Discrete Series of GLn Over a Finite Field. (AM-81), Volume 81 9781400881765

Table of contents :
TABLE OF CONTENTS
INTRODUCTION
CHAPTER 1 Partially Ordered Sets and Homology
CHAPTER 2 The Affine Steinberg Module
CHAPTER 3 The Distinguished Discrete Series Module
CHAPTER 4 The Character of D(V) and the Eigenvalue A(V)
CHAPTER 5 The Brauer Lifting
INDEX

Citation preview

Annals of Mathematics Studies N um ber 81

T H E D IS C R E T E SERIES OF G L nO V E R A F IN IT E FIE LD

BY

G E O R G E L U S Z T IG

PR IN C E T O N

U N IV E R SIT Y

PRESS

AND U N IV E R SIT Y

OF

P R IN C E T O N ,

TOKYO

NEW 1974

PRESS

JERSEY

C opyright ©

1974 by P r in c e t o n U n i v e r s i t y Press ALL

RIGHTS

RESERVED

Pub lis he d in Japan e x c l u s i v e l y by U n i v e r s i t y o f T o k y o Press; In o t h e r parts o f the w o r l d by Pr i n c e t o n U n i v e r s i t y Press

Pr in t ed in the U n i t e d States o f A m e r i c a by P r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n , N e w Jersey

L i b r a r y o f C o n g r e s s C a t a l o g i n g in P u b li c a t i o n da ta w il l be f o u n d on the last pr in te d pa g e o f this b o o k

T A B L E OF C O N T E N T S

I N T R O D U C T IO N ............................................................................................

3

CHAPTER 1 Partially Ordered Sets and Homology ..................................................

5

CHAPTER 2 The Affine Steinberg M odule...................................................................

30

CHAPTER 3 The Distinguished Discrete Series Module ........................................... 43 CHAPTER 4 The Character of D ( V ) and the Eigenvalue A (V ) ............................... 66 CHAPTER 5 The Brauer L i f t i n g .................................................................................... 80 I N D E X ................................................................................................................ 99

v

T he Discrete Series of GL„ Over A Finite Field

T H E D I S C R E T E SERIES O F G L n O V E R A F I N I T E F I E L D George L usztig

IN T R O D U C T I O N Since the fundamental work of Green [5] it has become clear that the central role in the ordinary representation theory of the general linear group

G L n(Fq)

sentations.

over a finite field, is played by the discrete series repre­

In this work we give an explicit construction of one distin­

guished member,

D (V ),

of the discrete series of G L n(Fq)

(here

the n-dimensional F-vector space on which

G L n(F^)

p-adic representation; more precisely

is a free module of rank

D (V )

(q —1 ) (q 2—1) ••• (q 11 1—1) over the ring of Witt vectors To construct cia l complex

X

D (V ),

mand (over Wp )

X

of

In fact

D (V )

T : A ( V ) -> A ( V )

additional homogeneity condition.

is

This is a

F. of a simpli-

is made out of affine flags in V

It turns out that

in A (V ) .

a certain endomorphism

WF

we consider the top-homology A ( V )

associated to V;

which are away from 0.

acts).

V

D (V )

is naturally a direct sum­

is defined as an eigenspace of defined geometrically, with an

The reduction modulo p of D ( V )

is

a modular representation of G L ( V )

which can be described as the top

homology of the Tits complex of

with values in a certain non-constant

coefficient system.

V

This can be used to determine the character of D ( V )

on semisimple elements of

G L (V ).

To deal with the non-semisimple e l e ­

ments we show that the restriction of

D (V )

to any proper parabolic sub­

group splits naturally in a tensor product of a module dim V,

D ( V ') , dim V '
3.

This can be proved by a method of Folkman ([2], Theorem 4.1) as follows. Consider the nerve

Nn

of the maximal covering (se e 1.5) of X(Sn ).

is enough to prove that H ^ N j j j A ) = 0 , Note that the k-simplices of N n hyperplanes in V,

0 < i < n—2 and

H q( N j p A ) = A.

are the sets consisting of k+1

whose intersection is non-zero.

no restriction, hence the ( n - 2 ) - s k e l e t o n of

It

linear

If k < n—2 this is

N jj is the same as the

( n_2)-skeleton of a certain standard simplex and the result follows. We wish to prove analogous results for the other sets (or direct s y s ­ tems) associated to V.

1.9

TH EO REM .

H 0(Sj; A ) = A

H i(SI ; A ) - 0

for all

where I - dim E > 2.

i such that 0 < i < £ — 1 and

P A R T IA L L Y ORDERED SETS AND HOMOLOGY

Proof.

13

Note that the homology is not changed if we change temporarily

the partial order in Sj constant).

to the opposite one (the coefficient system is

U sin g the acyclic covering lemma for the maximal covering

with respect to the opposite order we find that it is enough to prove that H i(N j; A ) = 0 ,

u < i < £ - 1.

H0(Nj; A ) s A,

where

complex whose k-simplices are those subsets of

Nj

E

is the simplicial

which have

elements and which span an affine subspace different from E. this is always the case, hence

Nj

has the same

(k+1)

If

k < £— 1

(£—l)-skeleton as a

certain standard simplex and the theorem is proved.

1.10 T HEO REM . H 0(Sm ; A ) ^ A,

H^(Sm ; A ) = 0 for all

i such that

0 < i < n—1 and

n = dim V > 2.

where

Just as in the previous proof it is enough to prove the following.

P R O P O S IT IO N .

Nni

Let

are those subsets of V

be the simplicial complex whose k-simplices

which have

affine subspace not containing

0.

(k+1)

elements and which span an

Then H - ( N i n ; A ) =- 0 for all

i such

that 0 < i < n—1 and H 0( N III; A ) ^ A, ( n > 2 ) .

I am indebted to M. Kervaire for supplying for me the following.

First observe that

Proof of the proposition.

follows from the fact that given we can find

v3 € V

are not collinear.

such that

Let ?

v x £ v2 in V 0,

and

Let

(vq, Vj , - • , v^)

?£ C ?

[

]

0 ,v 2,v 3

which span an a ^ xec^ ^a s is

as follows:

?£ con­

such that < vQ, V j , •••, v ^ > ^ 0

[ v 0 , v l f ••*,vk] n ( [ Q 1 , Q 2 , - ^ Q ^ l + Qf) - 0.

the affine span, and

V,

Q i > Q 2 ' * * ’ ’ Qn

of V

This

such that v 1 ^ 0 , v 2 ^ 0 ,

, v3 are not collinear and

For any £, 1 < £ < n define a subset

sists of all subsets

is connected.

be the set of all subsets of

affine subspace not containing 0. for V.

N jn

denotes the linear span.

Here


denotes

THE D ISC R E TE SERIES OF G L n OVER A F IN IT E F IE L D

14

We shall consider alternate simplicial chains on N j n . Let C 'k ^ III'A )

be the set of k-chains.

Given a k-chain

u,

define its sup­

port as

q. supp u - l ( v 0 , vx , •••, vk) 6 J |u(v0, v j , •**,vk) £ 0| .

Define a map

T£ : ®u6Ck^NIII; A ) I SUPP u C

^ Ck+1^NIII; A )

(1 < I < n) by the formula: (T£u)(Q(?, v0 , v l f •••, v k) -

if (vq, v lt -

, v k) e ?£.

On sequences which are permutations of

(Q £, v 0 , vi , ’ “ » vk^ define other sequences define

T^u

Tgu

using the alternacy conditions; on all

to be zero.

Note that ( v Q, V i , •••, v k) e ?£ implies to check

(Qg, v Q, vx , •••, vk) € 3\

It is easy

that if k > 1, d(Tj?u) + T ^ 0 u ) = u where u is inthe source of

Tj? (note that supp u C

Lem m a.

u(v0, v 1? •••, vk)

=>

U f C k( N m ; A )

Le t

supp u Cfl 5g+1 0 ...H

supp du C ?£).

be such that du = 0 and

Then there exist ux e C k( N l n ; A )

u2 f C k+i < N m ; A.) such that supp Uj C ?f>+ l U % 2 U ■ " U

and and

u — Uj = du2 (k > 1).

Proof.

We can write

u = u' — u",

where

supp u'C fPg, supp u" C 3£+ i U •••U supp z C

H

1 U ?2 U •••U ?n there exists

such that u = du2 (k > 1).

u2 f

with

k < n—2 lies in

f?1 U 3^ U •••U f?n. Otherwise, we could find a subset

( v 0, v 1 , •••, v^)

such that [ v 0 , v t , •••, v^] H ( [ Q x , •••, Q ^ ] + Q{?) ^ 0

for all

It follows by induction that Q j , Q 2 , *•*, Q n ^ tvQ, v i » "* » [v 0,v i,

( N l n ;A )

Hence to conclude the proof of the proposition

it is enough to prove that any k-simplex of N m

V

such

This contradicts the assumption that

of

£, l < 2 < n .

hence

V =

k < n—2 and the

proposition is proved.

1.11

TH EO R EM .

for all

Given a linear subspace

VC V

H^(SjV ; A ) = 0

we have

i such that 0 < i < m—1 and H q (Sj v ; A ) = A,

where

n = dim V

> dim V = m > 2.

Proo f. U sin g the

acyclic covering lemma applied to the maximal covering

we see that it is enough to prove that H 0( N i V ; A )

s

A,

where N IV

H^(NJV; A ) = 0, 0 < i < m—1 and

is the simplicial complex

defined as

follows:

The k-simplices of N IV

are precisely the sets

of

linear hyperplanes in V

such that

(k+1) N jV

(H Q fl Hj H ••• fl H ^) + V = V.

can be a ls o described as the simplicial complex whose

k-simplices are the sets

( L Q, L j ,

linear subspaces of V *

such that ( L Q+ L j + ••• + L ^ ) H

V

C V

(H qjH j ,--*,H^)

L^)

of (k+1)

is the orthogonal complement of V.

one-dimensional = 0 where

Note that N JV

nected (the proof similar to the one in 1.10 uses the fact that codim

> 2).

is con­

16

THE DISCRETE SERIES OF G L n O VER A F IN IT E F IE L D

Let

be the set of all sets

S

linear subspaces of V * Q 'l, Q 2 » ' " ^ a basis.

of one dimensional

such that ( L Q + L j + ••• + L k) fl

m ^e a set

vectors in

For any I, 1 < I < m,

consists of all elements

(L q , L j , •••, L ^ )

( L Q, L j , •••, L k)

T g : SueCk( N '; A )| s u p p

Let

whose images in V * / V

define a subset

^ ([Q \ >Q 2 > *“ » Q ^ - i l + Q £+ V 1 ) --, 0.

= 0.

form

ffg C J C ' as follows:

of ?

Jg

such that (Lq-hLj f •••+ L k)

Define a map

UC

?g! -> C k+1( N ' ; A )

by the formula 0 > ) ( [ Q g ] , L 0 , L 1 ( - , L k) =

if ( L 0 , L j ,

e fF£.

Tj>u using the alternacy condition; on other

T £ u to be zero.

Note that

(L q , L j , •••, L k) e fPg = >

([Q g ], L q , L j , •••, L k) e S’ . One verifies that for T g(d u ) = u,

LEMMA.

where

Let

L j , •••, L k)

On sequences which are permutations of

([Q ^ ], L 0 , L 1, •••, L k) define sequences define

u( L q ,

k > l

we have

1).

Proof.

Write

u = u '-u ',

where

u '.u 'f C ^ jA )

and

supp u " C 9'i+l U •••U 9'm. Then z = du = du" satisfies

9\ n (?£+1 U •••U 9 m).

Put

and it is enough to prove that (T g

z) ( L q,

L j , ••*, L k) ^ 0.

L 0 = [Q g], (L 1+ -

and

Uj = T g z — u", u2 = T'f u'. supp T g z C

supp z C Then du2 = u - u ^

U ■••U 9 m. Assume that

Then we must have (up to a permutation)

( L j , •••, L k) f supp z C 3^+1 U •••U 9 m- It follows that

+ L k) n ([Q '1, - , Q g +h_ 1T Q g + h + V i ) = 0

Hence also

supp u'C 9^,

for some h, l < h < m - £ .

P A R T IA L L Y ORDERED SETS AND HOMOLOGY

([Q {>] + L j +

for some

f L k) fl ( [ Q ' j , — , Q

h, 1 < h < m—I,

17

j-j_ x ] + Q(!+h+ V -*") = 0

which implies that

L 1, L 2, . . . ,L k) f ^ + 1 U... Ui Fm

([Q j],

and the lemma is proved. Applying the lemma repeatedly we find that given that du = 0 and (N IV ; A)

u e C^CNjy; A )

supp u C fP^ U ? 2 U ••• U 3>m, there exists

such that

u = du2 (k > 1).

such

u2 €

To conclude the proof of the theorem

it is enough to observe that every k-simplex of N JV with k < m—2 lies in

U J 2 U - * * U f?m. Otherwise, we could find

k+1

lines in V *

such

that

(L0+ L 1 + - + L k) n ( [ Q i . Q a . - . Q ^ l + Qg+V1) ^ 0 for all

£, I = 1, 2, •••, m.

L q + L 1 + ••• + follows that

It follows by induction that Q i> Q 2 > * “ ’ Q m £

+ V"k Since clearly

V*

C L Q + L 1 + •••+

= L q + L 1 + •••+ L^ + V which contradicts thehypothesis

k < m—2 (m = codim V^-)

and

the theorem is proved.

Remark. This theorem contains Theorem 1.9 as a special case. V

by

V «F

and take

SI V ( V © F , V).

it

V = V C V © F (codim V = 1).

Then

Replace

S j( V ) =

The reason we have proved Theorem 1.9 separately is that

its proof is easier than that of the present theorem.

1.12

THE ORE M .

= 0

Ho(Sn ; ^ n ) s V,

where

n

for all i such that

0 < i < n—2

and

= dimV > 3.

Proof.

U sin g the acylic covering lemma applied to the maximal covering

of Sn

(see 1.5) we see that it is enough to prove that

0 < i < n-2

and

H 0( N n ; £ n ) = V

defined in 1.8 and follows.

where

Nn

is the simplicial complex

is a coefficient system over

Given a k-simplex

(H 0> H x ,

H k)

= 0,

Nn

defined as

in N n ( H i linear hyper-

18

TH E D ISCR E TE SERIES OF G L n O VER A F IN IT E F IE L D

planes in V

such that H 0 H U 1 H . •• n H k ^ 0) we have

= H 0 fl H 1 fl ••• fl H^. natural inclusions. are the sets

The connecting homomorphisms are given by the

Let

N

(H Q, H x ,

be the simplicial complex whose k-simplices

H k)

of

k linear hyperplanes in V

section is arbitrary (possibly zero). a standard simplex.

over

N

over

with value

N

( i > 0) and

i > 0.

Let

V

V

is clearly

Hj ( Nn ;

Njj

^

denote the constant coefficient system

since

N

Then clearly

is a standard simplex.

natural embedding of coefficient systems # C V is defined.

N

on the simplices of

We have then clearly

at every simplex of N.

H 0( N ; V ) = V

and

can be extended to a

such that ^ = 9u

and ^j = 0 on the other simplices. for all

Nn C N

The coefficient system

coefficient system ^

H -(N ;5 )

We have

whose inter­

H ^ (N ;V )= 0

There is a

and the quotient S = V/ 3 ).

o p o s it io n

Proof. ? set

.

can be

This w ill again be similar to the proof of Proposition 1.10.

be the set of all simplices of N. C J

as the set of simplices

For any Z, 1 < I < n, ( L 0,

L k) e

Let

define a s u b ­

such that

P A R T IA L L Y ORDERED SETS AND HOMOLOGY

19

Lo + L i + ••■ + L k) n ([Q i ,Q 2, ••*,Q g_1] + Qg) = 0. Here Q i > Q 2 > " > Q n is a fixed basis of V * . C k( N ; S )

There is a natural notion of support for chains in

(we use alternating simplicial chains). Tg : { u e C ^ (N ; S) |supp u C 5gi

Define a map

C k+1(N; S)

by the formula ( T g u ) ( [ Q g ] , L 0 , L l f . . . , L k) = n C u C L ^ L ^ - . . , ^ ) )

L ^ ) e ffg.

if ( L q , L j ,

Onsequences which are permutations of ([Qg],

L q , L x , •••,L jc) define quences define FI : ( L

q

Tgu

Tgu using the alternacy condition; on other s e ­

to be zero.

+ ••• + L ^ ) * -> ([Qg] +

projection

L q

-f ••• + L ^ ) *

is direct since

We shall verify that for all d(T g u ) + Tg(du)

In fact, let

IT is the natural linear map defined as the dual of the natural

[Qg] © ( L Q + L x + ••• + L k) -> L Q + L j + ••• + L k; note that the sum

[Qg] + (L q + L x + ••• + L k)

(2)

(Here

( L Q, L t ,

c9(Tgu)(L0, L x ,

( L Q, L j , •• •, L k) e 3g.)

k > 0 we have u e C k(N; S) and

= u, where

L k) e 5g; we have

L k) = 0 ( ( T g u ) ( [ Q g ] , L Q, L x , •••, L k)) =

0 II

u

( L

q

, L ^ , • • •, L k ) =

where 0 : ([Qg] + L Q+ ••• + L k) * -> ( L Q + ••• + L k) * linear functions.

supp u C 5g .

We als o have in this c ase

u

( L

q

, L ^ , •• *, L k )

is given by restricting

T g ( d u ) ( L 0 , L 1, • • •,L k) = 0,

hence the equality (2) is checked on simplices

(L q , Lj^ , ••*, L k) e 3g.

other simplices, the right hand side of (2) is zero.

If

and [Qg] ^ L Q, L 1, •• *,L k , all terms of (2) are zero.

( L Q, L-^, ••*, L k) i $g Hence it is enough

to check that the left hand side of (2) is zero on simplices of the form ([Qg], L j ,

> ' " >L ^ -

For

20

TH E DISCR E TE SERIES OF G L n OV E R A F IN IT E F IE L D

We have i n x u(L, L t , •••, L k)

L where the sum is over all

L

such that

L ^ L 1, L 2 , •••, L ^ , ( L , L 1, •••, L ^ )

€ Jp£, and Tg(6»u)([Qg], L j , L 2 , •••, L k) = 2

112 0 2 u a . L p - . L ^

L

where

L

runs over the same set as above.

Here

2:

( q - l ) ( q 2- l )

■ (q ‘- l )

/3(1) = y ( l ) - 1.

In order to prove that j8(n) = q 1 ' 2 ' ’ ' ' h(n_1} to prove that for any

it is clearly sufficient

n > 2 we have the identity: n—1

X = q 1+ 2 + " - + ( n - l ) +

( _ l ) n—1q 1'l“2'*'"' *'Cl

1)

i= 1 X

(q n- i+1- l ) ( q n- i , 2 - l ) ; - ( q n- l ) + ( _ 1}n = 0 ( q - l ) ( q 2- i )

( q 1—l )

26

THE D ISCRETE SERIES OF G L n O VER A F IN IT E F IE L D

Now substitute: ( q n - H 1_ l ) ( q n - i » 2 _ 1 ) . , , ( q n _ 1)

(q n - i . 1_ 1} ( q n-i> 2 _ 1} ... m + 1 > m > 1.

O bse rv e that q i ( n _ m- l ) ( q m—i + 2 _ 1 ) ( q m - i i

3 _ 1 } ...

( q m + l _ 1)

- q i(n—m) ( q m ii 1 _ f ) ( q m—if2 _ f ) •.. ( q m—1) = q i ( n- m- 1) ( q m- i+2- l ) ( q m- i43- l ) - - ( q m- l ) ( q i - l ) We have

m+1

e"(n,nu 1) - £"(n,m) = ^ 1=1

( —l ) i+1 (q—1 ) (q2—1)

(q 11 * X- l )

.

29

P A R T IA L L Y ORDERED SETS AND HOMOLOGY

x q i(n—m - l ) ( q m—i 42- l ) ( g m- i 43—l ) - - - ( q m- l ) ( q - l ) ( q 2- l ) - - - ( q i- l )

,

(_ ! ) • " « l ( q_ i ) ( q 2

( q n—m—2 _ 1 ) q n—m - l

= q n - m - l ( ( q _ 1 ) ( q 2 _ 1 ) . . . ( q n - 2 _ 1) _ e ' (n_ l j m ) ) .

This implies clearly the desired formula by induction on formula is obvious for

m - 1 .)

m m.

(The

CHAPTER 2 TH E A F F I N E S T E IN B E R G M O D U LE

2.1.

Let

E

be an affine space of dimension I > 1 over a finite field

with q elements.

The affine Steinberg module associated to E

definition the free A-module A a ( E ) E

(see 1.14).

Let

in

E (dim E- i).

fied with the set of all functions

is by

defined by using the affine flags in

F l a g ( E ) be the set of all complete

E( E q C E 1 C •- - C E p _ j )

F

affine flags

Then A a ( E )

u : F l a g ( E ) -> A,

can be

identi­

satisfying the “ cycle

condition” ^ ( E E .

o

CE

j

C - C E ^ C E

jCE.

,

C -C E ^ ) = 0

1

for any given

i (0 < i < f —1)

of dimension

0,1, •••, i—1, i+1, •••, I —1 (this sum has

and

q + 1 terms for

and given

E QC E x C ••• C E - ^ C E^+1 C ••• C Ej?

q terms for

i= 0

0 < i < t—1).

Let

A ff (E )

be the groupof all affine

acts on

A A (E )

by the formula

isomorphisms t : E ? E.

A ff (E )

( t u ) ( E 0 C E 1 C . . . C E £ _ 1) = u ( t - 1( E 0) c r 1( E 1) C - - - C t - 1( E f _ 1))

where

t e A f f (E )

and

u € A a (E ).

Given a flag e ^ ( E Q C E x C ••• C E g _ x) we denote by all

t € A f f (E )

B e the group

such that t E Q = E Q, tEi = E i , •••, tE n- l = ^ n - l *

^ave

the following

THEOREM.

Assume that A

a complete affine flag in E. scalar) function

is a field of characteristic zero.

L et

E be

Then there exists a unique (up to a non-zero

u : F l a g ( E ) -> A

such that 30

THE A F F IN E ST E IN B E R G M ODULE

(i)

u£ 0

(ii)

tu = u for all

(iii)

u£ A a (E ).

31

t e B£

The proof will be given in 2.3.

COROLLARY.

If

A

is a field of characteristic zero,

A a (E )

is an

irreducible Aff (E )-m o d u le .

This follows by applying Frobenius duality to the Aff(E)-module

A a (E )

and the unit representation of B £ .

2.2

Bruhat decomposition in the affine case Let

F lag(V )

V

be an n-dimensional vector space over

be the set of all complete linear flags

in V(dim V -= i). Given ^ F l a g V , •••, tVn_ i = Vn_ i i. The group the formula

a? =

(Vi C V2 C •• -C Vn_ i )

let B^ = { t e GL(V)| tVj = Vj , tV2= V2 ,

G L (V )

acts transitively on F l a g ( V ) by

t(Vj C V2 C •••C V'n_ 1) = (tV, C tV2 C •••C tVn_ 1).

not transitive when restricted to B^,. The orbits of described by Bruhat’s theorem. Given

V 2 n Vi92 l ^ V 2 0 Vi? 2 (i2 Here we use the convention:

n)

B^

This action is on F l a g ( V )

are

Let us recall the content of this theorem. F la g (V )

= ( i j , i2 , •••, in) of (1, 2,

that

F(n > 2) and let

we define a permutation b ( V )

by the formulae: V n n Vi n- i

VQ = 0, VR =

H

^

H Vj ,

^ V n 0 Vi n M H’V V i Bruhat’s theorem states

= V.

are in the same B^,-orbit if and only if b ( V ) = b ( V ' )

the map b : {B^-orbitsi -» Spermutations of 1,2, ■••.n!

and that

is bijective.

We need an extension of this result to the affine case.

Such an exten­

sion has been found by Solomon [12] but we shall need a somewhat different approach.

The affine space

E

can be regarded as an affine hyperplane

not containing the origin in an (£fl)-dimensional vector space

V.

be the unique linear hyperplane in V

Note that

which is parallel to E.

Let

H

THE DISCRETE SERIES OF GL n OVER A FINITE FIELD

32

A ff(E )

can be regarded as the set of all

Given e = ( E g C E j C ■• ■ C E g _ j ) B £ on F l a g ( E ) .

(Note that

t e G L (V )

such that t (E ) = E.

our problem is to classify the orbits of

A ff(E )

acts transitively on F l a g ( E ) . )

£ gives rise to a complete linear flag [e ] = ( [ E Q[ C f E j l C - ’ -C [ E £ _ 1] ) in V and to a complete linear flag [ s ] H = ( [ E j ] fl HC [ E 2] H HC- - C [ E g _ 1] fl H) in H.

(Here

[E j]

is the linear span of E j

( E q C E ' C •••CE/(. _ 1) € F l a g ( E ) .

in V . )

Let

s' =

s' gives rise similarly to complete linear

[s'], [ e ' l H in V and H. Define a permutation (iQ, ij , •■■, ip) of

flags

(0 ,1 , •••,(’) by the formulae [ e ' 0] n [ E ] i [ e ' 0] n [ e = -j, o 10 U i 0- i [E '^ n i E j ] t [E 'jin iE j

j ] ( i ^ i g ) , •••,

[ E p ] n [ E ^ ] ^ [ E p l n t E ^ j ] , ( i p ^ i 0 , i 1( •••,ip_1) •

Here we use the convention [ E

1] = 0, [Ep] - lE'p] - V.

a permutation ( j j , j 2 ,

(1 ,2 ,...,()

jp) of

[E ^in tE - ] n H ^ [ E ', ] n [ E j

by the formulae

,]n H ,

[ e '2] n [E. 1n H / [ e 2] n [ E j2 _ t] n H ( j 2 ^ tE g ]n [E j ] _ Let

g h

0 , l , - - 4, f

and one of

), ••■,

^ [ E p ] n [ E j^ _ 1] n H ( j p ^ j 1, j 2 , •■ ■ ,!£ _!).

L , i , , i9 , •■•, ip b (s ') = ( . . • ); h ’h ’

Similarly define

this is a pair of permutations:

one of

t 1 ,2 ,

•••,?.

This invariant depends clearly only on the B £-orbit of £', not difficult to see that it distinguishes e\ e " are in the

B £-orbits (i.e.,

In fact, the entries of b(e )

satisfy

(4)

b (s ') = b (e ')

same B E-orbit). Note however that the invariant

b (£ ') cannot take arbitrary values.

(3)

and it is

j j f I iQ, i1 1, j2 f Si o ’ *1 ’ *2 ji >ix , j2 > i2 -

" ’ ’ if7 f ®*0’ *1 ’ *2 ’ j£ ^ l' l ■

must

TH E A F F IN E S T E IN B E R G M ODULE

33

To prove this, let d

= dim ( [ E ' J n [E - 1 O H ) / [ E ' J n [E ,] n H d Ja a Ja - 1

d a =■- dim ( [ E ' J n [E - ] ) / [ E ' J n [ E . ,] . Ja d Ja It is clear that 0 < dg < d'g 1. We shall prove the lemma by

b to be minimal with the above property.

TH E A F F IN E ST E IN B E R G M ODULE

Then we have: [ E a_ i l n

[ E 'a ] H [ E b] = 0.

35

L (line), L C H, [E 'a ] fl [ E b _ 1] - 0 and

Keeping fixed

[ E q ] , [ E j ] , •••, [E 'a _ 2L [ E 'g], •**, [ E ' ^ j ]

we consider all ( a —l)-dimensional affine subspaces E Q_ 1 of E which lie between and q

E'a_ 2 and

if a = 1.)

E a . (There are

There are exactly

[ E a _ 1] U [ E b] - 0;

(q+1)

such subspaces if a > 1

q values of

moreover the flags

E a _ 1 such that

( E q C E ^ C ••• C E a _ 2 C E Q_ 1 C

E'a C • • • C E f _ 1) corresponding to these values of

E a _ x are all in the

same B £-orbit (cf. Corollary 2.2), and they include the original flag In the case

a > 1 there is a unique value for

[ E a _ 1] fl [ E b] ^ 0 this is precisely for u and the fact that

u is B £-invariant we get:

if a > 1 if

a = 1

s ' = (E qC E'j C ••• C E a _ 2 C E a 2 © L C E a C ••• C E

a ( e ' ) - a ( e ' ) — 1. that q

such that

E'a 2 © L . Using the cycle condition

( -u (e ") q • u ( e ') = \ I 0 where

Ea l

x).

Hence

From this the lemma follows by induction on a.

correspondence with the permutations since the invariants

( i Q,

Note that these are in 1-1 , •••, ij?) of (0,1, •••,£),

j 1, •••,]£ are uniquely determined.

function as in the Theorem 2.1.

Since

Let

u be a

u is constant on B £-orbits and

since it vanishes on flags of type II (by the lemma) we can regard a map u : jpermutations of

E

(Note

is not a zero divisor in A . )

We now concentrate on flags of type I.

Let

s'.

u as

0, 1, •••,£! -» A.

( E q C E ' j C ,,T E a _ 2 C E /a C • • • C E ^ _ 1) be a subcomplete flag in

(the ( a —l)-dimensional subspace is missing).

permutation ( i Q, i j , •• •, i(?) of (0 ,1, •••,£) with

A sso c iate to this a iQ_ 1 > iQ by the formulae:

[E '0] u n [ E j1 0] A [ e '0]n u [E -10- 1J , - , [ E ' a - z ]n [ E .ia _ 2] ^ [ e ' a - z ] n [ E ,ia_ 2 _ t1 ] [E g ln tE j

] i [ E al n [E j _ ! ] , [ E a] n [E j ] ^ [ E a] n [ E j a —1 a—1 a a

[ E a+ l l n t£ ia+1l * [ E 'a+1] n

,

- , [ E ' £] n [ E if] ^ [E '£] n [ E i r l ],

36

T H E D IS C R E T E S E R IE S O F

(Convention:

O V E R A F IN IT E F IE L D

n

[ E _ 1] = 0, [Eg] = [E g ] - V . ) We assume that [E'^I fl [E j] + 0

= > [EC] H [ E j ] C H (E qC E j C

GL

■C E '

for all

i ^ a —1 and all

2 C EgC

with the missing

E'a

C a s e 1.

There are q

(a = 1).

l ’s

C E g _j)

j.

into a complete f lag by filling in

in all possible ways. possibilities for

E g = E'j n E j . The remaining (q —1)

complete flags of type I with invaiiant ( i Q, i j ,

follows that

(q —1) u(i0 , i-j, •••, ig) +

C a se 2.

i

In all

q cases we get

( i j , iQ, i2 , i 3 , •••, ig) for

ig) for the remaining ones.

Next assume a > 2;

E'Q. One of them is

possibilities give rise to complete

flags in the same B £-orbit (cf. Corollary 2.2).

and

We shall make

Ej fl E^

From the cycle condition it

, iQ, i2 , •••, ig) = 0, (i j > i2).

there are now (q+1) possibilities for

j p min ( i Q, i x , •••, ia _ 1)

(a > 2).

E'

^

One possibility is

E'a l

=

< E'

„ E ' H E - > (affine span). The remaining q values of E ' , 9—2. 9 1_ d *■ cl give rise to complete flags in the same B £-orbit (cf. Corollary 2.2). In a ll (q+1) c a se s we get complete flags of type I with invariant

ia ’ ia - l , i a + l , ' " , i f ) ia _ i , ig ,

( i Q, i i , — ,ia 2 »

for the first value of E'a _ 1 and 0 0 > H ’

ig) for the remaining q values.

follows that q u ( i0 , i 1, - - - , i a_ 2 , ia_ j , ia ,

* 8- 2’

From the c y c le condition it ig) 4 u ( i 0 , ^

ia_ 2 , i a ,

ia - l ' ia + l ' - ' if ) = °C a s e 3. [E j

ig l

] fl H

- min (ig, ij ,

is a line not contained in [E 'a 2],

= ( [ E'a _ 2] + L ) fl E which * i » i a_2>

From the cycle condition it follows that

(q—i ) u (i0 , ij , ■■■, i a _ 2 > ia - l ’ *a’ *a+l ’ + u(i0>i 1>- , i a _ 2 , i a , i a _ 1>ia 4 l , - , i g ) = 0 .

THE A F F IN E ST E IN B E R G M ODULE

37

We can collect the three c a se s in a single formula:

(5)

6 ■ u( i0 - il . - - i a- 2 . ia - l ’ ia - 1a+ l ’ “ ' ’ i^) ■+ u( i o ’ h >

ia -2 > *a’ * a - I ’ *at-l ’

if) = 0

where if

ia - 1 7 min(i0 , i2 ,

ia —1)

q —1 otherwise. Given a permutation w = ( i 0 , it , •■■, ip) of (0 ,1, •••,£) we define its length £(w), tions

as usual, as the minimal number of fundamental transposi­

(a, a+1), 0 < a < £—1 of which w

Wealso that ig ^ m(w).

can be the product.

define m(w) as the number of indices 0 and ig =

Let

min(i0 , i1(

i ).

Wq = (£,£—1, ■•■,1,0)

length f ( w 0) = 1 + 2 + •••+ £. (0 ,1, •••,?—1,£);

ig(0 < a

be the unique permutation of maximal

It is clear that

m(w0) = £.

we have £(w q ) = 0, m(wg) = 0.

Let

w'0 =

From (5) we get immedi­

ately by induction on the length that

u(w)

It follows that where w

=

u (w )= q

( —

q

)

~

m^w ) (1 — q)

u(w q )

.

£(wn)—£(w)t m(w)—m(wn) m(wn) —m(w) 0 0 (q—1) 0 u(w0)

is any permutation of

(0,1, •••,£).

last formula all exponents are positive.) to a scalar.

m(w )

(It is easy to see that in the

This proves the unicity of

The same proof shows’ the existence of u.

u up

The theorem is

proved. Remark. formula

Define a function u': {complete flags of type I in E i - » Z by the u ' (£ ') = number of elements in the B g -orbit of s',

recurrence formula similar to the one satisfied by

u' satisfies a

u:

u ( i 0 , i1; •••, ia _ 2 >^ - l ’ 1a ’ 1a+l ’ = 6 ■ u ( i 0 , ij , ••■, i a _ 2 >ia >ia - l ’ W l ’

^

’

THE D ISCR E TE SERIES OF G L n OVER A F IN IT E F IE L D

38

where 0 is as in (5) and B £-orbits of type I in F l a g ( E ) with permutations of

0,1, ...,£.

It follows that u and

x constant.

u(w )- u '(w ) = ( —1)

are identified

u' are related by

Choosing the constant to be equal to

the order of

B c we finally get the following formula for the function u £(ww ) of Theorem 2.1: u (e ') ■- ( —1) times the number of elements t eB £ such that t e ' = s',

if s ' has type I and

u (s ') = 0 if s' has type II.

2.4 Affine foliations Let

E

be, as above, an F-dimensional affine space over

that I > 2. of

E

An affine foliation O

of E

F.

is by definition a decomposition

into affine subspaces (called leav es) of fixed dimension

in such a way that any point in E

Assume

m (0 0 and

H (-)

denotes reduced homology, i.e.,

H j(-) =

H 0( - ) = k e r(H 0(-)-> A ). )

We now compare

S j(E )

and

S (E ,0 ).

Define a map a : S j (E ) -> S by

the formula ( E'

if

a(E' H / < E ' , £ > Here £

is any leaf of O

The map that

= a (E ") Let

£

first that

be a leaf meeting

hence a ( E ' ) < a ( E " ) .

E\

order.

In fact, let

E', E " eS j (E )

such

E', E " e SI V ( E , 0 ) . Then a ( E ' ) =E ' C E "

Assume next that

E', E " e S j (E ) \ SI V (E, $ ) .

We have a ( E ' ) = < E', £ > C < E", £ > = a ( E " ) ,

Finally, assume that E ' e S j(E ) \ S I V ( E , 0 ) , E " e

Then clearly a ( E ' ) e S j ( E / 0 )

hence a ( E ' ) < a ( E " ) = E " by

the definition of the partial order in S ( E , 0 ) . SI V (E,), E " e S j (E ) \ S I V ( E , 0 ) that a

.

meeting E'.

hence a ( E ' ) < a ( E " ) .

SI V ( E , 0 ) .

if E 'f S j(E )\ S IV (E,4>)

arespects the partial

E C E". Assume

E 'e St v ( E , 0 )

Note that the c ase

cannot occur when

E ' C E".

E' e

This proves

respects the partial order.

It follows that a

induces a simplicial map X (S j (E ) ) -> X(S)

hence als o a map a A : H £ _ 1(SI(E ); A ) -> H£_1(S; A).

and

We shall prove that

40

THE D ISC R E TE SERIES OF G L n OVER A F IN IT E F IE L D

Since a A ~ a g ® 1A

«A

is an isomorphism.

«£

is an isomorphism, and this would follow if we can prove that a A

an isomorphism whenever A

it is sufficient to prove that

Assume now that A

is a field.

is

is a field.

We know that dimA H|?_1(Sj(E); A) - ( q - 1 ) ( q 2- 1 ) ••• (q^—1) (cf. 1.14) and

dimA H£_1(S; A ) = dimA H g_m__1(SI(E/0); A).

dimA Hm_ 1(SIV(E, S>); A ) =

( q - l ) ( q 2 - l ) - - - ( q f - m- l ) - ( q £- m+1- l ) ( q f - m+2- l ) . - ( q £- l )

follows that a A

(cf. 1.14).

It

is a map between A-vector spaces of equal dimension,

hence it is sufficient to prove that a A

is injective.

To see this we first

observe that given any (2—1) simplex o = ( E mC E m+1 C •••C E j ^ j < E j _ mC E f _ m+1 C •••CE^_1) m < i < 0—1 and E'unique (0—1) simplex

in X(S) (where

is transversal to leaves, t

0—m < i < 0—1) there is a

= ( E 0 C E x C •■•C E £_1) in X (S j(E )) which under

a maps isomorphically onto o. (O < i< 0 -m -l)

E- contains some leaf,

In fact we have

and E- = E'j(0—m< i < 0-1).

H E\ _ m

E^ -

A ll other (0—1) simplices of

X (S j(E )) are mapped by a onto simplices of lower dimension. that given a chain u € Cg_j_ (X(S j(E)); A),

It follows

the image « A u e C^ _j(X (S ); A)

is given by the formula: («AuXEmC E m l C - - - C E p_ 1< E ^ _ mC E '£_ m+1C . . . C E ' E_ 1)

= u(EmnE'e_ mc Em+1nE 'F_ mc ...c

e^

de'

^

c

E £_mc E g _ m+1c - c E'g.

This shows that « A U = 0 if and only if u vanishes on all complete flags

( E 0 C E 1 C . . . C E £_ 1)

0 < i < 0-1

(£

in E

such that dim (E- H where

We say that d(e )

£ by the formula

d (e ) = (dQ, d x , •••, dg).

is

A (s ) =

Note that A ( e )

is

minimal if and only if A( e ) = 1 + 2 f •••+ m or if and only if d (s ) = (0, 0, •••, 0,1, 2, 3, level of £.

m).

We shall prove the lemma by induction on the

We know that u (s) = 0 when A( e )

that £ has non-minimal level.

Then there exists some

such that d j _ j < d- = d - f l . Consider E j+1

such that

E -_1

C

is minimal.

Assume now

i, 0 < i < f —1

all i-dimensional subspaces

E- of

E ^ . We have

2 u (E x C -C

k i_ 1 C E j C E i+1 C - . . C E ^ )

-

0 .

Ei In the above sum all flags except for the original flag (d Q, d1 ,

8,

have type

d - _ 1, d- —1, d-+ 1 , •••, dj?) hence have level equal to A(£) — 1,

so we can assume, by the induction hypothesis that u vanishes on them. It follows that

u(£) = 0 and the lemma is proved.

It follows from the lemma that a A

is actually an isomorphism.

Com­

bining this with the isomorphism (6) and identifying SI V ( E , 0 ) - SI V (V, V), we have the following THEOREM.

Let

V

b e a n (£4 1ydimensional vector space over

F,

V

an

m-dimensional linear subspace of V (f > m > 0) and E

an affine hyper­

plane in V

Let

E/V

be the

Then

E/V

is an

image of E

such that 0 i E

and V

is parallel to E.

under the canonical projection

affine hyperplane in V / V

V ^ V/V.

not containing zero and there exists a canoni­

cal isomorphism i/fE : A a (E / V ) ® PA ( V , V ) *

uniquely characterized by the property that

A a (E )

THE D ISCR E TE SERIES OF G L n O VER A F IN IT E F IE L D

42

= 0

Ei "k+1

(E o C E iC -C E ^ C E ^ C -C E ;,!) E 0 f E n - l \ E n - 2 - E i H [ E 0 ^ - - E kl i [ E k

_ l l , E kCEk+1CEk+2

E k+ 2 l l [ E k+ l ] ' - - ' E ; - l H [ E n - 2 ]

(cf. (7k+1) for f)

and (7k) for

Tf

is proved for

Next we prove (7n_ 2) for

0 < k < n—2. Tf

(here we assume

n > 3).

Let

( E 0 C E 1 C - - - C E n_ 3 C E n_ 1) be an incomplete flag in V ( 0 / E p .

We have

(q + D 2

( T f ) ( E 0 C E 1 C •- C E n—3 C E n—2 C E n—1 >

En —2

(qn- q n- 2 ) 2

=

f(E o C E j C - ’- C E ^ - j )

( E qC E j C- •-C E ^ _ j ) e Y E 0fE n_ 1\En_ 2(E 'l|[E 0L - - , E n_ 2 !!fEn_ 3]

(qn - l _ q n- 3) = ( - l ) n- X

(q)

2

2

(EoCE ; o - - C E n_ 2) E 0 f E n - l \ En - 2 - E i l l [E0 ] - - ” ’E n - 2 l l [En - 3 ]

e

; _

f ( E oC E i C-

CEn -l)= °

j

E n - 2 CEn - l

^

e

;_ i ( c f . ( 7 jj_ j ) f o r

and (7 n_ 2) for T f

is proved.

f)

46

THE DISCRETE SERIES OF GL n OVER A FINITE FIELD Finally we prove (7n_ 1) for

complete flag in V (0 / E p .

Tf.

Let

( E QC E x C •• • C E n_ 2) be an in­

We have

(q) 2

( T f ) ( E o c E i c ••• c

e

„ _ 2 c E n—i )

K-1

(qn- q n~ 1) 2

= ( - 1 )" - 1

f C E g C E j C •• • C

)

( E qCE^C- ••C E ^ _1) 6 Y E 0f V ' E n - 2 ' E i ,!tEt)1' - - E n - l l l t Eh - 2 ]

(qn - 1 - q n _ 2 ) = ( - l ) n_1

(q) 2

2

f ( E o C E i C - - C E n _ 1) =

(E jCE^C- •'C E ^ _ j )

E„

E 'll|[E0],E '/ E n_ 2

E ^ E;

E 2 ||[Ei],-->E n_il|[En_ 2]

0

(Cf. (70) f o r

f)

and the lemma is proved. It follows from the lemma that T : ? A -» 3 ^ into itself; the restriction of T letter:

3.2.

w ill be denoted by the same

T : A a ( V ) - A a (V ).

In this section we shall prove the following

P ro p o sitio n .

Let

(8 )

q

{(E

whenever Then

to A a (V )

takes the subspace A a (V )

C E

f e A a (V )

j

be such that

C •••€ E n_ j )

=

f ( E o C E ; c . . . C E n_ x)

E Q = Eq, [E.J = [E ^], •••, [ E n_ 1] = [ E /n_ 1L ( 0 ^ E ^ 0/ E p .

Tf = f.

T H E D ISTIN G U ISH E D D IS C R E T E SERIES M ODULE

Proof.

We shall first consider the case

ing (8).

n = 2.

Let

47

f e A a (V )

be satisfy­

We have (q -D (T f)(E 0 C E j ) = —

^ (E E

q

^

e

i

qC

f(Eo C E j ) E')£Y

\ E

q

, E j || [ E

q

]

(q-1 ) X

f(E 0 C E i>

( c f . (8 ))

K E 0 C E x)

(cf. (70))

E 0f E l X E o

Let now

n>3

E n_ i ) € (9k)

and let

f e A a (V )

be satisfying (8).

Let

(E q C E jO -C

prove by induction on k the following statement.

^e

( T f ) ( E QC E 1 C ••• C E n_ 1)

= ( - l ) n~ k - 1 £

f ( E QC E j C ■••C E k _ 1CE j {C < E ^ , E k> C . " C < E ^ , E n _ 1> )

Ek E k - l C E kC E n - l

( 0 < k < n —1)

E k ^E n —2

Note that (9

is precisely the statement

from the hypothesis (8).

= ( _ i ) n- k - l

and that (9Q)

Assume that (9k) holds for some

we shall prove the statement (9k+1). (Tf)(E0 C El C

T f = f,

follows

k, 0 < k < n—1;

We have

E n_ 1 )

£

f ( E 0C E 1C . . . C E k _ 1C E kC < E k E k> C - C < E j ( , E n_ 1 >)

Ek E k - l C E kC E n - l E k ^ E n—2

48

THE DISCR E TE SERIES OF G L n OVER A FIN IT E F IE L D

= ( - 1}

“

f ( E 0 C E j C - ••CE k _ j C EjjC E ^ + I c

2

E k+ 1

Ek

E k+l^En_ 2

Ek^Ek

< E k+ l - E k+ l > C -

= (_ l)n -k -2

C < E k+ l ’ E n - l > )

f ( E 0C E 1C . . . C E kC E ^ 1C< E ' k + 1 , E k + 1> C . . . C < E i { + 1 , E n _ 1

2

E k+ 1 EkC E L l C E n - l E k+ l ^ E n - 2 (cf. (7^)) and the proposition is proved by induction. the convention

(Here we have used

E _ 1 = 0 .)

Note that in case

q

is invertible in A,

any element

f e A a (V )

satisfy­

ing ( 8 ) must be zero.

3.3

In this section we shall assume that dim V = 2.

the set of homogeneous polynomial functions is clear that

P o l_ 9 (V ) q z

V -» F

L et

P o l q_ 2 (V )

of degree

is an F-vector space of dimension

be

(q —2). It

(q —1).

We

shall describe now two F-vector spaces which are canonically isomorphic to P o l n 9 (V ) 4

A

but whose definitions are somewhat more geometric.

First we recall (see 1.13) that ® ( V ) functions of V

is the F-vector s.pace of all

f which associate to any one-dimensional linear subspace

a vector

f(V j) e

such that

f( V j) = 0 (sum in V).

It is

VjCv

clear that dimF ® ( V ) = number of lines — 2 = (q+1) —2 = q — 1. ® ( _ 1 ) (V )

be the F-vector space of all functions

f(Av) = A- 1 f(v), A e F *

and such that

^

f : V \0 -> F

Let such that

f(v) = 0 for any affine hyper-

V

veEj^ plane

E j C V, 0 / E j .

Let

f 6 P o l q _ 2 (V).

f defines a function

THE DISTINGUISHED D ISCR E TE SERIES M ODULE

f:V \ 0 -> F

so that

f(Av) = A - 1 f(v ),A e F *.

Let

We can choose a basis

e x , e2

which clearly satisfies

an affine hyperplane in V ^ O ^ E j ) . E 1 = \e1 + (ie2 , (i e F L

49

Ej

be

in V

We have

fO i e l + ^ 2 e2) =

2

ai

ai f F

0< i< q-2

hence ^

f (ei + iie2) =

]£

ptfF

0

/ 3i ^

° y ( - 1 } (V ) which is clearly injective. In order to prove that this is an isomorphism it is then enough to prove that dimension at most vQ e E 1

be fixed.

( q —1).

To s ee this, let

In order to prove that

cient to prove that if f e identically zero.

^(V)

f(v) = 0.

^^(V) < q —1 it is suffi­

is zero on E ^ V q

Consider such a function

f(v Q) = 0 since

0 \ E x and let

E jC V ,

dim

has

f.

then f

must be

We must clearly have

From the homogeneity property of f

it

v£E1 follows that

f(v ) = 0 for any

parallel to E j . Let

v e V Wj

^ e VjXO.

Let

such that ^ 6 E j , 0 | E j . Then than vt . It follows that P o l n_ , ( V ) -» ? ( “ 1 )( V )

4 ^

Let now

f e ® (V ).

by the formula

E^

is the line through 0,

be an affine hyperplane in V,

f(v ) = 0 for all points

f(vx) = 0,

v

of E^ , other

and hence the assertion that

is an isomorphism is proved. We associa te to f a function

f ( v ) v = f[v]

(here

clear that f(Av) - A - 1 f(v),A 6 F * . hyperplane.

where

[v]

is the linear span of v).

Let

We shall show that

E x C V, 0 | E r

f(v ) = 0. veE^

f : V —0 -* F

defined It is

be an affine

In fact, let v *

be the

50

TH E D ISCRETE SERIES OF G L n OVER A F IN IT E F IE L D

unique linear function

VF

f[v] f f(ker v * ) = 0,

such that

v * ( v ) = 1 for v €

since f e ® (V ).

Applying v *

E 1. We have to this equality

veE1 we get

v * ( f [ v ] ) = 0 and hence

f(v )v *(v ) = ^

vfE^

vfE^

follows that the correspondence

f ( v ) 1.

It

vfE^

f -> f defines a map ® ( V ) -»

This is clearly injective, and since the two vector spaces involved have the same dimension, it must be an isomorphism.

We collect these results

in the following

PROP OS I TI ON.

If V

is a 2-dimensional vector space over F,

there

exist canonical isomorphisms of F -vector spaces

P o iq_ 2( v ) *

3.4

fp( - 1}( v ) ? 3)(v) .

In this section we shall assume that A = F.

the set of all A^f(EQ C E j Here

C

f e A F (V ) ••• C E n_ 1)

such that

Ap(V )^

as

f(AEQC A E j C ••• C \ E n_ 1) =

for all A e F *

and all

k is some fixed element in Z / ( q —1 ) Z .

Ap(V) =

Define

©

( E 0 C E 1 C ••• C E n_ 1) e Y. It is obvious that

Ap(V)(k).

k f Z /(q —1) Z

Note that the endomorphism T subspaces

of A p ( V )

leaves invariant each of the

A p (V)^k).

We now prove the following partial converse to Proposition 3.2.

PR OP OS I TI O N .

Let

f £ A F ( v / - 1 ^ be such that T f = f.

Then

f satis­

fies condition ( 8 ) of 3.2.

Proof.

Let

8 - ( E 0 C E x C ••• C E n_ 1 ), £ = ( E QC E x C ••• C E n _ 1 ) be two

elements of Y

such that Eq = Eq, [E ^ ] = [E ^ ], ••*, [ E fl_ j ] = [ E n_ 1 l.

We

T H E DISTIN G U ISH E D D IS C R E T E SERIES M ODULE

wish to prove that f (e ) - f(e).

2

Since

T f = f, this is equivalent to

f ( E o C E i C - C E n_ 1) =

( E 0 C " - C E n - l )eY

2

f(E qCE'^C-"CE'^_1).

(E 0 C E i C ' " C E ' n - l )fY

We define a function 0 : V \ [ E n_ 2] - F where

51

by 0(E'O) = fC E o C E i O . - C E ^ j )

E j ( l < i < n—1) is the unique i-dimensional affine subspace of V

such that Eq e E^ and E^||[E^ 1].

Then the equality to be proved can

be written as

This clearly fo llows from the following

LEMMA.

Let

V

be a v ect o r s p a c e of dim ension

a linear hyperplane in

0.

not containing cj> : V \V

j -> F

E ^ V \ V n_ !

Let

V and E^__j, E ^ - l

n > 2 over F, V n_ j

two affine hyperplanes in

E Q e E ^ _ 1 H E/^ _ 1 H V n_ 1. A s s u m e that

is a function such that 0(A E q ) = A_ 1 0 (E q ),

and

A 6 F*,

(q)

2

* (Eo> = 0

for any one-dimensional affine subspace

C V (E^ ||[E0], E j : V — [ E q]

F

( E Q, E^ , E^ , Vx = [ E q]

must be identically zero, cf. 3.3).

belongs to

(see 3.3).

Eq 6 V \ [ E q].

f e 5^

It follows that

^

0 ( E'0) = f ( E 0)

E oeEi ^Eo and similarly

(q -l) 2 E ofE 1XE0

From what we have

1\v)

(q -l)

and the Lemma is proved.

This map is

must be isomorphism.

S by hypothesis.

just proved, it follows that there exists for all

On the other hand,

Since the vector spaces involved have

the map fF^_ 1 ^(V) -> S

Our function

It is obvious

f e 5^- 1 ^(V) which vanishes on V - [ E q]

clearly injective (a function

dimension q - 1

^ ( V ) -> S

satisfying the

fixed).

is an F-vector space of dimension (q —1 ).

there is a natural restriction map 5^

•

E (/E 1 V E 0

be the set of all functions

hypothesis of the Lemma that

2

* < E o> =

* < E 0 > = f( E o)

such that f ( E ^ ) =

TH E DISTINGUISHED D ISCR E TE SERIES M ODULE

3.5. From 3.2 and 3.4 we deduce that

53

1 ^ A F (v /

T : A F (v /

^

is idem-

potent and its image is precisely 3 = { f €A f ( v / - 1 ^ ( f satisfies ( 8 )S . In this section we shall construct a canonical isomorphism 3 — ® ( V ) . We recall (cf. 1.13) that ® ( V )

is the set of all functions

to any complete linear flag

( V i C V 2 C ••• C Vn_ 1)

0 (V 1

C V 2 C ••• C V

in V

0

which associate a vector

t ) € V1 such that the following conditions are satisfied

(q+l)

2

(lo p

^ C ^ C .-C V ^ C V jC ^ C -.C V ^ )

= o

V.

1

(1 < i < n - 1 ) where ( V j C V j C - C V ^ j C V ^ C - C V ^ j ) incomplete flag (the empty fla g if vectors in

if

i > 1

or in V2

as follows.

Given

( E QC E 1

C V2 C •*• C Vn_

C ••• C E n_ 1)

Ei C V2 \0, •••,Efl_ i C V \ 0 .

x)

- f ( E 0 C E 1 C - . C E n_ 1 ) E 0

is any element of

Y

satisfies the conditions (10p

( l < i < n —1).

from the condition (7^_1) satisfied by

f.

we can clearly assume that dim V = 2. f € fp( —1 } ( V )

jp (-l)(V )

such that

E Q e Vj \0,

(This is independent of the choice since

satisfies ( 8 ) and is homogeneous of degree ( —1).)

E 0 6 V1\ 0

f e J , define

by the formula (a f) (Vj

where

Note that this is a sum of

i = 1.

if

We define a map a : 3* -> ® ( V ) , at e ® ( V )

n = 2 , i= 1).

is any given

If

f

We must check that at

i > 2 this follows easily

In order to check ( 1 0 ! ) for af In this case we must prove that if

(se e 3.3) then the function at defined by afO ^ ) = f ( E Q) E Q, lies in J)(V);

but this follows from the isomorphism © ( V ) 55?

constructed in 3.3.

Next, we define a map /3 : ® ( V ) -> 3*. by the formula:

Given 0 e ® ( V )

define /30 6 J

TH E D ISCR E TE SERIES OF G L n OVER A F IN IT E F IE L D

54

(fi)(E0 C E 1 C •••C En __1 ) E 0 = 0 ( [ E o] C [ E 1] C - - . C [ E n _ 2 ]),

( E q c E x C ■••CEn_ 1 ) e Y . It is clear that (dcj) fies ( 8 ).

is homogeneous of degree

(-1 )

and that /30

s atis ­

We must check now that /3 satisfies the conditions (7-),

(0 < i < n—1).

If

0 < i < n—1 this follows easily from the condition

(10i 1 1 ) satisfied by 2. r —

Then there is canonical isomorphism

of F -vector spaces fi : ® ( V ) “5 I f f A f ( V ) ( _ 1 ) ! T f = f 5 .

/3 rs uniquely characterized hy the property

(fi(f>)(E0 C E 1 C ••■CEn_ l ) E 0 = 4>(\E0K [ E 1] C - - C [ E n_ 2]) for all cf> f 5 ) ( V ) , ( E 0 C E 1 C - . - C E n_ 1) f Y.

3.6

We recall now the definition of the ring of Witt vectors

to the finite field

F. Let

p

mutative ring together with a

WF

be the characteristic of F. Wp

associated is a com­

ring homomorphism WF -» F , which is

uniquely characterized by the following properties (see [10], p. 48). a.

p is not a zero divisor in WF

b.

WF

is Hausdorff and complete with respect to the topology deter­

mined by the ideals

pmWF ( m > l ) .

c. The sequence 0 -* pWF -* WF

F

0 is exact.

It is well known that Wp must be in fact an integral domain characteristic zero, that there exists a unique map F*-> Wp

of

(denoted

THE DISTINGUISHED D ISCR E TE SERIES M ODULE

A->A)

such that A • p = Ajz(A, p e F * )

the natural inclusion Wp

F*

> F.

and the composition

(A ^ A

55

F * -> Wp ■-> F

is

is known as the Teichmiiller map.)

can be described explicitly as the set of all sequences

(A q ,A 1 ,A 2 , ■••) of elements in F

in which the addition and multiplication

are given by certain universal polynomials discovered by Witt (see [10], P. 49). If F

is the prime field of characteristic

p,

Wp

is canonically isomor­

phic to the ring of p-adic integers. If

F

> F ' is an imbedding of a finite field in another, there is a corre­

sponding imbedding Wp -> Wp-. naturally on Wp '

3.7

so that Wp

Moreover the Galois group

G a l(F '/ F )

acts

is precisely the ring of invariants.

In this section we shall prove the following

LEMMA.

field). M ® F

Let

L et

M

be a finitely generated free

W p - module

T : M -> M be a W^-linear map such that T

is idempotent.

a finite

(F

®

: M ® F ->

WF

Let

WF

WF

M' = Sx t M |lim ( 1 - T )1 x = 0 l , M" = |x< Ml lim T ^ O l i-->oo i-»oc

.

Then M = M' © M".

Proof.

We shall prove the lemma assuming first the following statement:

there exists

T e Endw (M)

such that T ' 2 ~ T ' f T ' T

TT

and

T ® lp

F

Wp

- T ' ® 1F . Then we have clearly WF mutes with this decomposition. and

T2 e Endw (ker T ' )

on ker T'.

Let

x 2 e ker T'.

Then

x e M;

M = ker (1—T ' ) © ker T ' and

Moreover there exist

such that

T - 1 — pTj

we can write uniquely

T

com-

Tj_ e Endw^ ( k e r ( l —T ))

on ker ( 1 - T ' )

and

T = pT2

x - x 1 + x 2 , Xj e ker (1—T ),

(1—T ) ix = ( p T j ^ X j + ( l - p T 2 ) ix 2 .

56

THE D ISCR E TE SERIES OF G L n OVER A F IN IT E F IE L D

Since

lim ( l - T / x

lim ( l - p T 2 ) ix2

hence x 2 -

0,

i -*CKj

( p T . ^ X j - 0 we must have

0 and lim

0.

This shows that

The reverse inclusion is obvious hence proves that position

M - M'© M".

Y0 - T, X Q

2

M' - ker (1—T')*

Similarly one

IVT ■= ker T ' and it follows that we have the direct sum decom­ The existence of T ' is a c la s s ic a l fact about

‘ li ft in g idempotents’ \ Define

M'C ker (1—T ').

The idea

is to construct

Y 2 - Y0 , - - , Y i ( 1

(i > 0),

T ' by a limit procedure.

Y, * Xj - 2 ^ ,

X .+ 1 =

. One checks by induction that

is divisible by

ry1 p

r\1 . It follows that Y itl - Y-

X ^ ( l —2Y j) is divisible by

hence '2 and it is easy to see that T T ,

/

Y-

1 converges to some T ' c Endw w p (M)

T T 'T 'T

and

T

®

- T ' ® 1F . The first part of the proof shows

wF

wF

that

T ' is in fact uniquely determined by these properties.

3.8

We use the notations of 3.5.

We shall apply the Corollary 3.7 in the

?(k)' We define A w (V)*

following situation.

p

as the set of all

f e A w (V )

F

F

satisfying the ‘ ‘homogeneity condition” f(AK 0

for all A f

F*

AH,

.

VM F

Let

(V )®

Here ®

k is some fixed A

k fZ /(q-l)Z

( V ) (k) and F

® F. WF

wp

T : A w (V ) -» A A

F

T :A w wp

” F

C E n l ) t Y.

We have A w ( V ) =

...

(V )W *

AEn .,) = A ^ C E j C - . - C E ^ j )

and all ( E QC E , C

element in Z / ( q - l ) Z . ...

•••

(V ),

( V ) ( - 1 ) be the restriction of the map

defined in 3.1 (note that all the subspaces

F

are invariant under

T).

F

It is obvious that we have a commutative diagram

I A f ( V ) 2.

dimp ( V ) - 1. In

58

THE DISCR E TE SERIES OF G L n OVER A F IN IT E F IE L D

I f : V \ 0 .WF i f(AE0) = A " 1f( E „ ), E 0 f V'. 0, A f F*| .

D (V )

This is a free Wp-module of rank 1 and there is an obvious isomorphism P : ® ( V ) -> D ( V )

® F

defined by / 3 (v )(E q ) =

where

WF

and

E q 6 V\0.

If

n y 2,

(q—l ) ( q 2 —1) ••• (q 0” 1—1). for dimF ® ( V ) ,

3.8

Let

E

v e ® (V ) - V

0

D (V )

is a free Wp-module of rank

This follows from the corresponding formula

see 1.14.

be an affine hyperplane in V

In this section we shall prove that Wp-isomorphism p E : D ( V ) * $ A ^

E

(E).

such that

0 \ E, (dim V > 2).

gives rise to a canonical Define pE

by the formula

F

(pE f ) ( E 0 : E 1

E n_ 2) = f ( E 0

for any complete affine flag It is clear that pE isomorphism. -i (q 11 —1)

K,

-

E„

( E QC E^ C ••• C E n_ 2)

in E

is an isomorphism if and only if pE

Since dimF ( D ( V )

E)

f 6 D (V ).

and any ® 1F

is an

® F ) - dimF ( A p ( E ) ) -= (q —l ) ( q 2 —1) •••

Wp it is sufficient to prove that pp.

®

w

lp.

T posing with the isomorphism D ( V ) W ® F = 'J (V ),

is injective. pF

F

map pE : ® ( V ) -> A p (V )

,

® lp

By com-

becomes the

Wp

given by

(Pe 0 ) ( E o C E 1 C "• C E n_ 2 ) E 0 = 0(1 E q] C [ E j l C "• C [ E n_ 2] )

e $>(V), (see 3 .7 ). Let

f ® (V )

i that cf>(V1

1 < i < n;

We shall prove by induction on

V 2 ” *-’ C V n_ 1) - 0 whenever

use the convention V n = V . )

assumption pE flag in V

be such that pE = 0.

0.

such that

Let

For

V T E

* 0.

i - 1,

this is just the

( V j C V 2 C — C V n_ 1)

fl E - O,

D E = 0

We have then by the cycle condition for yS:

(i

fixed,

be a complete linear for some

k, 1 < k < n.

THE DISTINGUISHED D ISCRETE SERIES M ODULE

o (V ,

V,

-

V k ._;

V k. ,

V.

....

Vn.,,

vk .2

vk

,

59

(q) = -

£

o (V ,

V,

•

Vj E ® I p

Hence cf) =--- 0.

V^_1

and

satisfy It follows

V fl H E 1=0 which is always

This proves that p E

is injective hence

*s inje c tive and we can state the following

THEOREM. V

V n ,)

hence the induction hypothesis can be applied.

that qS(V1 C V 2 C •■• C V n_ 1) - 0 whenever satisfied.

. .

V ^ _ 2 C Vj 2, and let

Assume 0 ( E.

E

be an affine hyperplane in

Then there exists a canonical isomorphism of

p E : D ( V ) 5$ A W ( E )

given by the formula

F

E,

for any complete flag

3,9

-

K n . . ) = f ( E 0 C E l C - C E n_ 2 C E )

(Eq C E j C ••• C E

2)

rn ^

an m > 0).

canonical W^-isomorphism i> : D ( V / V ) ® *

Wp

n over

F

and let

There exists a

Pw ( V , V ) S D ( V ) . F

Remark. This tensor product decomposition corresponds in terms of dimensions to the factorization

(q ~ l)(q ^ ~ l)

( q n _ 1 — 1)

=

( q - l ) ( q 2 - l ) - - - ( q " - " 1- 1 - ! )

—m—1) n ^( q nr- m i l x ( q,n n—m

3.11

1

. /„n—1

l)■ ■■(q r

Applying Theorem 3.10 repeatedly we get the following

C O RO LLARY.

Let

V

let V- C V; C - “ C V: n l2 \ n > i^ > •••> i 2 > i^ > 0.

be a vector space of dimension

n over

be a linear flag in V (dim^ V- = i_) * 1a a

F

with

There exists a canonical WF -isomorphism

and

64

THE DISCR E TE SERIES OF G L

0-

D (V /V-

) «

lk

/

®

W^f 'F I

OVER A F IN IT E F IE L D

n

Pw ( V / V : F

F

xa - l

, V: /V: )\ ? D (V ) la la - l \

yl < a < k

where we use the convention

V-

0.

This result describes the restriction of the GL(V)-module proper parabolic subgroup of G L ( V )

D (V )

to a

(the stabilizer of

V- C V - C - C V- ). h ~ 2 xk (V j C V 2 C ••• C V fl_ 1 ) in V.

Take for example a complete linear flag Then we have a canonical Wp-isomorphism

^ ; D ( V / V n_ l ) w ® /

Pw pC V /V a-r V ^ a - i A

\ 1 < a < n —1

describing the restriction of D ( V )

-

/

to a Borel subgroup of

G L ( V ).

In

terms of dimensions, this corresponds to the factorization of (q—1 ) (q 2 —1 ) ••• (q n 1 —1 )

3.12 all

Let

VC V

t € G L (V )

into the factors

1 , ( q - 1 ), (q 2 - 1 ), •■■, (q n“ 1 - 1 ).

be as in the Theorem 3.10.

such that

Let

be the subgroup of

1 1V = identity and t jV/V = identity.

clearly an elementary abelian p-group of order

This is

q m^n m^.

We have the following

PR O PO SITIO N .

For any V

C

V, 0 < dim V


V

such that t(V) = V.

(D ( V / V )

and

Pw ( V , V ) F

can be regarded as

G^-modules in a natural w a y.)

acts as the identity on D (V / V ). acts as zero on R J; ( V , V ) .

Note that any

te

It is then sufficient to prove that

t uU^

THE DISTINGUISHED D ISCR E TE SERIES M ODULE

Let

V

plete flag in V

2

tuV

V >V )

( V n_ mC V n _ m_| 1 I ••• C

and

such that \L + V = V (n—m < i < n—1).

V n -m C V n -m a : -

65

be an incom­ We have

CVn -l)

trtjY / V

2

u(t_ 1 v „n—m mc r 1v „n-m-: mjl1 c - - c t _ 1 v n— n l, )7

uuY V 2

2

( V n - m + l C' " CVn - l )

' ’n - n w l f

where

(V l _ , v n—m+1

u(s " 1v n- mc v'n_ m+1c . . . c v ' n_ 1) - o

V

C ■••CVln—11 )/

(t_ 1 Vn_ m f l C •*• C t- 1 Vn_ i )

runs over all distinct flags of the form

^or some

from the cycle condition satisfied by

1~ u.

The last equality follows The proposition is proved.

More generally, let v --- (V- C V- C •••CV- ) be a flag in V as in 3.11. h x2 Let G ^ be the parabolic subgroup of G L ( V ) consisting of all t e G L ( V ) such that t(V: ) = V- (1 < a < k). Let U ^ be the subgroup of G ^ con1a a ^ ^ sisting of all t r G ^ such that tlV- /Videntity, (1 < a < k+1) ^ a a —1 where V- - 0, V: = V. We have the following *0 *k+l

t acts as zero on D(V ).

C O RO LLARY.

teu y

Proo f. This follows easily by induction on k from the proposition. Remark. The conclusion of this corollary means that as a GL(V)-module, D (V )

satisfies the cusp-conditions hence it belongs to the discrete

series (s ee [ 6 ]).

CHAPTER 4 T H E C H A R A C T E R O F D (V ) A N D T H E E IG E N V A L U E A (V )

4.1 Let Let

V

be a vector space of dimension

t : V -> V

Tr (t| V ) 6 WF

be an automorphism of V. as follows.

G al(F V F ).

under

G a l(F V F )

under

G a l(F V F ).

see that

Tr(tjV )

F ' be a finite extension field of R e d e fin e

It is clear that the set

It follows that the set (which acts on WF 0 It follows that

subspace of V

invariant under

dim V = 1,

t :V ? V

so

Tr(t|V) = is invariant

is also invariant

+ ^2 + ‘

+ ^n

(see 3.6).

is independent of the choice of F'.

4.2 We say that an automorphism

any

hence

F

£Aj, A2, •••, A )

( A j , A2 , •••, A )

T r (t / V ) ^ 2 ' ’ ” '^ n

that the eigenvalues

Aj + A2 + ••• + An 6 Wp/. under

Let

n > 1 over a finite field

is invariant

It is easy to

Note that

Tr(t|V)

V.

t : V -> V

is anisotropic if there is no

t other than 0 and

is anisotropic.

V.

In particular, if

In this section we shall prove

the following

PR O PO SITIO N .

If t : V ^ V

is anisotropic then

Trw (t|DV) -

( - l )n _ 1 Tr(t! V ) .

F

P roof. The proposition is obvious when

Mi = D (V ) © ^

n - 1.

Assume now that

© c v D ( V n - 2) ) ® ( Vn© Cv D ( Vn - 4 ) ) ® -

and

66

n > 2.

THE C H A R A C T E R OF D (V ) AND THE E IG E N V A L U E A (V )

“2 *

and

M2

D^n^ are the eigenvalues of t on V.

set of eigenvalues of t on M- ( i - 1 , 2 ) .

S 1 U (A1 , A 2 ,•••, An) § 2 U (A1, A2 ,

Let

Sj

be the

It follows that

§2

(n even)

•••, An) •:=S x

(n o d d ) .

-

This implies that

V

Trw (t|D(V)) =

( —l )n _ 1 —i Trw (t| ©

F

F

K K n -i

Next we observe that Trw (t| © F

D(Vj)) + ( - I ) " " 1 T r ( t | V ) .

v.cv

1

D ( V - ) ) = 0 for

1 < i < n—1 because

acts on the set of i-dimensional linear subspaces of V point (since

4.3

t

v ic v

t is anisotropic).

without any fixed

The proposition is proved.

We shall prove the following:

PR O PO SIT IO N .

Suppose we have

t : V 5 V, t (V ) = V

such that t|V

= ( _ l ) m~ 1 ( N —1) where such that Vn_ m t V = V

N

V C V (0 < dim V - m < n) and is anisotropic.

Then Trw (t|Pw ( V , V ) )

is the number of linear subspaces

and t (V n_ m) = V n_ m-

Vn__m C V

68

THE D ISCR E TE SERIES OF G L

Proof.

n

OVER A F IN IT E F IE L D

From the exact sequence 1.13(e) we see that

2

TrwF ^ l pwF ( v - V ) ) =

( - 1 ) n _ 1 " i T r wF ( tlv ^ - pwF ( v i ' Vi n V ) ) + ( - l ) n

n —m 1 over F.

Consider the

virtual representation brCV)= ( —l ) n- 1 D ( V ) + ( —I ) " ” 2

© v„ _ l Cv

of

G L(V ).

©

D (V ).

V.CV

1

of

Note that G L ( V ) When

D(Vn_ j ) + — t- ( —1)°

©

D (V j)

v i Cv

operates naturally on the free Wp-module

1 < i < n—1 this can be regarded as the representation

1

G L (V )

induced by a representation of the maximal parabolic subgroup

GX , which is trivial on U y , for some fixed VjC V; s ee 3.12 for the i i definition of G y and U y . br(V ) can be considered as an element in i i Rw ( G L ( V ) ) . Under the natural map Rw ( G L ( V ) ) -> R F ( G L ( V ) ) , br(V) F F* 80

T H E B R A U E R L IF T IN G

goes to V,

81

regarded as a G L(V )-m odule in the natural way.

(T h is follow s

from 3.8 and the exact sequence 1.13(c).) We sh all now study the behavior of b r(V ) Let Let

VC V

W

be an m-dimensional linear subspace of V, (0 < m < n).

M be any free WF -module with a G L (V )-actio n .

^ V

(

under extensions.

f =0!

and

M/M'.

Then

M' and

Let

M' =l f f M|

M" are G^-modules since

v V is a normal subgroup of G ^.

Wp -modules

0

M' -> M -» M"

It is obvious that the exact sequence of

0 is split over Wp

general not compatible with the action of G ^ .

but the splitting is in

For example if M = D (V ),

we have

M ' = M = D (V / V ) ® Pm ( V , V ) , by Proposition 3.12 (cusp condiwF wf tion) and 3.10. It follow s that M" = 0 in this case.

P

r o p o s it io n

.

m=

Let

©

D(\A) (0 < i < n).

Then

vxiv (i )

M' =

©

D(VS/ V ) ® Qi(Vs , V )

r TT

TT

W-,-,

S

vcvs m \s\m + i

and (ii)

M" -;©

D (V j) ® /

VCV where

Q j(V g , V ) ^

©

_D C ^ A

/

Pw (V j,V ^ n v )

if

v.cv 1_ s

m < s < m+i,

F

V. + V = V

1

©

V V jC V / V

s

Q i(V s , V ) = ker /

© ViCV s

\v.+v=vs

Pw (Vj, 0) - Wp\ F

/

and Q^(Vg , V ) = 0,

otherwise.

if

s = m+ i

82

T H E D IS C R E T E SERIES OF G L

Proof . Let

f - (f^

6 ^

O V E R A F IN IT E F IE L D

n

D (Vp, f^a ^ e D(V^a ^) where a

parametrizes

v .1c v the set of i-dimensional linear subspaces of V.

Let

f(a )

(0---0, f(a ), 0---0)

Assum e that

be the element obtained from f by replacing

all the coordinates (except for the one on place a) V| fl V f-0, V,

where

V- = V^a ^. Write

V-

by zero.

Assum e that

as a direct sum U '© U v *pjy

y

1

(note that U ^ p .y = I t e U- |t (V -)- V-!.

Then we have

(cf. Proposition 3.12). Assume now that a ll

C V,

t < U - , hence / V

where

Let ? (V S)

V-

such that V- -t V = Vg.

Vi C

(f ^ ) a( j ( V s ) ’ f' ^

The group

In this case,

(f(a \ f 5 ( v y

D (V -) = D (Vg / V )

f D (V ^a b

) _0

w

for all

is the same as

acts on ? (V S) by permutations (it leaves

fC“ ) - o in D(VS/V).

Vg

invari-

lf and only if

“ ‘ From these remarks it follow s that

aeff(Vs )

M' sc/ [

©

for

e D OJs /' V) -

ant), and it is clear that

^

t f ^ = f ^

be the set of all i-dimensional linear subspaces

Hence a family

a family

Then clearly

t \ l ^ = lU y | T (a i Let now VS C V be such that

s = m 4 i. of Vs

V- = V^a ^.

D C V jA ® /

V iC V

I

J

yv.nvfco.v.

©

D (Vi+m/ V ) ® Q i(V i+m, V ) \

( Vi+m C V

F

)

\ y c v i4m

and M" : / ©

VViC^

D (V i )\ ® /

/

©

Vi+mCV

D (Vi+m/V)\ •

T H E B R A U E R L IF T IN G

Since the subspaces

Vi+m C V

83

such that V C

are in 1 -1 corre­

spondence with i-dimensional linear subspaces of V / V

the required

formula for M" follows. Next we observe that if Vj VjflVfO,^

1

1 1

Vs>

C O RO LLARY.

pw wf

( V , v .n v ) =

D(V-) fl

1

1

d (v . + v / v ) < s pw ( v ,v . n v ) 1 Wp F

and collect together terms corresponding

The required formula for M' follows.

Le t

be an algebraically closed field of characteristic

fl

-> Q

zero, and let ©

wF

We put Vg = V- + V,

to the same

such that

then

D(V-) = D(v-/v. n v) 0 (cf. 3.10).

is a linear subspace in V

Then the G L (V)-m odules

be a ring homomorphism.

are irreducible for all

1 < i < n except when we have

W

V.CV

F

1

simultaneously

q=2, n>2, i = l

in which case

0 D(\A) ® fl V l CV WF

has

two irreducible components ( one of which is the unit representation).

This is an easy consequence of the proposition.

5.3

We shall now prove that, given

V C V (0 < dim V = m < dim V = n)

there exists a natural exact sequence of WF -modules (12)

0 -> (D (V )X -» /

0

D(vn_ i j —

r ® v D(Vi ) )

VVn - l CV (the symbol

Mf has been defined in 5.2).

In fact, according to Proposition 5.2 it is sufficient to construct natural exact sequences (13)

0

pw (VS ,V) -* Q S_ 1 (VS ,V ) ->

> Q s _ m+1 (Vs .V) - Q s _ m(Vs ,V ) ^ 0

F

for all

Vg C V

such that V C Vg

tensoring with D(VS/ V )

and

m < s.

((12) follows from (13) by

and taking direct sums over all

Vg .) Now the

84

T H E D IS C R E T E SERIES OF G L n O V E R A F IN IT E F IE L D

natural exact sequence (13) is provided by 1.13(e) and our assertion follows.

It is obvious that the exact sequence (12) is compatible with the

action of G ^ .

(14)

It follows that

( —l ) n - 1 ( D ( V ) ) + ( —l ) n - 2 /

0

V

D (VV l ) \ + •■•+ ( - 1)7 - CV

n —1

©

D (V j)V = 0

Vvicv

)

/

in the Grothendieck group Rw ( G ~ ) . wp *F A V Let now

G be a finite group and assume that G

in the vector space

V.

We shall show that the correspondence

defines a group homomorphism br : R F (G ) -> Rw (G). ~

for any exact sequence G-action we have

is acting (linearly)

~

V -* br(V)

We have to show that

F

0 -> V -> V -> V / V -» 0 of F-vector spaces with

br(V ) = br(V ) + b r(V / V )

in Rw (G).

This follows from

F

(14)

and Proposition 5.2(ii). We can state the following

Th

e o r e m

.

Let

G

be a finite group and let F

be a F -vector space on which

G acts.

sequence of free Wp -modules with

be a finite field.

Let

V

Then there exists a canonical

G action

V^n^,V^n ^ , * * * , V ^ \ n = dimF V )

and a canonical exact sequence of G -modules.

0

->

® F -> \Kn - 1 ) ® F -> — > WF wp wF

® F ^ V -

0 .

Moreover the correspondence v - ( - l ) n _ 1 V (n) + ( - l )n - 2 V ( n _ 1 ) + - - - + ( - 1 ) ° V (1) defines a group homomorphism br : R p ( G ) -> RW^ ( G ) . identity map of R p ( G ) ,

where d : Rw (G )

R p (G )

We have

d © br =

is the decomposition

F

homomorphism induced by the canonical projection WF

5.4

F.

We shall now determine the character of b r(V ) on an arbitrary element

of G (where V is an F-vector space with a G-action).

It is clear that

T H E B R A U E R L IF T IN G

85

Trw (g|br(V)) - Tr(g|V) F

whenever

g e G

has order prime to p.

On the other hand we shall prove

that T t y p C g ’ M b r ( V ) ) - Trw (g ' |b r(V ))

whenever a power of

g', g" e G p and

the subgroup group.

G' of

are such that

g' has order prime to p,

g'- g =- g • g . We restrict rhe action of G

generated by

We can find a G'-invariant flag

spaces of V

such that g

is the

and

g

g";

V = V- + V- /V- 4 h x2 h

G

note that

V; C V- C •••CV; h x2 \

on V

l2

••• + V/V- in R^CG'). b

to

G' is a cyclic

of linear sub-

identity on V- ,V- /V- , ll

have

g" has order

. We

h

Ak

Since

br

is

agroup

homomorphism (cf. 5.3), we have br(V ) -

br(V: ) + br(V- /V- ) + • • • + br(V/V: ) H

l2

h

in Rw ( G ' )

xk

F

hence Trw ( g V ' l b r ( V ) ) = Trw Cg"g"" I br(Vi ) ) + T r w Cg"g""! br(Vi /V - )) F

F

1

F

+-

2

+ Trw ( g V I b r C V / V j ) ) F

= Trw (g'lbrCVj ))+ T rw (g | b r ^ F

H

1

F

2k

/V { 2

1

) ) + - + Tr(g | br(V/V- ))

- T r w (g '|b r(V )). F

This shows that the function

g -> Trw (g| br (V ))

is precisely the Brauer

F

lifting of the modular character of V

as a G-module (se e [5], [9]).

The fact that this is a virtual character of G was first pointed out by Green ([5], Theorem 1) and proved using Brauer’s characterization of characters in terms of elementary subgroups [ l l .

(This has been called

the Brauer lifting by Quillen [9], who showed how it can be applied to homotopy theory.)

5.5

In this section we show how Theorem 5.3 implies the following

k

86

T H E D IS C R E T E SERIES OF G L

(Swan [17], Theorem 3).

THEOREM

O VE R A F IN IT E F IE L D

n

G

Let

be a finite group and let F

Then the natural homomorphism Rw (G ) -* R~

be a finite field.

F

duced by the inclusion

Wp -* Q p

in-

isomorphism.

Note that Swan proves a more general result, where by ( A ,K ) A

(G )

yF

any semilocal Dedekind ring, and

K

(Wp , Q p )

is replaced

its field of fractions,

but this is not needed here. We define a map R n (G ) -* Rw (G )

Proof.

VF

Q p -vector space with a G-action. M e R0

in M and we send

vF

as follows.

F

(G )

Let

We choose a G-invariant Wp -lattice

to L e Rw (G).

to prove that given two G-invariant Wp -lattices we have

L = L'

in Rw (G).

L

We have to check that

F

this is well defined, i.e., independent of the choice of

pL C L ' C L,

M be a

L, L '

L.

It is sufficient

in M such that

Using the isomorphism

F

Rw (G ) -» R ^ F

(G )

(see 5.1) we see that it is enough to prove that

L = V

F

in R ^

(G).

We have an exact sequence

F

enough to prove that

L / L ' = 0 in R L

0 -> L ' -» L

(G).

Since

L / V -* 0. p L / L = 0,

It is then it follows

F

that

is an F-vector space with a G-action.

L /V

V ^n _ 1

can find a sequence

U sin g Theorem 5.3 we

V*-1^ of free Wp -modules with

G-action and an exact sequence of F-vector spaces compatible with the G-action:

0

® F -> V ^n _ 1 ^ ® F -> — > V ^1 ^ ® F -> L / V -* 0 . wp wp wp

In order to prove that L / V that

0 in

® F = 0 in R ^ (G), W

® F = Wp

R w (G ) Wp

it

is then enough toprove

i, 1 < i < n.

F

We have an exact sequence y (i)

for all

( G)

0 ->

= 0 in RL (G).

® F It follows that

0 hence

L = L'

in

WF

hence our map RQ (G ) -» Rw (G ) vF F

is w ell defined.

check that this is just the inverse of R W^ ( G ) -> R q f ( g ) is proved.

It is easy to

anc* the theorem

T H E B R A U E R L IF T IN G

1.

C O RO LLARY

Let

L, L ' e R w (G )

87

be such that

Tru; ( g | L ) = T r w (g|L'),

F

g e G.

for all

Then L - L '

F

F

in R w (G). F

C O RO LLARY

2.

The map br : R p ( G ) -> Rw (G )

is a ring homomorphism.

F

In fact, we must check that

br(V ® V ' ) --= br(V ) b r (V ')

in R w (G )

for any

F

F-vector spaces

V, V ' with G-action.

According to Corollary 1, it is

sufficient to check the equality of the corresponding traces, which is obvious.

5.6 Let

V

be a vector space of dimension

known that the Steinberg module

StF ( V )

The

F [G L

(V )]-m o d u le

F.

It is well

(s ee 1.13) is a projective inde­

composable module for the group algebra

THEOREM.

n > 1 over

F [G L (V )].

is projective and indecom-

V ® S t F (V) F r

posable. provided q 4- 2 .

Proof.

We can clearly assume that

n > 2.

The fact that

V StF ( V )

is

F

projective follows from the fact that StF ( V ) there exists a unique projective = V ® S t F (V ).

is projective.

WF [G L (V )]-m od u le

It follows that

M such that M ® F Wp

M has the following character: t 6 G L (V ) .

Trw (t|M) = Tr(t| V ) ■ Trw^(tjStw^ ( V ) ) , We have (15)M = D ( V ) © / ©

Vj p ) \W ® . -. .e©// DCV/Vp ® Stw ( V

\^ 1 CV as

Q F [G L(V )]-m odu les.

WF

1

)

V

0

D (V / V n_ j ) , CV

vv« -

® Stw (Vn. WF

"F

This can be checked by a character computation

(the character of the Steinberg module is known, se e [15]).

For example

the equality of the dimensions of the two sides of (15) is the identity

88

THE DIS CRE T E SERIES OF G L n OVER A FIN IT E F IE L D

n . q U 2 + ...+( n - l )

y

q l +2+ ■■-4 ( i - l ) ( q i+ 1 - 1 ) ( q i+2- l > • ( q n- l )

iTo Replacing q by q _1

n

V

and multiplying with q 1 + 2 " + ( n- 1) this becomes

( l ) n *'i—1 (qi f l - l ) ( q i l 2 - l ) --(qn- l )

i= o

(c' n“ 1- 1>

which is precisely the identity expressing the vanishing of the alternating sum of dimensions of the terms in the exact sequence 1.14(c). now that V ® St~ (V ) F

posable.

is not indecomposable.

Assume

Then M is not indecom-

*

Since the GL(V)-modules

®

D(V/V-) ® St(V;) (0 < i < n - 1 ) w

V .^ V

F

are irreducible when the scalars are extended to any field of characteristic zero

(q

2) we must have then the identity

(16)

1 + 2 , K+(nn -- 1l )K £V mq1+21....

q > - 2 ....... ( i - l ) ( q i+1- l ) ( q i t 2 - l ) - - ( q n- l )

i= 0 where

0 < m < n and cr(0 < i < n—1) are numbers equal to 0 or

such that a Q = 0. tive,

M1 f-0,M

(In fact, we can write M -

and such that

the dimension of IV^

D (V )

©M2 with

IV^

1,

and

projec­

occurs in M2 - It is w ell known that

must be of the form

m - q 1+2+

+ ^n 1^).

The

identity (16) is im possible since its two sides have distinct p-adic valu a­ tions.

The theorem is proved.

Remark.

The case

n = 2 of the theorem is due to V. Jeyakumar [7].

5.7 According to 4.10 the G L(V)-m odule

©

D(V^) considered in 5.2

v.cv can be defined naturally over the discrete valuation ring © F • C Kp • (1 < i < n).

T H E B R A U E R L IF T IN G

Let A ( F n)

KF

be the subfield of Q F

for all

Then 0 p -

n > 2.

We have

Kp = Jim^ Kp

lim^ 0 p n and © F

a unique prime ideal (= p ' f F ) D (V )

F , n 0 F ,n

bre

F

(V ) -

generated by

89

x, x e F * Let

and by

0 p = Kp fl Wp .

is a discrete valuation ring in Kp

with residue field

F.

Let

with

DCV)^

0 F . The formal alternating sum

( - 1 )11- 1 D ( V ) e

F

( - 1)° ©

n- 2 (-i):

VjCV

Vn - l CV can then be regarded as an element in R/c> ( G L ( V ) )

F

(cf. 5.1).

get a natural Brauer-lifting homomorphism R p ( G ) -> R q all finite groups

G.

F

(G )

We thus

defined for

It is easy to check that all the proofs given in the

case of the Witt ring Wp

remain valid when

Wp

is replaced by O p .

There is a natural commutative diagram br,c R F (G )

R«

(O

Rw

(G )

br

which shows that the map br^

5.8

Let

k be an integer.

F

is a refinement of br.

Consider the cla ss function X v ^ : G L ( V ) ^ K p

defined as follows: (17)

where

* v>k(t) = ( - l ) n+j Tr(tk|Van) ( q m- l ) ( q 2 m- l ) - - ( q ( j “ l ) m - l ) t