E R SE C T IO N COHOM OLOGY

(2.18.3)

The vector space (2.18.2) is zero unless

i = j (mod 2) , the eigenvalues of (F ^ )* p (i+ j)d /2 ^

i = j (mod 2 ).

If

on (2.18.2) are a ll equal to

By one of the main properties of intersection cohomology, we have a canonical nonsingular Poincare duality pairing

(N ote that X

, is com plete and it sa tis fie s (2 .3 .2).) w Hence the vector space (2.18.1) may be canonically identified with

and w e see that the theorem follow s from (2.18.3). 2.19.

Assume that G has connected center.

If w f Z T

L ,n

’

w 'eZT „ , L ,n

f

then w - 1w ' sa tisfies

A C C w - ^ 'r k t ) ) = A (t)A “ ( t ) , ( t f T ) ,

for some character A>2 of T , depending on w, w '.

Since the center of G

is connected it follow s that w _ 1w r € WT M. JL,n Thus, there exists a unique element w1 of minimal length in Z L n and we have Z L

= Wj • WL n •

We choose a coherent liftin g (s e e 1.23) Z L n of Z L n to J l(T )F . For any w e Z L n , we denote w the corresponding element in Z L n . (2.19.1) L et

b be the sm allest integer > 1 such that F ^ : G -» G is an

in tegral pow er of

F '. ■j M

(2.19.2) We shall assume chosen, once for a ll, a square root p in Q£ .

of p

54

C H A R A C T E R S OF R E D U C T IV E GROUPS O V E R A F IN IT E F IE L D

Let

M ttt . be the set of isomorphism cla sses of irreducible L ,n ,w ,i r

Gf -modules (over Qg ) appearing as components of the G F -module • c_ l H1(X w , p , (w 0.

We shall denote Ml^ the (G F ,F^)-module

Hi (X w ,5 :.L ) and by M*. its w w,n w,p G F -module.

n the union (not necessarily

Note that

p-isotypic part (p e

n w ; ) , as a L ,n ,w ,i7

Ml is not, a priori, F^-stable. w,p

We have the

the follow in g 2.20.

P R O PO SIT IO N .

G

We assume that we are in the setting of 2.19. L e t


, where 0

Tj' /

with the c y c lic group of order b

acts on G F

by

= F ^ ( g ) . There

exists a map p > p* from § L n to the set of isomorphism classes of irreducible G F -modules ( over Qg ), and a map p -» A^ from § L the set of algebraic numbers in

to

, a ll of whose com plex conjugates

have absolute value 1 , such that the follow ing hold. (i)

For each p e § L n , the restriction of p> to G F

(ii)

If p e & T ; then Ml is i^,n,w,i w ,p

a (G F ,F ^ )

is p its elf.

-stable and moreover it admits

stable filtra tion each of whose s u cce ssiv e quotients is

isom orphic as a G

-module to p and, as a G

as ( A ') ~ 1p~*c^ 2F c* ) to p\ In particular, P

F ^

-module ( with cf> acting acts on M*. w,p

as

(A p k p id b /2 times a unipotent transformation. (iii) w eZ L

If x e G f , r > 1 is an integer such that r = 1 (mod b) and , then

^ ( - l ) iT r ((F dr)*x *,H j;(X w ,3:^ n) ) = ^ ( - l ) i ’

1

V ^

< P ,M ^ >

{ A p V ^ T r ^ x - 1 ,p )

L ,„

(Given two GF -modules M,M', of fin ite dimension over Q g , we set < M ,M '>

p , = dim Horn F < M ,M ').) G

G

2. L O C A L IN T E R SE C T IO N COHOM OLOGY

Proof.

Let

d\

be the algebra

be the algebra

w

End

).

w ,p

gf

End

G

w

55

) and let &\

w ,p

, (p ( F ^ )” 1 • h • F d maps i^,n,w,i w into its e lf; it is clearly an algebra automorphism, sin ce F d normal­ izes

G F . It follow s that h -> ( F ^ )-1 • h • F d maps the center of 2*. into w its e lf and its restriction to that center is given by a monomial matrix (with

respect to the decomposition of the center into direct sum of its intersec­ tions with the simple components of S 1. ).

On the other hand, the endo­

morphism h -> ( F d) - 1h F d of 2*. is unipotent, by 2.18 (ii). w

Its restriction

to the center of 2 1. is both unipotent and monomial hence it is the w identity. It fo llow s that the endomorphism F d of M*. maps each summand w M\ into its e lf, i.e. h -» ( F d) -1 • h • F d is an automorphism of the w,p algebra 2 1. for each p e t&r „ „ • . Th is being a simple algebra, has w,p j-v,n,w,i ° the property that any automorphism of it, and in particular ours, is of the form h -> A-1 hA where A automorphism of

is an invertible element of 2 1. , i.e. an w»P commuting with the action of G F . We then have

A ( F d)~ 1 h (F d) A -1 = h for a ll h e

w,p

, hence F dA“”1 : M\ ~>M\ is w,p w,p

a Qg-linear combination of endomorphisms of

^ defined by the

various elements of G F . In particular, any G F -submodule of M1.

is

* / W,P stable under F A- 1 . Consider a filtration f of M1. by G F -submodules, w,p stable under A , which cannot be refined in a nontrivial way to a filtration by G f -submodules, stable under A . The nonzero su ccessive quotients F' of this filtration are then irreducible G -modules, since A commutes with the G F -action.

This filtration is stable under F dA_1

G f -submodule is ) and under A

(sin ce any

(by definition) hence also under F d .

Now F d^ : M\ -> M*. commutes with the action of G F (by the w,p w,p definition of b , (2.19.1)) hence it acts on each nonzero su ccessive

56

CH A R A C TE R S OF R E D U C T IV E GROUPS OV E R A F IN IT E F IE L D

quotient of the filtration a 1, a 2) '**’ a N

f as a scalar aj times identity.

Let

the scalars thus associated to the nonzero su ccessive

quotients of f , (N 2 = dim S*. ) . The eigenvalues o f h -» •h •F ^ 3, w ,p (3*. -> S* ) , are then the scalars a -/ a -' (1 < j, j / < N ) ; but this v w,p w ,p r J v J linear map is unipotent by 2.18 ( i i ) hence aj/ay = 1 • Thus, we see that F^k acts on M\ as a scalar times a unipotent transformation. By the w ,p results of D eligne and Gabber already used in the proof of 2.18, this scalar must be of the form A ^ p ^ i /2 where A^ e

is an algebraic

number a ll of whose complex conjugates have absolute value 1 . A priori, AL/ depends not only on A , but a lso on w , and i such that p e & X-J T JHyWyX • Applying again 2.18, we see that A^ is, in fact, independent of w and i Thus to each p e

n we have associated an element A^ e QJ.

s elect, for each p e S T

ij,n

We

, a b-th root A' e Q? of A . P

P

l

Returning to the filtration

f of M1. (p e „ tTT •) considered , w,p r L,n, w ,i above, we extend the action of G F on each nonzero su ccessive quotient of f to an action of G F , by letting e G F (It is clear from the definition of A of G f

X'

act as

(A p “ 1p~ ^ //2 F^ .

that this is a w ell-defined action

.)

If p is also in € r „ r x»,n,w ,i

we may consider a filtration V J

of M1. , w fp

with properties sim ilar to f , and we extend, as above, the action of G on each nonzero su ccessive quotient of f

F /

to an action of G

Given a nonzero su ccessive quotient of f and a nonzero su ccessive quotient of V , we attach to them a root of 1 as follow s.

We choose a

G F -isomorphism between them; it w ill take the action of on the first space to a b-th root of 1 times the action of (f> on the second space. The collection of b-th roots of 1 thus obtained, each multiplied by p ( i - i ) d /2 ^

pj*e c j s e iy the set of eigenvalues (with their m u ltiplicities)

of the linear map h -> ( F ^ )-1 hF^ , from Horn ^'(M * G

W

,p

, M\ , ) into itself. W ,p

By 2.18, this linear map is equal to p ( i - i )d /2 times a unipotent trans­ formation, hence a ll the b-th roots of 1 considered above are equal to 1 ,

2. L O C A L IN T E R SE C T IO N COHOM OLOGY

hence a ll the G

F

-isomorphisms considered above are automatically 'V p ' /

G

57

-isomorphisms.

Th is shows that the G

^

-module (which we c a ll p )

constructed above on the nonzero su ccessive quotients of f on M1. w }p depends only on p , and on the function p -» but not on w , i . Thus, we have constructed the maps p -> p ->/?, satisfyin g (i) and (ii). To i * check ( iii), we note that F acts on each nonzero su ccessive quotient of the filtration of M*. in ( i i ) as ( A ' w ,p P that cfj - cf> since r = 1 (mod b) .) Hence

p

= ( A ') r p ^ f//20 . (N ote P

T r ( ( F dr)* x * ,M ^ ) = 2 < P’ MW > F ^ V rpidr/2Tr( W be an automor­ We form the semidirect

product W of W with the infinite cy c lic group with generator y , so that in W, we have the identity

y • w • y~

= y ( w ) , (w E^

and an integer c > 1 such that

r f = 1 , I j Y j I ' f 1 = S(y i) : E 1

We now define

Ej

(y1 fW j) .

F : E -> E by

TCej® e 2® ••• ® et) = F L(e t) ® e 1•••®et_ 1 , ( e ^ E j ) .

It is clear that T ct = 1 and T y T -1 = y (y ) : E required

W-module.

irreducible. L et

W

E (y 6W ). This gives the

Thus, we are reduced to the case where (W, S) is

We may assume that the order of y : W ^ W is

c > 2.

be the sem idirect product of W with the c y c lic group of order

c with generator y , or equivalently, the quotient of W by the subgroup generated by y c . Our assumption implies that E extends to a

-module

(over Q ). If (W, S) is of type A n(n > 2 ), D 2n+1(n > 2) or E 6 , then c = 2 and we may define r : E -> E to be the action of the longest element in W. Hence we have the required

W-module (over

Q ).

Assume now that (W, S)

60

C H A R A C T E R S OF R E D U C T IV E GROUPS O VER A F IN IT E F IE L D

is of type D 2n^n - ^ ’ anc* c = 2 . Then

WQ is isomorphic to a Weyl

group of type B 2n and therefore, a ll its irreducible representations are defined over Q . Since E is assumed to be extendable to

(over

Q )

it is automatically extendable over Q . F in a lly, we assume that (W, S) is of type

and c = 3.

the semidirect product W' of W with symmetric group (W, S) in the standard way.

Consider

acting on

It is w e ll known that W/ is isomorphic to a

Weyl group of type F4 . If we identify the c y c lic group of order 3 generated by y with a subgroup of

, we get an imbedding of

into W ' .

Our assumption is that E is the restriction to W of an irreducible *** ^ — Wc -module E (over Q ). But one checks that amongthe three possible such E , there is exactly one which is the restriction to

of an irre-

E such that y • w • y~ 1 = y(w ) : E -> E , yc = 1 : E -> E , for some c > 1 . Since the construction

E

E(u) is canonical, and y is a Coxeter group

automorphism of (W, S) , the map y : E -* E gives rise to a Qfu1//2,u“ 1//2] -linear map y(u) : E(u) -» E(u) such that

y C u l^ y C u )- 1 = Ty(w ) : E(u) - E(u) , y(u )c = 1 .

Hence the H-module E(u) becomes an H-module, denoted E (u ), in which Ty acts as y ( u ) . (Compare [ L 12, 1.5 ].) Now, the linear map y : E -* E above is determined up to multiplication by to - y , has the e ffect of changing

±1 . Changing y

: E(u) -> E(u) to - T

L e t a -> a be the involution of the ring

Q[u1//2, u~1

. such that

u1^2 = u~1//2 . The follow in g properties (3 .3 .1 )-(3 .3 .4 ) are checked as in [ L 11? 1], [ L 12, 1].

(In these formulas, Tr

means trace over Q[u* //2,u“ 1 //2].)

(3.3.1)

T r (T w ,E (u )) f Z [u 1/2] , (w eW ) .

(3.3 .2)

T r ( T ~ L E (u )) = T r (T w ,E (u )), (w eW) . w

(3.3 .3)

T r (T

, E (u )) = T r ( T _ x,E (u )), (w SL n) and to its automorphism y defined by (2.15.1). (3.4.1) From now on, we shall always assume, unless otherwise specified, that the map F ' : G > G in 2.1 is the Frobenius map for a rational structure on G over a fin ite subfield of k.

It follow s that y : WT *

L ,n

-» WT „

L ,n

is

ordinary (see 3.1). Since

p

1 /o

— has been chosen in Qg (2.19.2), we have a w ell-defined

ring homomorphism h ^ : Qfu1/2, u-1 /2] -> Qg, taking u1/2 to pdr/2 , ( r is an integer > 1 ).

L e t H (pdr) = H ® Qj?, H(pdr) = H ® Q £ be the

algebras over Qg obtained from H , H using the homomorphism h^f . We shall denote the basis elements L et

0 q be given by (2.9.5).

TW®1

of H(p

) again by T

, (weWL fl).

R ec a ll that 2.13 and 2.14 define an isomor­

phism of Qg-algebras: (3.4.2)

,(9

H(pdr) -5U End

Q °) .

Gf

For each E : x

(T h e representations

E

n which

are not extendable do notcontribute

to the trace.) 3.7.

Let

JK(Gf ) be the Grothendieck group of virtual G F -modules of

fin ite dimension over

(3.7 .1)

Qg.

For any Ee (WL n)^x ,

R2 - |WLi nl ^

X

we define

M ) ‘ T '< W . ® Hc (X w .y . 3:w i y , - , „ ) .

i> 0 ( ( wiy )

€ % )• Th is is an element in JR(GF ) ® Q . It depends on the “"w

ch oice made in 3.6 for an extension

GF < V

rpidr/2T r (^ “ 1’ p ) =

L,n (3.8.1) = 2

(-1 ) i-Tr((Fdr) * x * ,

S*

)) .

i> ° Using (2.8.1), the last expression can be rewritten as /2(£(w1z)-£(w 1v)-£(z)+£(v))dr t1 u w .^ m w .v - ^ w v Air

S

P

1

1

d J P v , Z (P

j= )(- D

x

veWL , n

i>0 x T K ( F * ) * x * , H ‘ (X WiV, y ^ v ) . n)) which, by ( 2 .10.1), is equal to

(3.8.2)

2

P

y2(£(wl z>-(kw1v)-f!(z)+£(v))dr . _ 0n „6 1 1 Pv z (p dr)T r(2 F V (w v ) - : y °-,SP

vfWL , n where x f G have

’ corresponds to x eG

under (2.9.9).

~ -1 /2(£(w1)-£(w1v)+?(v))dr • = p 1 1 r. T , (on (W V) r Wj v ’

hence, by (3.6.3), the expression (3.8.2) is equal to

1 By (2.16.1), we 09 °) ’

69

3. G L O B A L IN T E R SE C T IO N COHOMOLOGY

yitffw .zl-Pfzlld r (3.8.3)

2

P

.— __ __~ . Pv z (p dr) 2 T r ( X F ; V E)T r (T y v ,E(pdr) ) .

vfWL,n

’

(T h e last sum is over a ll

E

E e (WL n)^x .)

Thus, the expressions (3.8.1) and (3.8.3) are equal. We shall now define polynomials Qz v e Z [ u ] , (z ,v e WL n) by the equations 1 if y —v /

(

0 if

xs,xx ( v , v ' e WL n) .

V

4 v'

Note that the matrix of polynomials (Py z ) , (v ,z e WL n)

is triangular riangular with 1 on diagonal, diagona so that Qz

are w e ll defined.

We shall multiply (3.8.1) by

|GF '| (3-8-5) Dg'Cp

_^(£(W z)-2(z)+2?(v'))dr ) •P 1

x

Pw

(pdr)d im (E ')

L ,n

where v ' t W L n , E f (WL n)g X , P ^ L n ’ and sum over a ll and over a ll

x eG

v ', z t W L

. The result is

D E '(P dr>

V

JL,n

r 1? (z )+T(v^)+iJd r/2„ - l/2^ wl z ^

z )+2^ v/))dr w

JL/, n i>0

(3.8.6) x Q z .v < P dr) < P - Mjw1z ) - > GF ^ ( T y v ^,E'(pdr) ) ( A ^ ) r .

(H ere we have used the orthogonality relation 1 , if P = P ' (3.8.7)

|Gf

|_1

^ xf GF

Tr(c£x“ ^p)Tr(x