Topics in the Theory of Algebraic Groups

TOPICS IN THE THEORY OF ALGEBRAIC GROUPS

Notre Dame Mathematical Lectures. Number 10

Topics in the Theory of Algebraic Groups

James B. Carrell Charles W. Curtis James E. Humphreys Brian J. Parshall Boris Weisfeiler

University of Notre Dame Press Notre Dame London

Copyright© 1982 by University of Notre Dame Press Notre Dame, Indiana 46556

Library of Congress Cataloging in Publication Data

Main entry under title: Topics in the theory of algebraic groups. (Notre Dame mathematical lectures; 10) 1. Linear algebraic groups. 2. Algebraic varieties. I. Carrell, James B. II. Series. QA1.N87 no. 10 [QA564] 510s [512' .33] ISBN 0-268-01843-X Manufactured in the United States of America

82-17329

Preface In the fall semester of 1981/82 Professor A. Bialynicki-Birula from the University of Warsaw was guest professor at the mathematics department at the University of Notre Dame.

He gave a course entitled

"Algebraic group actions on varieties".

In conjunction

with his visit, Professors J. B. Carrell, C. W. Curtis, J. E. Humphreys, B. Parshall and B. Weisfeiler came for a week each and presented survey lectures on topics in the theory of algebraic groups.

In view of the superb

quality of both the course and the surveys, it was decided to make them available to a wider mathematical audience. This is the raison df§tre of this book.

Due to political

upheavals in Poland at the time, it was impossible for Professor Bialynicki-Birula to include a survey on the material of his lectures.

This is most unfortunate, of

course, and rather ironic for it was Professor Bialynicki-Birulafs course which served as catalyst for this volume. On behalf of our colleagues of the Mathematics Department of the University of Notre Dame, we wish to thank both the National Science Foundation and Professor 0. T. O'Meara, Provost of the university, for their

vi generous financial support.

Preface Professors Curtis, Humphreys,

Parshall and Weisfeiler were in addition supported by research grants of the National Science Foundation during the preparation of their lectures; Professor Carrell was similarly supported by the Natural Sciences and Engineering Research Council of Canada.

Finally, a personal word of

thanks to all the speakers for their visit and the cooperative way with which they prepared their manuscripts.

Alex Hahn Warren Wong

Notre Dame, Indiana July 1982

Contents

HOLOMORPHIC C* ACTIONS AND VECTOR FIELDS ON PROJECTIVE VARIETIES by J. B. CARRELL

1

1.

C* actions on protective varieties

3

2.

The B-B decomposition

7

3.

The Homology Basis Theorem

13

4.

A generalization of the Homology Basis Theorem

17

5.

Holomorphic vector fields and the cohomology ring

22

6.

Borelfs Theorem and holomorphic vector fields

25

7.

Holomorphic vector fields with one zero

29

8.

A remark on rationality

31

9.

Closing remarks

33

References

35

A DUALITY OPERATION IN THE CHARACTER RING OF A FINITE GROUP OF LIE TYPE by C. W. CURTIS 1.

Duality in the character ring of a finite Coxeter group

39 41

2. 3. 4.

Truncation and Duality in the character ring of a finite group of Lie type

48

Main theorems on duality

59 f

Applications to Springer s Theorem and a conjecture of MacDonald

64

Some additional results

68

References

71

ARITHMETIC GROUPS by J. E. HUMPHREYS

73

5-

1.

Lattices in Lie groups

73

2.

Finite generation and finite presentation

80

3-

Normal subgroups

86

4.

Normal subgroups (continued)

91

References

97

MODULAR REPRESENTATIONS OP ALGEBRAIC GROUPS by B. J. PARSHALL

101

1.

Elementary theory

102

2.

Induced modules

108

3.

Infinitesimal methods

113

4.

Rational cohomology

121

5.

Lusztig's conjecture

127

References

131

ABSTRACT HOMOMORPHISMS OF BIG SUBGROUPS OF ALGEBRAIC GROUPS by B. WEISFEILER

135

1.

Motivation

136

2.

Evident restrictions

140

3.

Isotropic semi-simple groups over fields

141

4.

Short remarks on OfMearaTs method

147

1

5.

Tits

results on homomorphisms of Lie groups

149

6.

Rigidity, strong rigidity, and superrigidity of lattices

152

7-

Quotient groups of lattices

163

8.

Lattices in PSO(n,l) and PSU(n,l)

167

9-

Concluding remarks

174

References

176

Holomorphic C* Actions and Vector Fields on Projective Varieties JAMES B. CARRELL

In this series of talks, I will discuss two ways of relating the topology of a smooth projective variety

X

(over ffi ) with the fixed point set of a one dimensional group of automorphisms (either X .

1 = 0., SL or I* = G m ) on These ideas are summarized in the following diagrams:

/-.x JPixed point set X I [of a £E* action on XJ

I Integral homology I (groups Hs(X,Z) J

Zeros of a holomorphic vector field (2) on X with isolated zeros

Complex cohomology ring H"(X,ffi)

If X

admits a (C* action with

X

finite and nontrivial,

then X

also has a holomorphic vector field with isolated

zeros.

The connection between the diagrams (1) and (2) is

not clear, however, and seems to be one of the basic open questions in this area (c.f. §2.5).

1

2

Holomorphic C* Actions and Vector Fields

This paper is divided into two parts, the first four chapters deal with

E

actions, and the next five with holomorphic

vector fields.

I have tried to keep the presentation on

a nontechnical level. have been included.

Several examples but very few proofs A few unsolved problems have also been

mentioned. I would like to thank the University of Notre Dame for support under the Kenna Lectureship Series.

J. B. Carrell

1. ffi* ACTIONS ON PROJECTIVE VARIETIES

A good place to begin a discussion of

£

actions is

with the fact that a holomorphic representation of ffi* on a finite dimensional complex vector space V , say

p : IE* •*•

GL(V) , induces a holomorphic action of ffi* on V , that is a holomorphic map

u : ffi x v -»• V

and p(x1X2,v) » u(X1,u(X23v)) . X-v

such that

y(l,v) = v

(We shall often write

for y(x,v) when speaking of affi*action.) The fact

that

p

is a linear representation means that each

X e ffi*

preserves lines through the origin in V , so u descends to give a holomorphic action of ffi* on 3P(V) ,

V :ffi*x jp(v) -*• I>(V) . A basic result about finite-dimensional representations of ffi* says that

V

decomposes uniquely into a direct

sum of weight spaces V, 9 kg 2, i.e. V = © Vk(k e S) , # where v € V. if and only if u(X5v) = Xkv for all X € ffi The k e S of the

1C

such that V, £ {0} are called the weights action on V .

By a holomorphic variety

1C

action on a complex protective

X , we mean a holomorphic map

n : OJ* x x •*• X

satisfying the properties mentioned above. It is well known that any holomorphic action of a one parameter subgroup

1C

on fl33Pn arises through

X : ID* •* 3PGL(n,E) , hence up to

4

Holomorphic C* Actions and Vector Fields

projective transformation a

I*

action on ffinPn is of the

form

x.cz 0 ,z 1 ,...,z n ] = cx a °z 0 ,x ai z 1 ,..,x an z n ]

(i.D where

aQ,a.,,...,a

c Z2 .

One frequently encounters the situation in which

X

is an invariant subvariety of a ffi3Pn with respect to ffi* action of the form (1.1) on map

01 x x •* X

defines a

1C

Example 1. The variety ffiUP3 with action Example 2.

Q23Pn . In this case the natural action on

X.

V(Zg5 + Z^Z*0 + Z-^zb

in

X-[Z^Z-^Z^Z-] - [X3Z0,X10Z13X5Z2,Z3] .

Grassmannians. Any ffi* action on IDn

permutes k-planes through the origin, hence defines a 03* action on the Grassmannian see that the image of Plucker imbedding is

Gk(ffin) .

Gk(ffin) in 03

IP(Affi )given by the representation X •* Akx . Notice that any ffi action on i.e. points

x

so that

IP(AkEn)

under the

invariant with respect to the ID*

n

action on

It is not hard to

X-x = x .

k

exterior power

CE3Pn

has fixed points,

Indeed the connected

components of the fixed point set are the linear subspaces of

Q3IPn

which correspond to eigenspaces of the induced

linear action on ffin .

Clearly any closed invariant

J. B. Carrell

subset

K of ffi]Pn has fixed points: namely if

x eK

then

lim X-x and lim X-x are both fixed points in K . X-K) X^« For convenience we set xn0 = lim X-x X-M)

and x

= lim X-x °° X->»

What is suggested by this construct is to consider the m*

connected components of the fixed point set X these are labelled X.

(suppose

X,,...,X ) and for each such component

its "plus and minus cells"

X. and X" , namely

X* - {x e X : XQ c X±} , and X~ - {x€ X : x^ e X±}

These "cells" turn out to be the fundamental objects that lead to connections between the topology of X and the ffl» topology of X . We will frequently refer to them simply as B-B

cells after A. Bialynicki-Birula who first proved

the main structure theorem fo'r them

CB-B] (which will be

discussed in §2). Example 3- Let ffi* act on ffilP2 by CZ0,XZ1,X2Z2] .

X-CZ0,Z-L,Z2] =

Clearly the fixed points are [1,0,0] ,

[0,1,0] , and [0,0,1] . Then, [1,0,0]+ = E3P2-V(Z0) , [0,1,0]"*" = V(ZQ) - {[0,0,1]} and [0,0,1]+ - [0,0,1] . In each case the plus cell is an affine space-

Holomorphic C* Actions and Vector Fields

Example 4. Consider the action on X = Gp(ffi ) induced by the action ii on 03 where vectors

X- (z0,z.,,z2,z.-) = (X zQ,X z-^X aQ > a.^^ > a2 > a^ .

z

p» x

z

o)

For a pair of independent

jh

u,v e ffi , let denote the 2-plane they

span. We will compute <e.,,e«>

9

where

denotes the standard basis of ffi4 . lim X-V for 2-planes of the form

{e. : 0 £ i £ 3}

It suffices to consider V = < auneun ^-cue., , 3neun + 1 1 - -

B-j_e.. + 32e2+ ^QeQ> where X-V

(an-a,)

,

(a, -a..)

Since

aun > axn > ad0 > JaQ , it follows that lim X-V = X-^0 = <e- L ,eo>. To give an invariant characterization of <e..,e~> , we recall the definition of Schubert cycles in G2(E ) . that

(See also CKL]).

If k-i* b 2

are

1

+

» 0 ( 2 , 4 ) - 0(1,4) - 0 ( 2 , 3 ) .

<e-,,eo>

T

= 0(2,4) .

Gp(E )

ID .

It follows that

The Schubert cycles

o(b-.,bp)

on

are ordered by inclusion via the lexicographic

order on the

(b.,,bp) .

By a similar argument,

This is shown in the diagram below. <e.,e.>

= 0(1+1, j+1)

if

0 0(2,3)

A useful feature of this diagram (sometimes called the Hasse diagram) is that one can see a free homology basis of any 0(1,3) rnamely itself and the Schubert cycles that precede it in the diagram. 2. THE

B-B

DECOMPOSITION

The structure theorem of Bialynicki-Birula [B-B] describes the structure of the plus and minus cells on X when

X

is a complete, smooth variety with

over an arbitrary algebraically closed field. Sommese CCS-,] and Fujiki

G

action Carrell and

[Pu] showed that this theorem goes

through without change to compact Kaehler manifolds .

g

Holomorphic C* Actions and Vector Fields

Recently, however, Sommese has found an example of a compact Moishezon manifold

X with a

E

action for which the B-B

decomposition X = UX. exists but not all of the canonical j maps X. •> X. , x •* XQ , are continuous CS2] • For a smooth projective variety X

X with fixed point components

X^,...,

the theorem says the following: THEOREM 1. (i) For each i = l,...,r , the natural

map

p± : X* ->• X± , x * XQ , is the projection of a holomorphic

fibre bundle whose fibres are all ffi* equivariantly isom. morphic -to a fixed £ d . (ii) 1C

In fact, if x € X, , then p^Cx) is

equivariantly isomorphic to

TX(X)/TX(XI) with ffi*

action induced by the representation

X H- dXx of C* in denotes the differential of the map

GL(T x (X)) .

y •* X • y

(dXxY at x.) (iii)

Zariski closure.

X^

is a Zariski open subset of its

Hence X.

a closed subvariety of X

(the topological closure) is containing

X. as a Zariski

open.

IP* X-j^ , of X"'

(iv) There exists a unique component, say + so that x£ is Zariski open in X . X][ is

called the source of X . A completely analogous result holds for the minus

J. B. Carrell

9

decomposition of X . that

X"

The distinguished component

is Zariski open in X

X. so

is called the sink of X .

We will always label the sink as Xr . COROLLARY. is rational.

Suppose either the source or sink of X

Then

X

is rational i.e. X

is birationally

n

equivalent toffi3P . For a proof see isolated source vector space

CCS.,3 •

x , then

X

*n "the case, say, of an is a compactification of the

N ({x}) .

An important, but easy to establish, fact is that if X

is a smooth invariant subvariety of JD3Pn , then there

exists a Morse function of increasing on the

f on X that has the property

3R

orbits in X .

In fact, let V

denote infinitesimal isometry associated to let f

n denote the Pubini-Study metric on by solving the equation

restricting

i(V)Q = dP

on

S

cffi*,and

D3IPn . One finds D2IPn

and then

P to X . Let O

"ET 7 ,...,£ 7 ~I — J?L4 J Un II

as long as coordinates

[Zg,...,Z ] have been chosen so that

X-[Z0,...,Zn] = Cxa°Z0,...,xanZn] .

The following are not

hard to verify using the contraction identity (i) f = P | X

Q

To I /Y/Ii7l ^Ij la..I 7\ £. J_ _L J.

i(V)fl = dP :

is a Morse function on X whose critical

10

Holomorphic C* Actions and Vector Fields

submanif olds are (ii)

f

X-, , . . . 3X ;

is strictly increasing on the

H+

orbits of

nonfixed points; (iii) then

if

X is not contained in a hyperplane of ffiIPn ,

X]_ = XnCsource of

DJ3Pn )

and

Xp = Xn(sink of ffi3Pn ) ;

and

(iv)

the Morse index of f on X±

is dim^N"^) ,

x e X., i where N~(X. x i ) denotes the subspace of T_(X) x generated by vectors of negative weight (it is actually a subspace of the normal space to X. at x ) . In the compact Kaehler casea assuming

XD3* 7* 0 , there

is a Morse function satisfying (i),(ii), and (iv) due to Prankel

CPr]

and Matsushima.

Its importance here is in

guaranteeing f that f there i is no * sequence of points lle in in X - X so that (x.) and (x) component for i=l,...,k-l and lie in the same component.

(XI)Q

anc

*

xI la...,x, K tne same (xir)« also

J. B. Carrell

11

Examples of such "quasi-cycles" are known in the non Kaehler case (see CJu] and

CS23) •

The Prankel-Matsushima a different manner in Example 5.

Morse function is applied in

[At] .

(G/B) .

Let

G

be a semi-simple

algebraic group,

B

torus in

W = NG(H)/CQ(H) be the Weyl group of

H

in

B

G .

and

a Borel subgroup,

G/B

a fixed maximal

It is well known (see e.g. [H]) that

a smooth projective variety and that on

H

by left translation:

(G/B)H = {gB : g e NQ(H)} Thus the correspondence

and

H

G/B

acts holomorphically

v(h,gB) = (hg)B . gB

g •> gB

is

Moreover

depends only on

g e W.

sets up a one to one IT

correspondence between

W

and the fixed point set

and we may unambiguously refer to subgroup

X : ID* •* H

under the action

G/B

y(t,gB) = (X(t)g)B

associated to

source of

X

is

A one-parameter

is called regular if

By a theorem of Konarski of

wB .

X

is

(G/B) ,

(G/B)ffi

= (G/B)H

of ffi* .

[Kon] , the plus decomposition B

invariant provided the

eB . This can be used to identify the

associated plus decomposition and the Bruhat decomposition. In fact, for each (1.2)

w e W,

(wB)+ = B(wB) (the B To see this note that

orbit of

BwB c (wB)

wB e G/B)

by Konarski!s

12

Holomorphic C* Actions and Vector Fields

result.

Since the plus cells

Bruhat cells

(wB)

are disjoint and the

B(wB) cover G/B , the proof of (1.2) is

complete. Another treatment of the Bruhat decomposition using 1C

actions appears in

[A-,] .

We now turn our attention to possibly singular projective varieties

X

invariantly imbedded in a ffilPn . For

example we can now consider actions on Schubert cycles and, more generally, on the generalized Schubert varieties which are closures of the plus cells.

Although the

X. B-B

decomposition is no longer always locally trivial, one can single out a natural class of actions (which always exist in Schubert varieties) on which the still nice enough.

B-B

To do so, suppose X

decomposition is is endowed with

an analytic Whitney stratification whose strata are ID* invariant.

(For example, the canonical Whitney stratification

of X

is always invariant

on

is called singularity preserving as

X

singularity preserving as x eA

implies

means that

XQ

XQ e A

CW]). The Whitney stratification (resp.

X •*• « ) if, for any stratum

(resp.

A,

x^ c A ) . Intuitively, this

is just as singular as x

Example 6. Let

X -*• 0

is.

Y

denote the cone in ffilP^ over a 2 smooth algebraic curve X c ffilP with vertex x - [0,0,0,1] € ffiE^ -ffilP2. The natural action of ffi*

J. B. Carrell

on Y

13

induced by the action

[ZQ,Z13Zp,XZ-] on ffiH?^ has source Y

can be stratified with strata

X

and sink

{x} and Y - {x} and

this renders the action singularity preserving as Since

{x} is an isolated singular point on

is not singularity preserving as stratification of Y . Y - {x} and

{x> .

X •* «

X •> 0 .

Y , the action

for any Whitney

Note that although the cells

{x} of the plus decomposition are locally

trivial affine space bundles, the minus cell

x~ = Y - X

is not .

The next theorem partially answers the question of what structure a singular invariant subvariety must have. The proof will appear in CCG] THEOREM 2.

If

X

is a

D3

invariant subvariety

ffiIPn whose ffi* action is singularity preserving as

X -> 0

with respect to some invariant Whitney stratification of X then for each connected component X. of X ffi* , 9

the natural projection

p. : X. •> X. renders xl" a j j J j topologically locally trivial affine space bundle. The fibres are biregularly (and equivariantly) isomorphic to m +) . some ffiJ1 (depending only on X. 3.

THE HOMOLOGY BASIS THEOREM

Recall that the classical Basissatz of Schubert

14

Holomorphic C* Actions and Vector Fields

calculus basis for

CKL] says that the Schubert cycles form a homology G, (ffin) .

To be precise, fix a flag ffi1 c ffi2 c

XV

... c JCn in

JDn .

Then for any increasing

(a,,...,a.) of integers so that

k-tuple

1 < a., < a2 < ... < a, £ n ,

set

(1.3)

0(a1,...,ak) = {V € Gk(En) : dimffi(V n ID*1) ^ 1}

The ft(a.,,...,ak) are projective varieties called Schubert cycles (or Schubert varieties) whose associated homology classes in H.(Gk(ffin),ZZ ) we denote by The Basissatz says: For each m with the of

[Q(a.j,... ,ak)H with

[&(a,,...,a. )] .

0 £ m £ k(n-k) ,

J. 1(a. - j) = m

form a basis

H2m(Gk(ffin),ZZ) . Even showing that

n(a.,,... ,ak) is a projective

variety is somewhat complicated (see e.g. [KL]).

However,

by a calculation similar to that in Example 4, there exists a ffi action on

Gk(ffin) so that

\ X. = fl(a.,,...,ak)

for some component

X. . Consequently, by the theorem of J Bialynicki-Birula, n(a-L,...,ak) is automatically a subvariety of Gk(fl3n) . A more interesting fact, however, is that there exists an analog of the Basissatz for any smooth (and many singular) — play # projective variety with ffi action in which the X.T j a role similar to the role played by the Schubert cycles

J. B. Carrell

15

(with respect to a fixed flag) in Xffi

is isolated, the

xt

Gk(ffin) .

In fact, if

form a homology basis of

For this reason, we sometimes refer to the X. as generalized Schubert varieties.

HB(X,ZZ)

(and X" )

Before stating this

generalization of the Basissatz, let us mention that using the Prankel-Matsushima Morse function in

CPr]

(see also CKob]) that

(i) bk(X) = IjtVx.^j5 «J (ii)

X

X,,...,X over

.

where

X

j=

has torsion if and only if

THEOREM 3 CCS2] . Let X variety with

f , Prankel showed

1C

dlm

]RNx(XJ)

XE

=

does.

be a smooth projective

action having fixed point components

Let m.

(resp. n.)

denote the fibre dimension

p. : X. •* X. (resp. q. : X" •* X. ) . Then there j j J J J J exist canonical plus and minus isomorphisms (1.4)

(C of

,k : •jHk.2llljUJ,B) -Hk(X,2Z)

and

(1.5)

vk : ®jHk_2n.(Xj'2;) *Hk(X'ZZ) J

By dualizing these isomorphisms to cohomology over ID and using the Hodge decomposition Hk(X,D3) = ® Hp(X,flq) p+q=k one obtains the following result. COROLLARY [CS23 .

The plus and minus Isomorphisms

16

Holomorphic C* Actions and Vector Fields

induce isomorphisms

p-m. q-ni. TT* : H p (X,fl q ) + § H J ( X j a f i °)

(1.6) and (1.7)

(Xj ,Qq-nJ )

y* : HP<X f a*> - 9^'^ By taking dimensions (over

IE ) we get

-.-

= I

J

J

which is a result obtained by several authors: independently by Luzstig and Wright

[Wr]

for isolated fixed points via

Morse theory and independently by Pujiki

[Pu]

and Iversen

using mixed Hodge structure. There are several consequences that relate the source and the sink to each other and to

X.

(a) H°(x,nq) s H°(x1,oq) = H°(X (b)

(c)

ir

there exist exact sequences 0 -> K* -> Pic(X) -^ Pic(X1) ^ 0 0 -> K" -> Pic(X) -> Pic(Xr) •> 0

J. B. Carrell

K+

where X

17

(resp.

K" )

generated by the

in

X?

is the

ZZ-module of divisors in

(resp.

X~ )

which are divisors

X. Another relationship between (d)

X and

X

Index(X) - Llndex(X.) J J

The proofs of (a) - (d) are contained in is also proved in 4.

is

CCSp] .

(d)

CPu] .

A GENERALIZATION OP THE HOMOLOGY BASIS THEOREM One can ask whether the homology basis theorem is

also true for singular invariant subvarieties inffi3Pn. The answer is, not surprisingly, no in general.

However,

for actions which we call "good", the answer is yes. Among the spaces with a good action are the generalized Schubert varieties

X.

in a smooth

of plus cells in

X

X

which are themselves unions

(i.e. there exist

!,,...,!.

so that

xt = xt u ... u X* ) due to the fact that the plus cells J _1 ^ in xt are xl" ,...,xt and the fact that, since X is J x H k smooth, the X. are locally trivial affine space bundles. 1 k The strategy for extending the (plus) homology basis theorem is to single out a class of actions with plus cells being locally trivial affine space bundles for which a plus homomorphism with natural properties can be defined.

The

18

Holomorphic C* Actions and Vector Fields

proof then uses the Thorn isomorphism.

It seems to us that

the class of good actions does not give the optimal generalization . X. of Xffi* , let r . denote the J J closure of the graph of p. : X. •> X. in X x x. 9 and let J J j j g. : r. •*• X. be the projection. 0 J J For any component

DEFINITION.

An action

ID* x x + X

if, for each connected component

is good as X -* 0 ffi* X. of X , the following j

conditions hold: (i) the projection

p. : X. •*• X. is a topologically J J J locally trivial affine space bundle, and (ii)

X. has an analytic Whitney stratification such

that for each stratum

A,

closure{(p.(x),x) e X. x x|x

where

A"1" = {x e X : X

e A+}

e A) .

The condition (ii) means one can unambiguously write r. H

for gT (I) c r. . It is easy to construct a space X J J with a point XQ in the source X., of X having the property that Y = E3P1 x 1E3P2

g" (XQ) 3 closure{ (XQ,X) :x e XQ} . Let with the action

X- (lzQ9z^1'9 Cwo,w]L,w2] ) =

,Xz..];[w,w5W]) , and let X be Y with the point

J. B. Carrell

19

([0, !];[!, 0,0]) blown up. Now take

XQ = ([1,0]; [1,0,0]) .

The reason for condition (ii) is to allow us to construct a wrong way map to define XQ

H k (X..,ZZ) -> HHk+2m (r..,ZZ) g :: H(X..,ZZ) (r..,ZZ). . If we try

g( cycle) = closure

"~ cycle) , then the point g"~(

in the above example will certainly cause a problem.

We must therefore be able to stratify

X.

so that the set

of bad points in each stratum is a subvariety of the stratum and then consider only cycles on transverse to the strata.

X.

that are

Thus a nice complex of transverse

cycles is obtained on X. that admits a wrong way chain J map into the chains of r. . In the example above we may " JC * stratify the components of X with one stratum each. Nice

0-cycles and 1-cycles in X, will avoid XQ . When a wrong way homomorphism g# exists, the plus homomorphism is defined as the composition (1.8)

Hk(Xj,ZZ)-^H

(r^ZZ)* H

(X,S)

J

u

where the latter map is induced by the projection

r. •*• X . j

We then have THEOREM 4 [CG]. If the action ffi* x X •* X as

is good

X -> 0 , then the plus isomorphisms (1.8) are valid for

all k .

Moreover, for almost every

the class of 1

Z

k

cycle

PT,( ) is represented by the

p" (z) on X .

z on X. , J k cycle

20

Holomorphic C* Actions and Vector Fields

Examples of actions that are good as (i) if X

is smooth, then any

union of plus cells in X (ii)

any

X

X •»• 0 :

X* in X j with the induced E

in which each

X. J

that is a action;

is smooth.

It is hoped that a more general setting in which the plus isomorphisms are valid will be found.

At the present,

all the examples we know of singular varieties with a plus isomorphism have a good action.

Hopefully, it will

eventually be shown that the plus isomorphisms are valid whenever the plus cells are locally trivial affine space bundles. Example 7. Let vertex

Y be, as in Example 6, the cone with

o p x e DOT - ID3P

over a smooth curve

2 X in ffi]P .

Then the action defined in Example 6 is good as not good as

X -*• « .

X -> 0

but

The plus isomorphism takes the form

H0({x» * HQ(Y) , H±(X) * H±+2(Y) , 0 s i • 0

We close this chapter with two questions. 1. In the case of a good action, how does the mixed

22

Holomorphic C* Actions and Vector Fields

Hodge structure on on

X

relate to the mixed Hodge structure

X1*?

2.

If

X

has a not necessarily good action with

isolated fixed points, do the odd homology groups of

X

vanish? 5-

HOLOMORPHIC VECTOR FIELDS AND THE COHOMOLOGY RING

It is a basic fact that the cohomology ring of a smooth protective variety field

V

X

with isolated zeroes

To be precise let sheaf

admitting a holomorphic vector

Z

is determined on

Z.

denote the variety with structure

0Z = flx/iCV)^1 where

contraction of holomorphic i(V)

Z ^ 0

i(V) : flp -> flP"1 denotes the p-forms to

(p-l)-forms.

Then

defines a complex of sheaves

o. which is locally free resolution of

0^

since

V

has

isolated zeros. It follows from general facts that there exists a spectral sequence with H°(X,0Z) .

E"p*q = Hq(X,Qp)

The key fact proved in

[CL-j,]

abutting to is that if

X

is compact Kaehler, then this spectral sequence degenerates at

E., as long as

finiteness of

Z

Z / 0 .

and

As a consequence of the

i(V) being a derivation, we have

J. B. Carrell

23

THEOREM 6 CCL23. If X

is a smooth projective variety

admitting a holomorphic vector field

V

with

Z = zero(V)

finite but nontrivial, then (i) Hp(X,flq) =0 2p

P

if p / q

P

H (X,E) - H (X,Q ) and H (ii)

(consequently

2p+1

(X,ffi) = 0 ) , and

there exists a filtration H°(X,0Z) - Pn = P^

where

n = dix X , such that F-jF.1

= ... =P1 c P

j_+-|

and

=F0.

having the

property that as graded rings

s

(2.D For example, if V

V2p(x-ffi)•

has only simple zeros, in other

words if Z is nonsingular, then

H (X,0Z) is precisely

the ring of complex valued functions on algebraically,

Z.

Thus,

H (X,0Z) can be quite simple. The difficulty

in analyzing the cohomology ring is in describing the filtration

P.

Example 9- For each holomorphic action of ffi* , one also has the infinitesimal generator, i.e. the holomorphic vector field

V

with respect to

obtained by differentiating the action X:

24

Holomorphic C* Actions and Vector Fields

Clearly, the fixed point set of ffi coincides with zero set of V .

One can easily show that the infinitesimal E3P11 in local affine

generator of the ffi* action (1.1) on coordinates

^ • Z-^ZQ,...,^ =zn/zo

at the flxed

point

[1,0,..., 0] is the holomorphic vector field (2.2)

V = IJ. 1 (a ± -a 0 )c 1

on Let us continue this example by exhibiting the filtration. The holomorphic vector field (2.2) on zeros if of

aQ < a.. < ... < a .

n

JC3Pn

has isolated

Also, the cohomology ring

n

DJIP , © H "l3I3P ,03), has the structure of a polynomial

ring, on one generator of degree two, truncated at degree 2n .

That generator is in fact the cohomology class of

the closed two form ft on singular and finite, H (ZjO™) o Z .

(^z\ *ffifor

each

Z

? £ Z

is nonso

is the ring of all complex valued functions on

We will let

value at

Q3IPn . Now since

(XQ,...,An) denote the function whose

[1,0,..., 0]

is

XQ

etc.

Then it can be shown

that P0 - < (!,...,!)> * H°(ffi]Pn, 03) Px and

=

(a0,...,an) is sent to ft under the isomorphism.

J. B. Carrell

25

(2.1) . In general,

For example, the linear independence of the for

0 < i £ n

(aQ,...,a J

follows from the van der Monde determinant

det

n (a. -a.)

a

n

1^ 4

Example 10. Vector fields on the Lie algebra of

H .

-1-

J

G/B .

Let

We call a vector

k

v e h

denote regular

if the set of fixed points of the one parameter group exp(tv)

of

H

acting on

H

exactly

(G/B) .

G/B

Set V =

H

zero(V) = (G/B) .

by left translation is

exp(tv) | txB() so that

Clearly, the zeros of

V

Z-

are all

simple. 6.

BOREL'S THEOREM AND HOLOMORPHIC VECTOR FIELDS

For of

W

w € W

on

k.

and W

v e k , w.v

will denote the action

thus acts effectively

h* in the Usual way: w-f(v) = f(w every character line bundle (gb,a(b

)z)

-v)

on for

k

and on

f e fe* .

To

a e X(H) , one associates the holomorphic

La = (G x QJ)/B on and where

a

G/B

where

(g,z)b =

has been extended to

B

by the

26

Holomorphic C* Actions and Vector Fields

usual convention. the

Now

da e h*

and since

da , for all a e X(H) , span

well defined linear map the condition

h .

for any a e X(H) .

also denote the algebra homomorphism

algebra of ft* . Iw)

W

3 , where

3 : R = Sym(ft*) -*

acts on R , so denote by

f(0) = 0 ) .

Let 3

Sym(fe*) is the symmetric

the ring of invariants of ¥

such that

Thus there is a

0 : /i* + H (G/B,E) determined by

3 (da) = ^(1^)

H*(G/B,JC) extending

G is semi-simple,

in R

I,, (resp.

(resp. f e I,,

Borel proved that 3 is a surjective

homomorphism whose kernel is R I,, . Consequently, since R I™

is a homogeneous ideal,

3 induces

an isomorphism of graded rings

3:

(2.3)

The purpose of this section is to show how Borel!s theorem relates to vector fields. Note that H (G/B,0«) ej = w C for any vector field on G/B generated by a regular vector in ft .

We will begin with a more detailed description

of

H°(G/B,0Z) .

Define a linear map ^v : h* + H°(G/B,(?Z)

by

¥y(w)(w) = -wa>(v) .

algebra homomorphism let

I¥

such that

Then

can be extended to an

¥y : R -»• H (G/B,0Z) .

denote the ideal in R f(v) » 0 .

¥y

The ring

For any v e h ,

generated by all R/Iy

$ e ITFW

is only graded when

Iv is homogeneous, i.e. only when v = 0 and R/IvW = R/RI,w+T . However R/IV is always filtered by degree. Namely, if

J. B. Carrell

27

p - 0,1,...,

set

(R/Iv)p = Rp/Rp n Iy

{f e R : deg f .

Notice that

where

Rp -

Iy c ker¥y ; for if

(j> e I,r and 4>(v) = 0 , then for all w e W , #__(*) (w) = w » (w#)(v) - (v) = 0 . In fact it is shown in EC] that for

v

in a dense open set in h9

(2.4)

¥

induces an isomorphism

?v :R/Iy + H°(G/B,0Z)

preserving the filtration, i.e. ?y((R/Iv) ) = F . Consequently, for each

p , the natural morphism

F-.p -* P

is onto. The first step in the proof is to identify elements in

H (G/B,0L7) that determine the Chern classes c., JL (L a ) for a e X(H) . To accomplish this we recall the theory of V-equivariant Chern classes. on

X

is called

V-equivariant if the derivation

lifts to a derivation map satisfying s e 0X(L) .

A holomorphic line bundle

V(f) = i(V)df , V

(i)

CCLp]

A

f e 0X ,

V e H°(X,0™) Z .

that

V e P.. and has image

c., (L)

under the isomorphism

(2.1) , and (ii)

Z * 9-

A

defines a global

section of End(0Xv(L) ®0 n 0^) s 0 ; i.e. Z X Z It is shown in

V : 0V •*• 0Y

V : 0X(L) + 0X(L) ; i.e. a ffi-linear

V(fs) = V(f)s + fV(s) if

Since

L

every line bundle on

X

is

V-equivariant if

28

Holomorphic C* Actions and Vector Fields

The calculation of

c

i(La)

is

provided by the following

lemma Lemma. of

V

Given

to

V

0(La ) so that in H°(0/8,0,, ii) , V a (wB) = where v € k is the regular vector corresponding

-da(w~ -v) to

a e X(H) there exists a lifting

V. In other words, ty (da)w = -(w-da)(v) = -da(w~ -v)

so, since the da

span

of the proof is outlined in appear in

^v(^ ) c P^ •

k ,

[C] .

The remainder

Complete details will

[A23 .

To prove Borel!s theorem (2.3), note that we have, for each regular

v c k , a commutative diagram

H2(G/B,E)

where

1

is an isomorphism, and

Consequently

3

is surjective.

a commutative diagram for each

¥

Moreover, this results in p > 1

-" F~/F.

(2.5)

is surjective.

J. B. Carrell

where

29

V

is surjeetive and

i

is an isomorphism.

3 : R •*• H*(G/B,C) is surjective. one must show that

ker 3 = R I™ .

To complete the proof, But because

dim R/R Iw =

dirnH0 (G/B,0Z) = |W| , it suffices to show that and this is surprisingly easy. then

^v(f) PO -

3(f) = 0, and Borel!s

theorem is proved. 7-

HOLOMORPHIC VECTOR FIELDS WITH ONE ZERO

So far we have considered only vector fields with simple isolated zeros, i.e. vector fields with the maximal number of zeros.

At the other extreme are vector fields

with exactly one zero. at

p € X

Suppose

and let V = £a.3/3z.

coordinates near

p.

so the cohomology ring

V

has exactly one zero in holomorphic local

Then

H (XjO,,) o L = l[zn±,... ,zn I/, \a^,...,a n H*(X,ffi) is the graded ring

associated to a certain filtration of lECz.,,... ,z ]/(a.,,... ,a ) Let's consider a basic example. Example 11. Let V on ffi]Pn generated by matrix

be the holomorphic vector field exp(tM)

where

M

is the

(n+l)x(n+l)

30

Holomorphic C* Actions and Vector Fields

The unique zero of coordinates

hence

V

is

C-, *•••,£„

[1,0,...,0] , and in the affine

at

[1,0,...,0],

H°(03]Pn, 0^) = ffiU-^/U^4"1) .

cohomology ring of ffi3Pn .

This is already the

In, fact using the theory of

equivariant Chern classes, it is shown in corresponds to

c.,(0(l))

under the isomorphism of Theorem 6 H (JDIPn, 0Z)

The existence of the grading on the fact that the

D3

CCLo] that c-j

action

follows from

A- [ZQ,Z, , . . . ,Z ] =

[Z0,AZ1,. . .,AnZn] on ffi3Pn has the property

dA-V - A-1V

which implies that the functions that define the ideal i(V)fl

are homogeneous (with respect to the action) and

hence that

H (Q3]Pn, 0Z)

is graded.

In general we know

the following THEOREM 7 CACLS] .

Let

X

be a protective manifold

having a holomorphic vector field with only isolated zeros but having zeros.

Suppose there exists a

01

(A,x) •*• A-x

X

for some integer

k 7* 0 .

Then

on

so that

H°(X,0Z)

d A - V = \f

action

is a graded ring in which the

filtration by degree coincides with the filtration of Theorem 6.

Consequently,

cohomology ring of

X

H (X,(?z)

and

P.

H*(X,Q3) , the

with complex coefficients, are

]. B. Carrell

31

isomorphic graded rings. Applications of this theorem to the algebraic homogeneous spaces

G/P

will appear in a later paper.

In the

case a regular unipotent one-parameter subgroup of

G/P G

will

generate a holomorphic vector field with exactly one zero and this subgroup will imbed in an Jacobson-Morosov Lemma SL(2,ffi) Thus

provides the

H" (G/P,02)

CJa] .

SL(2,JD) c G

by the

The maximal torus in this

02* action of Theorem 7 where

can be viewed as an analytic ring.

k = 2. Its

relations will be reflected in the structure of an infinitesimal neighborhood of the zero.

It would be interesting

to know if the generalized Schubert cycles on

G/B , i.e.

the closures of the Bruhat cells, admit an intrinsic characterization in the ring duals of these classes in explicitly in

CBGG] .

H (G/B,0Z) .

The Poincare

H* (G/B,ID) are calculated

We will return to this question in

§9. 8.

A REMARK ON RATIONALITY

The condition of Theorem 7 that vector field

V

some integer

k / 0

X-^CtJ-X"

1

k

= 4>(X t)

$ : E •* Aut(X) Aut(X)

and a

X

admit a holomorphic

02* action so that

dX-V = XV

for

is equivalent to requiring "that for all

X e 02* , t e 0] , where

is the one parameter subgroup (of the group

of automorphisms of

X ) generated by

V .

When

32

Holomorphic C* Actions and Vector Fields

the identity component

AutQ(X) is semi-simple and

a unipotent one parameter subgroup, i.e. a GdQ [H] , the Jacobson-Morosov Lemma existence of an

<J> is

action

CJa] guarantees the

SL(2,ffi) c AutQ(X) in which

$(03) =

(fo l]:t £ ^l ' If (X*x) "* X"x denotes the E* action on X induced by the maximal torus in SL(2,ffi) , then X"1 = $U2t) .

Using this fact, it is possible to

prove a result of Deligne THEOREM 8 Suppose X such that

Aut(X)

CD] . is a smooth projective variety

is semi-simple.

Suppose that there

exists a holomorphic vector field on X

generated by a

Ga. Q

action whose fixed point set is rational (as a projective subvariety of X ) . Then Outline of proof.

X

is rational.

By a theorem of Sommese

if AutQ(X) is semi-simple, then any

E

CS.,] ,

c AutQ(X) has

fixed points on X .

It follows, by Blanchard's theorem

CM, p. 25] , that

can be imbedded in some ffilP so

X

that each g e SL(2,Q3) c Aut(X) transformation.

is induced by a projective

By Theorem 7.1 of

CCS«] , V

is tangent

to the fibres of the plus cells in X , hence the sink of X that X

is contained in zero(V) . X

N

Therefore, assuming

is not contained in any hyperplane of

= X n L = zero(V) n L

X

HIP ,

for some linear subspace L of

ffi]P . It follows that the sink

X

of X

is rational,

J.B.Carrell

so

X

33

is rational, by the corollary to Theorem 1.

The question of whether the existence of a holomorphic vector field on X having isolated zeros implies X rational has been considered by Lieberman in and by Deligne

CD] .

is

[L-.ljCLp] ,

By the induction argument in [Lp]

one can reduce this problem to showing the Conjecture; A smooth protective variety that admits a holomorphic vector field with exactly one zero is rational. 9-

Borel's Theorem

CLOSING REMARKS

R/RIy

H

H*(G/B,ffi) has another

interpretation due to Kostant can be seen to be the ring (nonreduced) variety cone in

Q.

[Kos] .

Namely,

R/RI,,

EC firm] of functions on the

h n n , where

n

is the nilpotent

A problem of Kostant is to understand in an

intrinsic manner how Schubert calculus works in

EC firm] •

The isomorphism of Theorem 7 may shed some light on this problem since we now have available the fact that

H"(G/B,ffi)

is isomorphic to H (G/B,0Z) for the vector field associated to any regular element in

n•

In the same spirit as Kostant,

one may ask

Question.

Suppose V

is a holomorphic vector field

with one zero having an -associated

E*

action so that

34

dA-V = XCk

Holomorphic C* Actions and Vector Fields

£ 0). Find intrinsically the elements in

H^CX.O™) associated to the *

xt by Poincare* duality. J

J. B. Carrell

[A..]

35

REFERENCES ^ E. Akyildiz, Bruhat decomposition via G action, Bull. Ac ad. Pol. Sci. . Ser. Sci., Math., 28(1980), 551^547. _ , Vector fields and cohomology of G/P , to appear.

CACLS] E. Akyildiz, J.E. Carrell, D.I. Lieberman, and A.J. Sommese, On the graded rings associated to holomorphic vector fields with exactly one zero, to appear in Proc. Symp. in Pure Math. [At]

M. Atiyah, Convexity and commuting Hamiltonians , Bull London Math. Soc.3 14 (1982), 1-15-

CBGG3

I.N. Bernstein, I.M. Gelfand, and S.I. Gelfand, Schubert cells and the cohomology of the spaces G/P , Russ. Math Survs . , 2JJ (1975), 1-26.

CB-B]

A. Bialynicki-Birula, Some theorems on actions of algebraic groups, Ann, of Math. 9J3 (1973), 480-497-

CC]

J.B. Carrell, Vector fields and the cohomology of G/B , Manifolds and Lie groups , Papers in honor of Y. Matsushima, Progress in Mathematicsa Vol. 14 Birkhauser, Boston7 1981.

CCG]

J.B. Carrell and R.M. Goresky, Homology of invariant subvarieties of compact Kaehler manifolds with ID* action, to appear.

CCL.,]

J.B. Carrell, and D.I. Lieberman, Holomorphic vector fields and compact Kaehler manifolds, Invent . Math. 21 (1973), 303-309-

_ and _ _ , Vector fields and Chern numbers, Math. Annalen, 225 (1977), 263-273-

_ , Vector fields, Chern classes and cohomology, Proc. Symp. Pure Math . 3 Vol. 30 (1977), 251-2PT [CS-]

J.B. Carrell and A.J. Sommese. Scand. Jj-3 (1978), 49-59-

E* actions. Math.

36

Holomorphic C* Actions and Vector Fields

_ Some topological aspects of W actions on compact Kaehler manifolds. Comment . Math. Helvetici 51 (1979), 567-582. CCS.,]

_ SL(2,E) actions on compact Kaehler manifolds, to appear in Trans . Amer. Math. Soc.

CD]

P. Deligne, Letter to D.I. Lieberman

CPr]

T.T. Frankel, Fixed points and torsion on Kaehler manifolds, Ann, of Math. 70. (1959), 1-8.

CFu]

A. Fujiki, Fixed points of the actions on compact Kaehler manifolds, Publ. R.I..M.S.., Kyoto University 15 (1979), 797-826.

CH]

J.E. Humphreys, Linear algebraic groups , SpringerVerlag, Berlin - New York (1975) -

CJa]

N. Jacobson, Lie algebrasa John Wiley and Sons, New York - London (1962).

CJu]

J. Jurkiewicz, An example of algebraic torus action which determines the nonfilterable decomposition, Bull. Ac ad. Polon. Sci. , Ser. Sci. Math. Astronom. Phys"- 2TT19 76) 667-674.

CKL]

S. Kleiman and D. Laksov, Schubert Calculus, Am. Math Monthly 79 (1972), 1061-1082. ~

CKob]

S. Kobayashi, Trans formation groups in differential geometry, Springer- Ver lag, New York Tl972) .

CKon]

J. Konarski, Properties of projective orbits of actions of affine algebraic groups, to appear.

CKos]

B. Kostant, Lie group representations on polynomial rings, Amer. J. of Math. 8£, 327-404.

CL^]

D.I. Lieberman, Rationality and holomorphic vector fields, to. appear. _ Holomorphic vector fields on projective varieties, Proc. Symp. in Pure Math 30 (1977), 273-276.

J

J. B. Carrell

37

CM]

Y. Matsushima, Holomorphic vector fields on compact Kaehler manifolds . Regional Conference Series In Math. No. 7, A. M.S. (1971) -

CMS]

J. Milnor and J. Stasheff, Characteristic Classes, Annals of Math study no. 76. Princeton University Press, Princeton N.J. (1974).

CS.,]

A.J. Sommese, Extension theorems for reductive group actions on compact Kaehler manifolds. Math. Ann. 218 (1975), 107-116. , Some examples of

E*

actions,

to appear. CW]

H. Whitney, Tangents to an analytic variety, Ann. of Math. 81 (1965), 496-549.

CWr]

E. Wright, Killing vector fields and harmonic forms, Trans . Amer. Math. Soc. 199 (1974), 199-202.

A Duality Operation in the Character Ring of a Finite Group of Lie Type CHARLES W. CURTIS

INTRODUCTION

The subject of these lectures, while perhaps not a major theme in the representation theory of finite groups of Lie type, nevertheless cuts across the representation theory of these groups in interesting and sometimes unexpected ways.

The main results reported on here are

due to Alvis (El], C2]), following an earlier paper C73 by the author.

Some of Alvis's main results were obtained

independently by Kawanaka, and a homological interpretation of the operation has been given by Deligne and Lusztig CIO] In order to describe the contents of this paper, we first require some terminology. group, and let

ch(IDH)

denote the ring of complex valued

virtual characters of H. the

Let H be a finite

The elements of

ch(EH) are

2Z-linear combinations of the elements of Irr H,

the set of irreducible characters afforded by the simple 39

4-0

Characters of Groups of Lie Type

left

EH-modules.

The operations of addition and multi-

plication of characters correspond to the operations of forming direct sums and tensor products of the corresponding a

EH-modules.

A duality operation in

7L -automorphism of

Ch(ffiH)

of order

2,

ch(EH)

is

which pre-

serves the inner product (f,g)HH - iHl"1 I f(x)IO£T xcH of class functions on

H.

Such an operation clearly per-

mutes, up to sign, the elements of

Irr H.

example of a duality operation is the map £ € ch(EH), where

£

A familiar £ -*• "£,

is the complex conjugate of

£.

This map corresponds to the operation of forming the contragredient module of a given

EH-module.

Another

example is given in § 1, for finite Coxeter groups, and consists of multiplying a character by the sign character.

The duality operation described in § 2, for virtual

characters of a finite group permutation of

Irr G

G

of Lie type, defines a

(up to sign), with corresponding

characters not necessarily having the same degree. degree of

£ € Irr G

j£* 6 Irr G,

and its dual

always have the same

differ only by a power of

p,

c* e ch(EG),

The

with

f

p -part, and hence

where

p

is the character-

istic of the finite field associated with

G.

In § § 1-4, we have given a self-contained exposition,

C. W. Curtis

41

often with complete proofs, of the main results.

In § 5,

we survey without proof some other results, with references to the literature. It is a pleasure to acknowledge the hospitality and interest of the Notre Dame Mathematics Department during the time these lectures were presented. 1.

DUALITY IN THE CHARACTER RING OP A FINITE COXETER GROUP.

Let

(W,R) be a finite Coxeter group with dis-

tinguished generators

R = {r-,...,r },

with the pre-

sentation W = .

e : R •* ID* defined by

preserves the defining relations of

eCr.^) = -1, 1 £ i < n, W,

and therefore

can be extended to a homomorphism

e : W -* E* which we shall call the sign representation of example, the symmetric group with generators

defining relations ^riri+l'

= (n,n+l)

(r±r.) - 1 if

=» 1> 1 ^ i ^ n.

For

S +., is a Coxeter group

r1= (12),...,r 2

W.

and

|i - j| > 1, and

In this case

e(cr),

for

a € S , is the usual signature of a permutation, and is 1

if

a

is even, -1

if

a

is odd.

42

Characters of Groups of Lie Type

It is easily checked that the map li € ch(IDW),

is a duality operation in

y -»• ey, ch(EW)3

for and it has

traditionally been used to organize the character tables of the Coxeter groups. Our first aim is to give a geometric interpretation of

e.

It is a standard result (see C33) that

W

can

be identified with a finite group generated by reflections on a real Euclidean space n = |R|.

Each reflection

pointwise fixed, and if

r eW a

E

of dimension leaves a hyperplane

H

is a vector orthogonal to

H,

then

where on

E.

(

, ) is the positive definite scalar product

The vectors

{+a}

orthogonal to the hyperplanes

fixed by the reflections in

W,

with their lengths

suitably normalized, are permuted by the elements of and are called the root system There exists a set properties that

II of roots

{(^,...,0 }

$ associated with {ou, . . . ,an>,

W,

W.

with the

form a basis of E over

K , and every root a can be expressed in the form n a = Z c. a. 3 where the coefficients {c.} are either 1 1-1x x all non-negative or all non-positive. Such a set of roots is called a fundamental system, and there exists a fundamental system II such that the distinguished

43

C. W. Curtis generators

tr.K

reflections

{r

< i =

which are themselves Coxeter groups, and

u

are called parabolic subgroups of

W.

In this set-up, the sign representation is defined by

e(w) = det w,

vector space for

E.

the determinant of

w 6W

We are interested in one more formula

e, which arises from the action of

unit sphere

S

on the

in the Euclidean space

define a natural triangulation of of the root system.

S

The action of

W

E.

on the We shall

defined in terms

W

on the simplicial

complex defining the triangulation yields a representation of W (1.1) group with

on the rational homology

H#(S),

and we have:

PROPOSITION.

Let

(W,R)

be a finite Coxeter

n = |R| > 2.

Let

r

be the abstract

simplicial complex whose simplices are the cosets {wWj : w e W,J c R} W,

of all proper parabolic subgroups of

with order relation (defining the faces of a simplex)

given by the opposite of inclusion. (i)

the geometric realization

homeomorphic to the unit sphere (ii)

Then:

S

the rational homology

in H#(D

|r|

of

r

is

E. =S = : H ^

is

44

Characters of Groups of Lie Type

given by: H^CD • 0 except in dimensions and

0

and n-1

HQ(r) = Hnn(r) s $ as rational vector spaces. (iii)

the rational homology group

the trivial representation of W, group

H r n_i( )

HQ(D

affords

and the homology

affords the sign representation

We shall give a sketch of the proof.

e.

For, more

details see Carter C4] or Bourbaki C33- Let II = {a.,...3a } be the fundamental system of roots such that the reflections and let C

}-

.

C = {£ € E : (£,0^) > 0

coincide with for all

R,

o^ e n}.

Then

is an open simplicial cone called the fundamental

chamber of subsets J

{r

W

acting on

Cj c (J, where

E.

€E:

C

(J ig" the closure of

ranges through subsets of

U

The walls* of

n,

are the C,

and

defined by

(5,a ± ) - 0, a± e J , ( C 5 a ± ) > 0, a± e E - J

It can be proved that for each subset stabilizer of of

Cj

in

J ,

the Levi decomposition is given by B - UH, with

V, - U, L. - H,

and is part of the definition of split The parabolic subgroups of bolic subgroups ments of

G.

{PTd}., 05 d CJTl

G

(B,N)-pair.

are the standard para-

and their conjugates by ele-

In the case of

GL (IP ) , they are the

stabilizers of all flags in the underlying vector space on which

GLn(3P )

acts, where a flag is a chain of

subspaces W--L c W0tL c ... c w S , s * 1.

Thus the Borel subgroups are conjugates of the stabilizers of complete flags (with

dim W. = i, 1 <s i £ s

and

s = n - 1).

The definition of the duality operation involves restriction and induction from parabolic subgroups

PT, d

J £ R. The usual definitions, however, have to be modified to take into account the unipotent radicals

VTd,

that we really are setting up relationships between characters of

G

and characters of the Levi subgroups

so

C. W. Curtis

Lj

of

53

Pj,

for

J £ R.

The process can be described in

a general way as follows. subgroup of

G,

V < E z G. £.

and

Let

Let

V a normal subgroup of

X be a left

Then the restriction

characters of

where

G be a finite, H,

£[„ = resH£

is a sum of two-

H,

H

having

V

£|H

from irreducible

in their kernels, and C" n

is the contribution from those that do not. £' £1

can be viewed as a character of

hand, let

trivial representation of

since

V

V 5 H,

V;

then

X

affording the

inv^(X)

and, in fact, is a

afforded by

for each character

of

H

natural map

with

X V

is a flJH-

E( H/V) -module

acts trivially on inv,,(X) . Let

the character of H/V

acter of

Note that

H/V. On the other

inv,_(X) be the sub space of

module since

so that

EG-module with character

£,1 is the contribution to

characters of

H a

£(H/V)

be

invv(X) . Finally,

H/V, let

X

denote the char-

in its kernel defined by the

H •+ H/V.

These ideas are connected by the

following result, whose proof is left as an exercise. (2.1)

PROPOSITION.

Keeping the above notations,

the following characters of H/V

coincide :

?' = CrH/vx - (5L)/w/vx E U,XG)X. H (H/V) »H (H/V) X£lrr(H/V)

54

Characters of Groups of Lie Type

An easy lemma from the Felt-Thompson paper (Cll3, p. 783) implies that for an element C Q (x) n V = {!},

x c E such that

we have

which is a kind of reduction theorem for character values on certain elements in terms of characters of the smaller group

H/7. From this point on, we shall often be following the

approach of Alvis [13 (see also Curtis C71). (2.2)

DEFINITION.

Let

type, with Coxeter system operation of truncation homomorphism of character

G be a finite group of Lie

(W,R),

and let J £ R. The

Tj : ch(EG) •*• ch(IDLj) is a

E- modules9 which assigns to each virtual

\i e ch(EG)

the virtual character

TTuu € ch(ELjT), where TJTuU) = |V J"1 J The fact that of

Lj,

TjU

Z y(v*), ^ €JLT. is in fact a virtual character

and connections with Proposition 2.1, are given

by:

(2.3)

PROPOSITION.

ter afforded by some

" H(Q)

with Zariski dense image.

After composing with the inclusion into

H(Q ),

rigidity Theorem forces the image of

to be relatively

T

compact in the Q -topology, for each prime p.

the Super-

This means

that the powers of p in denominators of matrix entries in a(T)

are bounded.

But

T

is f.g., so the denominators in

question can involve only finitely many primes. these statements,

a(T) n H(Z)

Combining

has finite index in

This is a major step toward proving that

T

a(T).

is arithmetic.

We should mention a further striking consequence of Margulis1 methods:

With G and

T

as above, each noncentral

80

Arithmetic Groups

normal subgroup of

T

is of finite index.

Earlier results

of this type mostly depended on having a positive solution to the congruence subgroup problem.

2.

FINITE GENERATION AND FINITE PRESENTATION

Given an arithmetic group whether

T

F,

it is natural to ask

is finitely generated (f.g.) and, if so, whether

it is in fact finitely presented (f.p.).

These questions

can sometimes be answered positively by exhibiting generators and relations; but in other cases only a qualitative or indirect proof is available.

And in a few situations,

negative answers turn up. (2.1)

Consider a very classical example: the group

T =

A

PSL2(2), VIII].

or its close relative

Let S=

£°

with respective images

r = SL2(Z), cf. [19, Ch.

J), T- (5-^ s,t,u

in

= (J j) m

T.

Note that

?,

T = SU.

A

From linear algebra one knows that U (or equivalently, by S and T). elements T

s, t

T So

is generated by S and T

is generated by the

of respective finite orders 2,3.

In fact,

is the free product of the cyclic groups they generate. A

To see this, it is easier to work in T. It has to be shown a e e b l n that A = + T ST S...T S can never reduce to + I (where a,b € {0,1} a

= b = 0,

and so

e. e {1,2} ). By rearranging, we may assume e e i n A = j ^ S T . . . S T . Now it is enough to show

J. E. Humphreys that no nontrivial word in the semigroup generated by \0

3J and

for each while

ST

* (l l) reduces to + I.

Z = fe °\ in this semigroup,

(ST2)Z

= /*

Z have like sign.

\ .

But note that (ST)Z = - fa*c b*~)

By induction, all entries of

It follows that if Z has a nonzero entry

off the diagonal (which ST and ST true for these longer words. (2.2)

ST =

both do) , the same is

So we can never reach + I.

Nielsen found a finite presentation for

which Magnus later reduced the case of

SL-(Z),

SL (20

for

In a modern guise, this fits into the computation of by Silvester-Milnor: SL (Z) (n £ 3) elementary matrices

to

n £ 3. KpZ

is generated by the

E..(i * j ) , where

E.. has 1 in the

(i,j) position and on the diagonal, but 0 elsewhere, subject only to the relations:

(EjM»Ek£) = 1 if j * k, i^£;(E1 .,E.k)

if i,j,k are distinct; (Ei2E2l"lE12^=

= E±k

two relations alone define a central extension SLn(Z),

with kernel

Iu

The

flrst

St (Z) •>

K2Z « Z/2Z. (The covering group is

called the Steinberg group.) Other Chevalley groups GKZ) such as

Sp2n(Z) were sub-

sequently studied by Klingen, Wardlaw, Behr, HurrelbrinkRehmann, culminating in the explicit presentations of Behr [6].

He views G(Z) as an amalgamated product of rank 2

parabolic subgroups, via the action of G(Z) on a simplicial complex introduced by Soule*.

Then the rank 2 cases

Spit(Z), and G2(Z) can be plugged in.

S

82

Arithmetic Groups

Relatively few groups over other arithmetic rings have been treated as explicitly as these groups over Z: mainly SLp

over rings of integers of imaginary quadratic

extensions of C(G) -> T

__

TA, TA __

+ T. +!•

J. £. Humphreys

39

The question then becomes:

Is_ C(G) trivial (and, If not,

how big is it)? In case G is simple and simply connected, G and F.

have straightforward descriptions, due to strong

approximation; e.g., FA

is just the product of the groups

G[(A ) taken over the integers A of all completions

(3.3) The case

(* = SLp

can serve as a microcosm of the

congruence subgroup problem for simple algebraic groups. In this and the following lecture I want to sketch some of the key points in Serre [25]. results: When

Consider first the "negative"

|S| = 1, C(G) is_ infinite.

This involves

three separate cases: (1) The "rational" case A = *& (cf. (3-D above). (2) The "imaginary quadratic" case, e.g., A = 2[i]. (3) The "characteristic p" case, e.g., A = FqCt] ( H-L(Xr), where

and the orbit space (3-5)

a

X/r

with a map of

X = SL2(C)/SU2(G)

has a compactification

Xp.

It still has to be deduced that C(G) is infinite.

Cases (!) and (2) can be argued together:

Let

C«

be the

A

intersection of C(G) with the closure of C(G)

were finite,

of Bass-Milnor-Serre

Cp would be also.

T

in G.

If

Then the arguments

[3,§l6], which depend just on the

finiteness of the congruence kernel, would imply the finiteness of

H^I^Z) = Hom(rab,2),

contrary to (3-4).

J. £. Humphreys

91

In [3] it is essential that the characteristic be 0, e.g., to get the splitting of short exact sequences of finite dimensional Kr -modules, or to apply results of Lazard on p-adic groups. For case(3), one can argue that

\T\ = c,

since the

congruence topology has a countable basis of neighborhoods of 1.

On the other hand, the result of (3-4) on A

implies that

Ta

A

T

maps onto the second dual

V

of an infiA

nite dimensional vector space V over F ; here |C(G)| ;> 2C.

forcing

|V| =2

,

(Alternatively, Serre [26,11,2.?]

uses the action on the Bruhat-Tits tree to show more directly that the set of S-arithmetic subgroups of

T

has

cardinality c.)

4. (4.1) When

NORMAL SUBGROUPS (CONTINUED)

Retain the notation of (3-2): G, K, S, A, TA, G. etc. G = SLp

and

|S| = 1,

the congruence kernel C(G) is

infinite and the congruence subgroup problem has therefore a strongly negative solution.

Serrefs proof involves a

close study of the group structure of

T., G,

and of the way

TA (or its subgroup T) acts on a related topological space, but requires no delicate arithmetic information. |S| £ 2,

When

deeper arithmetic considerations enter into the

solution, which is positive or "almost" positive:

92

Arithmetic Groups

THEOREM (Serre). C(G) = y

Let

G - SL2, |S| :> 2.

Then

(the finite group of roots of unity in K)

if K

is a totally imaginary number field and S the set of all archjmedean valuations.

Otherwise

The simplest case occurs when

C(G) = 1. K = Q, S = {p,»}. (This

had been studied earlier by Ihara and Mennicke.) The assumption

|S| £ 2 crucially affects the

structure of the group U of units of A, which has the form y x an ' . In particular, U now has elements of infinite order. The proof of Serrefs theorem involves showing that

(4.2)

C(G) lies in the center of G.

Here an essential role is

played by an auxiliary family of normal subgroups of for a nonzero ideal q of A, E

I\:

is the normal subgroup gen-

erated by "q-elementary11 matrices in

T . (It is unclear

at first whether E has finite index or not.)

Now the

proof goes in steps. (1) Any subgroup N of finite index in

r.

includes

some E . (We may assume N is normal, of index n, so q = nA

will do if char K = 0.

In characteristic p, the

choice of q is a bit more complicated and uses the fact /u 0 \ that the set of u € U with \ e N has finite index JnN.)

V° M

(2) A non-central subgroup H of G normalized by an S-arithmetic subgroup N includes E

for some q.

(H must

f. £. Humphreys

93

contain some matrix

J with

ac * 0 ,

otherwise its

Zariski closure in SL^ would be a proper, normal, but noncentral subgroup.

Now (1) yields an

E , c N,

and some

delicate manipulation of matrix entries using elements of infinite order in U yields the required (3) Set C = F /E ^ rA/Eq'

then the image of

[u

_ j

in

If

E

c H.)

u c U, m « |y|,

r

^/Ea

centralizes

C .

Y

(The proof is not easy: it requires Cebotarev density, Artin reciprocity, etc.) (4)

Let

C = lim C q. T« ^ H

acts (via inner automorph-

isms) on C , hence on C, and this action extends (via to an action of G on C.

(2))

This action is trivial, whence C

is abelian (and f.g. since T

is).

action contains ju

(3)» hence is infinite. But

-]

^

(The kernel of the

G is almost simple.) ^

(5)

A

C(G) = lim C

C(G) is central in G. (4.3)

(profinite completions), and thus (This follows from

(4).)

Now the proof shifts gears, applying the theory of A

Moore [16] to the central extension

__

1 •*• C(G) •* G •*• G •*• 1.

This theory implies that G is isomorphic to the "universal covering" of G (relative to G), with C(G) isomorphic to the relative fundamental group

•n'1(G",G).

As a result of

94

Arithmetic Groups

Moore's calculation of fundamental groups, C(G) is of the form asserted in (4.1).

Moreover, C

turns out to be finite

and cyclic, of order dividing m in the totally imaginary case but trivial otherwise; so the index of Eq in finite after all, and

C = C(G).

TA A

is

As a further byproduct of

step (2) above, we see that for any subgroup N of finite index in

I\,

the normal subgroups of N all have finite

index or lie in

{+ 1} .

For example, Nab

contrast to what can happen if

is finite (in

|s| =1).

It should be emphasized that Moore's determination of relative fundamental groups involves the whole arsenal of class field theory.

So by the time Serre concludes his

argument he has invoked a considerable amount of arithmetic in order to answer what might seem to be a straightforward group-theoretic question. (4.4)

When ^ is a Chevalley group (simple, simply con-

nected, split over K) of rank ^ 2, the solution of the congruence subgroup problem is "almost" positive in the same sense as above.

For example, take S to be the set of

archimedean valuations of a number field K. if K is totally imaginary; otherwise

Then

C(G) = 1.

C(G) =y This

situation was studied independently by Mennicke and by Bass-Lazard-Serre when (j - SL

or

Sp«

G = SLn (n^3) over Q, then for —• by Bass-Milnor-Serre [3]. Matsumoto

[14] completed the treatment of Chevalley groups by making

J. E. Humphreys

95

heavy use of the results of Moore [16].

(For a partial

exposition, see [10].) (4.5)

The congruence subgroup problem for other simple

algebraic groups over K has not yet been fully solved, but there has been substantial recent progress.

The most

likely conjecture goes as follows (for (} absolutely simple, simply connected): pletion K over S).

Let r

of K for each

be the rank of G over the comv c S,

and set

r = Z r

In case S contains non-archimedean valuations,

require G to have positive Ky-rank for each such v. is a kind of non-compactness.). when

(sum

r > 2,

(This

Then C(G)ought to be finite

as for Chevalley groups of rank £ 2.

More-

over, C(G) ought to be trivial unless K is a totally imaginary number field and S its set of archimedean valuations; in this case C(G) ought to be

y (or conceivably a

quotient of y). Here is a quick summary of some recent work in this direction. In [23] Raghunathan showed that C(G) is finite if K is a number field and G has K-rank at least 2 (while C(G) has a p-subgroup of finite index in the function field case).

He also showed that each normal subgroup of an S-

arithmetic group (when G has K-rank £ 2) is either finite and central, or else includes an S-elementary subgroup (whose index is finite in the given arithmetic group).

As

96

Arithmetic Groups

in the earlier work, it is essential to show that certain extensions are central.

Building on this work, Deocihar C93,

has gotten more precise results in the case of quasi-split groups (including Du). Bak-Rehmann [2] have made a detailed study of non-split groups of type A.

In particular, they solve the congruence

, subgroup problem for many groups

SLp(D)

and "most" groups

SL (D), n ^ 3, where D is a finite dimensional central division algebra over a global field. More recently Bak Ell has announced a more comprehensive solution of the problem for classical groups (other than DJ.) of rank at least 2.

This involves a reduction to

the cases treated in C23, and uses heavily some techniques of algebraic K-theory.

(Cf. his monograph, K-theory of

forms, Ann, of Math. Studies 98 (1981).) Independently, Prasad and Raghunathan [21] have made considerable progress on the congruence subgroup problem and the related "metaplectic" conjecture. One final remark:

It is known that a non-split simple,

simply connected group G of positive K-rank contains a simply connected split group of the same rank (constructed by Borel-Tits).

It is worth asking whether the respective

congruence kernels can be related directly, since the latter is known explicitly.

Such comparisons with split

or quasi-split subgroups already play a role in the work of Deodhar and Prasad-Ragunathan.

J. £. Humphreys

97

REFERENCES 1.

A. Bak, Le problSme des sous-groupes de congruence et le problSme m^taplectique pour les groupes classiques de rang > 1, CJ.R. Acad. Sci. Paris 292 (1981), 307-310.

2.

A. Bak, U. Rehmann, The congruence subgroup and metaplectlc problems for SL « of division algebras (.preprint)

3-

H. Bass, J. Milnor, J.-P. Serre, Solution of the congruence subgroup problem for SL (n £ 3) and Sp2 (n ^ 2), Inst. Hautes Etudes Sci. Publ. Math. 33 (196?), 59-137-

4.

H. Behr, Uber die endliche Definierbarkeit verallgemeinerter Einheitengruppen, II, Invent. Math. 4 (1967), 265-274.

5-

H. Behr, Endliche Erzeugbarkeit arithmetischer Gruppen uber FunktionenkSrpern, Invent. Math. 7 (1969), 1-32.

6.

H. Behr, Explizite Presentation von Chevalleygruppen uber ZZ , Math. Z. 141 (1975), 235-241.

7.

H. Behr, SL^dP Ct]) is not finitely presentable, pp. 213-224 in: Homological Group Theory, L.M.S. Lect. Note Ser. 36, Cambridge U. Press, 1979-

8.

A. Borel, Introduction aux groupes arithmetiques3 Hermann, Paris, 1969.

9-

V.V. Deodhar, On central extensions of rational points of algebraic groups, Amer. j\ Math. 100 (1978), 303-386.

10.

J.E. Humphreys, Arithmetic Groupsa Lect. Notes in Math. 789, Springer, Berlin, 1980.

11.

J. Hurrelbrink, Endlich prasentierte arithmetische Gruppen und Kp liber Laurent-Polynomringen, Math. Ann. 225 (1977), 123-129.

12.

G.A. Margulis, Discrete groups of motions of manifolds of nonpositive curvature, Amer. Math. Soc. Transl. (Ser. 2) 109 (1977), 33-45 [Russian original appears in proceedings of 1974 Intl. Congr. Math., Vancouver]

Arithmetic Groups

13.

G.A. Margulis, Arithmeticity of irreducible lattices in semi-simple groups of rank greater than 1 [Russian], appendix to Russian translation of [22], Mir, Moscow, 1977-

14.

H. Matsumoto, Sur les sous-groupes arithmetiques des groupes semi-simples deployes, Ann. Sci. Ecole Norm. Sup. 2 (1969), 1-62.

15-

O.V. Mel'nikov, Congruence kernel of the group SL«(S), tL Soviet Math. Dokl. 17 (1976), 867-870.

16.

C.C. Moore, Group extensions^of p-adic and adelic linear groups, Inst. Hautes ftudes Sci. Publ. Math. 35 (1969), 5-70.

17.

G.D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), 171-276.

18.

G.D. Mostow, Existence of nonarithmetic monodromy groups, Proc. Natl. Acad. Sci. USA 78 (1981), 5948-5950.

19-

M. Newman, Integral Matricesa Academic Press, New York, 1972.

20.

O.T. O'Meara, On the finite generation of linear groups over Hasse domains, J. Reine Angew. Math. 217 (1965), 79-108.

21.

G. Prasad, M.S. Raghunathan (to appear).

22.

M.S. Raghunathan, Discrete Subgroups of Lie Groups 3 Springer, Berlin, 1972.

23-

M.S. Raghunathan, On the congruence subgroup problem, Inst. Hautes Etudes Sci. Publ. Math. 46 (1976), 107-161.

24.

U. Rehmann, S. Soule, Finitely presented groups of matrices, pp. 164-169 in: Algebraic K-Theory (Evanston 1976), Lect. Notes in Math. 551, Springer, Berlin, 1976.

25.

J.-P. Serre, Le probleme des groupes de congruence pour SL2, Ann, of Math. 92 (1970), 489-527-

26.

J.-P. Serre, Arbres, amalgames, SL2, Asterisque 46 (1977)

J. E. Humphreys

99

27. U. Stuhler, Zur Frage der endlichen Prasentierbarkeit gewisser arithmetischer Gruppen im Funktionenkorperfall, Math. Ann. 224 (1976), 217-232. 28. U. Stuhler, Homological properties of certain arithmetic groups in the function field case, Invent. Math. 57 (1980), 263-281. 29-

R.G. Swan, Generators and relations for certain special linear groups, Adv. in Math. 6 (1971), 1-77.

30. J. Tits, Travaux de Margulis sur les sous-groupes discrets de groupes de Lie, Sjm. Bourbaki 1975/76, Exp. 482, Lect. Notes in Math.~5^"7, Springer, Berlin, 1977-

Modular Representations of Algebraic Groups BRIAN J.PARSHALL

These lectures provide an introduction to the modular representation theory of semisimple algebraic groups. Sections 1 and 2 assume only a basic acquaintance with the theory of algebraic groups and with the standard language of representation theory.

Later sections, however, employ

the theory of group schemes, and so make more demands on the reader.

Nevertheless, it should become clear that the

study of positive characteristic phenomena is ideally suited to the approach defined by these techniques. Many important topics as well as many proofs are omitted or only barely sketched due to lack of time. The reader may consult the papers in the bibliography for further information. It is a great pleasure to thank Warren Wong and Alex Hahn for inviting me to visit Notre Dame and for making my stay there so pleasant. 101

102

Representations of Algebraic Groups

1.

ELEMENTARY THEORY

Throughout we will fix an algebraically closed field k. Unless explicitly stated to the contrary, we assume that

k has positive characteristic

(1.1) RATIONAL MODULES. group defined over

p.

Let G be an affine algebraic

k. A finite dimensional kG-module

V

is said to be rational if the associated homomorphism Pv: G -»• GL(V) is a morphism of algebraic groups. rational G-module we mean a kG-module

V

By a

which is a union

of rational finite dimensional submodules in the above sense. ¥e let M«

be the category whose objects are the

rational G-modules and whose morphisms are the G-module homomorphisms. (1.2)

EXAMPLES/REMARKS.

(a) MQ

is an abelian category

which is closed under the formation of tensor products9 direct limits, and duals of finite dimensional modules. Further,

it is easy to see that

MQ

possesses enough

infective objects. (b) The coordinate ring

k[G] of G is a rational

G-module relative to the left translation action of

G:

(g»f)(x) = f(xg), f 6 k[G], g,x e G. We may also view k[G] as a rational right G-module by using the right translation action of

G:

(f-g)(x) = f(gx).

(c) k[G] has a well-known commutative Hopf algePor the most part, all modules are taken to be left modules.

B. J. Parshall

103

bra structure, with comultiplication A: k[G] •»• k[G] 8 k[G], counit MQ

e: k[G] •> k, and antipode

r\: k[G] •*• k[G],

is isomorphic to the category of comodules for

Then k[G]

(see [14; 1.1] for more details). (d)

Let Q be the Lie algebra of

Lie algebra of all k-derivations the identity (18 D)A = AD.

Then

D

G, that is, the of

3

k[G]

satisfying

becomes a rational

G-module, called the adjoint module, as follows.

Take

g.D, g e G, D e 3, to be the derivation defined by (g.D)(f) = g-(D(g"1-f)).

It is easy to check that g.D

satisfies the required identity to be an element of (1.3) NOTATION.

We now take up the case when

(connected) semisimple algebraic group. we will assume that

G

G

Q.

is a

For simplicity,

is simply connected.

We list

below some of the standard notation that we will use throughout: B = T.U

Fixed Borel subgroup with maximal torus T, unipotent radical U

B" = T.U"

Opposite Borel subgroup

$

Root system of

$

Positive roots defined by B

U = {ou ,... ,otp}

Fundamental roots in

W

Weyl group

w

Long word in



W-invariant, symmetric, positive definite bilinear form on E = Z$ 8 E

T

in G

$

W

104

Representations of Algebraic Groups

av

coroot 2a/ , a e $

A

Weight -lattice in E spanned by the fundamental dominant weights (i)-^,...,^- (where ^,aY> = 6..»

I £ i,j £ A) A

Dominant weights

p

o^ +...+ Co.

A* r

r X e A+ satisfying 0 j

-^*IB3 necessarily an

inclusion by a). homomorphism since

S(X)

Similarly, there is a nonzero P-module

S(X) •*• -X*|B, whose image is contained in V is a cyclic B-module, generated by a nonzero

-X*-weight vector. d)

Thus, by c), S(X) £ V|£.

an Lj-submodule of

V

by b).

It follows therefore that Lj-module.

Hence, S(X)UJ s (V|p)UJ,

S(X)

J

is an irreducible

B. J. Parshall

(2.2.5)

113

ITERATED INDUCTION.

If P1,...,Pn is any sequence

of parabolic subgroups containing B and if V is a P ... P P P P rational B-module, let VJ 1'""' n denote v| 1|B| 2 - . - l B l n, the result of successively restricting to to P.. i

Then -

P

l'"Pn = G'

[16], w

«0

V|

1 * " * "* n

= V|

The reader wil1

G

B

then inducing

as Pjj-modules —— - provided that "

find a proof of this fact in

Let us merely point out an application.

Let

= S

Q ---P S Q be a reduced expression for the long word *! N w . Setting P. = P/g j, we have P.,...PN = G. Now if V is a rational B-module which extends to a rational P.module, we have from (2.2.1) and the isomorphism kCP./B] = k x P, that V|DD = V. Thus, we get the following extension

theorem [16]: A rational B-module extends to ci rational G-module iff ±t extends to a, rational P-module for each minimal parabolic subgroup P SB.

3.

INFINITESIMAL METHODS

The theory of group schemes aims to rehabilitate in positive characteristic the classical algebraic group-Lie algebra correspondence.

Below we will indicate several

applications of this point of view.

First, we introduce

some preliminary terminology, mostly taken from [20]. Let MJE denote the category of k-functors: object in

M,E

is a functor from the category

commutative k-algebras to the category

E

an M^

of sets.

of The

114

Representations of Algebraic Groups

category

Sch/k

to the k-scheme

of k-schemes embeds naturally in X

we associate the functor (still denoted

X) R + X(R) = Homk(Spec R,X), R 6 Ob(Mk). between

Sch/k

————

and

M, E •—"J£

M.E •*• M^E

Intermediate

is the full subcategory M?P«

sheaves in the fppf topology on morphism

M,E:

M, E

of

•-""tC"~~

The inclusion

admits a left adjoint

*>: M.E •»• KE

(X •*• X) commuting with finite projective limits, called sheafification.

The point behind the introduction of

here is roughly to enlarge

Sch/k

MkE

suitably in order to

facilitate many natural constructions (especially of a group-theoretic nature). Next, recall that a k-group to the category

Gr

of groups.

affine k-group scheme)

G

G

An affine k-group (or

k[G]).

An affine algebraic

in the classical sense defines an affine k-group,

still denoted R 6 Ob(Mk). G(R)

is a functor from M.

is a k-group which is represen-

table (by its coordinate ring group

G

G, by setting

G(R) = Homk_al (k[G],R),

The Hopf algebra structure on

k[G]

with a group structure in a well-known way.

endows The

standard notions of rational representations, etc., all apply in the more general setting of affine k-groups. We cannot enter into further details of the above here.

The reader may wish to consult [20], especially

Ch. II,§1, and Ch. II,§§1,2,3, for more details. formalism involving

The

MfcE. will enter only in a technical

way below which the reader may wish to ignore at first

B.}. Parshall

115

reading. (3-D

PROBENIUS KERNELS.

Assume

G

Let

G

be an affine k-group.

is defined over the prime field

a: G •*• G

k , and let

be the Probenius morphism (1.5).

a closed subgroup scheme of

G.

Now let

H

For a positive integer r,

we define the rth infinitesimal thickening of

H, denoted

HG , by means of the pull-back diagram

Thus,

HG

is a closed subgroup scheme of

H = {e}, the trivial subgroup, we denote G^ r

G. HG

a)

Let

G = SL . Then

G

When by just

and we call it the rth — Probenius kernel of

EXAMPLES,

G.

is given by

Q P CR> = {[ajj] e SLn(R) |a£ = «±J). b)

Assume

Hopf algebra

G

is connected.

k[G-,]* of

of G.

For r = 1, the dual

k[Gn] is isomorphic to the

restricted enveloping algebra

V(g) of the Lie algebra

In particular,

M_ is isomorphic to the category "*! of restricted g-modules. c)

Let

G

be a semisimple, simply connected alge-

braic group defined and split over see that

*\f

G/Gp = G

for all

sheafification of the k-group

be

r.

k . It is easy to

Here

G/Gr

f\f

G/Gr

is the

defined by

116

Representations of Algebraic Groups

(G/G r )(R) = G ( R ) / G r ( R ) , R e Ob(^). Gr/Gg = Gr_g for r > s

Similarly,

(cf. [23; 3-6]).

It is then

easy to extend the argument of (1.6) to see that the map X + S(X)|Q , X e A*, is a bijection between A* and r representatives from the distinct isomorphism classes of irreducible rational G -modules.

(In a different form,

this result is due to Humphreys [26] for r > 1.) More generally, for an arbitrary closed subgroup scheme H of _ 0.

We remark that the morphism (*) is an epimorphism in case If

G H

H n L

is a reduced algebraic group and

is a closed subgroup scheme of for H x

L, we can express the

G

(3-2.1) as

H

V| |H

a V|HnL!

Now assume that k, and that

H

and

has an open orbit x

G L

Q

V 6 Ob(MH), let V by making

Hx

n = 0

case of

for V 6 Ob(ML) !!.

are closed subgroups such that

^ G/H.

in

f

Choose in

L

f e Q(k) and let

G(k).

Hx = x~ Hx, be the stabilizer of x

and if we write

is reduced and of finite type over

be a representative for

where

G

G(k) = L(k)f(H(k)),

Let x

in

L L.

= Hx x

L,

For

denote the rational H-module obtained

act on V

through the morphism Hx + H (hx-»-h),

Now we have (3.2.2) THEOREM [18],

Assume that

G/H - ft has codimen-

118

Representations of Algebraic Groups

sion >_ 2.

Then for a rational H-module

V, we have

X

In addition to the proofs of (3-2.1) and (3-2.2), [18] gives a version of (3-2.2) valid for the higher derived functors of induction. (3-3)

APPLICATIONS.

Let

G

be a semisimple, simply

connected algebraic group, defined and split over a)

PARABOLIC INDUCTION THEOREM.

Let

two standard parabolic subgroups such that One verifies that the P^-orbit K

8

PJT

k .

and

PvA be

J* U K = n.

of wO P T/P, in G/PT d d J

is open and that codim(G/P- - fl) >_ 2. Therefore, we_ w P obtain from (3-2.2) that V|G[p = V °|p npwo| K for V 6 Ob(Mp ) b) integer

[18].

THE STEINBERG MODULES.

For each positive

r, define the rth Steinberg module

S((pr-l)p).

St(r) to be

Below we sketch a proof that

St(r) = -(pr-l)plS-

The argument is given in several

steps: 1)

dim St(r) = prN, N = 1$ + I. This follows from

the r = 1 case (in view of (1.6)) where one argues directly, cf. [18] for details. 2)

Since St(r) is G -irreducible (3-D, reciprocity G gives an inclusion 1: St(r)L —*- -(pr-l)p|Rr of G r r modules. 3) Applying (3-2.1) for L = Bp, H - U~, we obtain,

B. J. Parshall

119

using (2.2.2) and the fact that Br x u~ is the r G trivial group, that -(pr-l)p |Br | a k[lT], so has r r rN dimension equal to p . Thus, i is an isomorphism by 1) above. G = -(pr-l)p| r , so r r BG r r j: StCr)!-,^ i BGr = -(p -l)p|r 'B BG

4) Similarly,

-(pr-l)p|B r!Q

there is an isomorphism

which factors as St(r)|BQ £ -(pr-l)p |B|BQ 5 -(pr-l)p|B r. r r By (2.2.3b), a is an injection on its B-socle, so a is an injection.

It follows that B is an isomorphism, which

clearly proves our claim. c)

BUNDLE COHOMOLOGY. Consider the following commu-

tative diagram, the square being a pull-back:

Here a,af,b

are the natural inclusion morphisms.

(3.2.1), we have that (since

a^

V 6 Ob(EB).

By

Rnb*oc* = ar*oRna* = ar*oRnbxoai

is exact [l4;4.2]) for all n >_ 0. If we apply the above to

Let

V(r) ® -(pr-l)p,

using b) above and the fact (2.2.1) that the exact functor - 8 St(r) commutes with induction, we obtain immediately that

120

Representations of Algebraic Groups

(3.3-1)

R n a*(V (r) ® -(p r -l)p) = R n a*(V) ( r ) 8 St(r), n >_ 0.

(This argument was first discovered by E. Cline and reproduced in [18] in detail.) Using the easily derived fact that

Rnax(V) = Hn(G/B,Lv)

for example)

(sheaf cohomology, cf . [19]

we obtain from (3-3.1) the following impor-

tant result discovered independently by H. Andersen [5] and W. Haboush [253: (3-3-2)

THEOREM.

For a rational B-module

V, we have an

isomorphism of rational G-modules: Hn(G/B,Ly)(r) 0 St(r) s Hn(G/B,Lv(r) ^ ^(pr.1)p), n^ This theorem has several important consequences. (3-3-3)

COROLLARY.

Let X e A+.

for all positive integers

Then

Hn(G/B,L_x) = 0

n.

This result, due to G. Kempf [36], follows easily from (3-3-2) using the ampleness of the line bundle

L_ ,

cf. [5] for more details. (3. 3.4)

COROLLARY. For X 6 A+,

ch -X|G is given by

Weylfs formula (1.7) (with X* in place of X). This is a well-known consequence of (3-3-3), cf. for the argument. (3-3.5)

COROLLARY. For

V 6 Ob(Mg), there is an injec-

B. J. Parshall

tion

121

Hn(G/B,Lv)(r) -»• Hn(G/B,Lv-(r)) of rational G-modules

for all n ^ 0, r ^ 0.

This result, which had been conjectured earlier by Cline, Scott, and the author, was proved by Andersen in [5]. It follows easily from (3-3.2).

4.

RATIONAL COHOMOLOGY

The rational cohomology of affine algebraic groups was first studied by Hochschild in the early 1960fs.

It plays

an important role in the representation theory, and in this section we will survey some of the work done to date. (4.1) Let

BASIC DEFINITIONS. FQ

Let

G

be an affine k-group.

be the functor from the category

M«

to the cate-

gory of k-vector spaces which assigns to each V B Ob(M~) G 4 the space V of G-fixed points. It is trivial to see that MQ

possesses enough injectives, so we can speak of

the nth right derived functor V 6 Ob(MQ), write

RnFQ

of

FQ, n >_ 0.

For

Hn(G,V) - RnFQ(V), the nth rational co-

homology group of_ G

with coefficients in

V.

Similarly,

we can define the rational ExtG-groups, Ext^(V,-) = RnHomG(V,-), V 6 ObCMg). Now let let

W

H

be a closed subgroup scheme of

be a fixed rational G-module.

have that

HomH(W,-) = Koj, where

G

and

By reciprocity, we

J = |H

is induction

More precisely, if we view V as a k[G]-comodule with structure map Ay: V + k[G] 8 V, then VG = {v 6V|Ay(v) =

1 3 v}.

122

Representations of Algebraic Groups

from MJJ to Mg J

and

K = Hora

Q(w>-)-

takes injective objects in

Ifc

is immediate that

MIT to injective objects in

MQ, so there is a Grothendieck spectral sequence (4.1.1)

E^

= Ext^(W,RtJ(V)) «>Ext|+t(W,V)

for all rational H-modules

V.

(4.2)

In this section

SOME BASIC RESULTS.

G

will be a

fixed semisimple, simply connected algebraic group, defined and split over

k .

(4.2.1) TRANSFER THEOREM [19]. module.

Let V

be a rational G-

Then the restriction map on cohomology induces a Hn(G,V) •* Hn(B,V), n >_ 0, of cohomo-

natural isomorphism logy groups.

To prove this result, we will use the spectral sequence (4.1.1) with

H = B.

It is not hard to see that

H (G/B,LZ) for every rational B-module

Z.

Z - V e Ob(MQ), we obtain from (2.2.1) that V 8 RtJ(0^), where B-module.

£

Taking

W

Now if RtJ(V) =

denotes the one-dimensional trivial

It follows from (3-3-3) that

for t > 0.

RtJ(Z) a

HG/B^) = 0

to be the trivial G-module, our

spectral sequence collapses to give the desired result . (4.2.2) REMARKS: V e Ob(MG) and Extg(V,-X),

a)

The same argument shows that for

X 6 A+, we have that ExtQ(V3-x|G) =

n ^ 0. See [19]-

B. J. Parshall

b)

123

One can also show for

r

a positive integer and

V e ObfM^) that there is an isomorphism Hn(G,V) •*• Hn(BGp,V), n ^ 0. (4.2.3) THEOREM [193- Let that -X

V 6 ObCMg), X 6 A+.

Suppose

is not strictly greater (in the partial order >^

(1.3)) than any weight G

Ext£(V,-X| ) =0,

n

of

T

in V.

Then

n > 0.

For the proof see [19;3-2].

This result has the

following important consequence (also taken from [19]): (4.2.4)

COROLLARY.

Let

a) -X|G 8 -y|G = 0 for all positive b)

-X| 8 -y

(4.2.5) REMARK. that

X,y 6 A+.

is G-acyclic, i.e., Hn(G,-X|G 8 -y|G) n. is B-acyclic.

Recent work of Wang Jain-pain [45] shows

-X|G 8 -y|G

(X,y 6 A+)

has a G-filtration with

sections isomorphic to induced modules at least as long as

p

-co|

is sufficiently large.

(o> 6 A ), It follows

from (4.2.4) therefore that an arbitrary tensor product -X-J0 8...® -*n!G

U±

6 A"1")

is G-acyclic.

Using the

parabolic induction theorem (3.3a)a D. Vella has determined (unpublished as yet) conditions which guarantee that -X| , X e A , has an L-filtration with sections isomorphic to induced modules -o>| where

L

(co a dominant weight for

L),

is a Levi factor of a suitable parabolic sub-

124

Representations of Algebraic Groups

group of

In particular, this means -X|G is L-acyclic.

G.

The above results have an interesting (though formal) application, observed first in [42] (and motivated by similar results in characteristic 0), to the representation theory.

For V,W

finite dimensional rational B-modules,

we define X(V,W) =

I 1-0

(-1)1

dim ExtJ(V.W). B

(It is not hard to see that the Ext-groups here are finite dimensional and vanish for i

sufficiently large, cf .

+

[19].) Now for X,u 6 A , (4.2.4) implies that X(y*,-X|G) - 6, (Kronecker delta). A ,11 ch -X|G -

I x(y*,-X|G) ch -y|G, X eA+ additivity of X, we get that (4.2.6) (4.3)

ch S(X) -

FURTHER RESULTS.

Thus, and so, by the

I . X(U*,S(X)) ch -y|G. Let

G

be a semisimple, simply

connected algebraic group, defined and split over kQ. (4.3-D

GENERIC COHOMOLOGY . For q - pd, let

denote the subgroup of points.

G

G(q)

consisting of GF(q) -rational

Then, given a rational G-module

V, we can con-

sider the classical discrete cohomology groups

Hn(G(q),V).

The "generic cohomology" arises from the stability of these groups.

More precisely:

B. J. Parshall

125

(4.3-1.1) THEOREM [19]. rational G-module.

Let

V

be a finite dimensional

For a fixed

n, the cohomology groups

Hn(G(q),V) achieve a stable value

Hnen(G,V) as

d -» «> Hn(G,V^r')

which is given in terms of rational cohomology by for r » 0.

Besides the proof of this result, [19] contains arithmetic conditions on G v

r

and

n

d

n

which guarantee that (r)

H!Lv^ gsn > ) ~ H (G(q),V) = H (G,V ). The above result has also been extended to include the twisted groups [9]. We next introduce a variation on the Kostant partition function

P.

Namely, for n « 0,1,...,

P (X) denote the number of ways sum of

n

positive roots.

P = PO + P-, + ...

X

and X e A,

let

can be written as a

Thus, we have that

. In the following result, we assume

is of simple type. (4.3.1.2)

THEOREM [22], [23].

condition <X+p,a^> £ p, where root in

$*.

Assume also that

Coxeter number of $.

Let X 6 A* aQ

satisfy the

is the maximal short

p >^ 2h, where

h

is the

Then for 0 _ 1,

0

n odd

det(w)p (w(X+p)-p), n=2m is even. In view of (4.3.1.1), the above formula calculates

G

126

Representations of Algebraic Groups

also the generic cohomology of (4.3.2)

S(X) in a range of degrees.

INFINITESIMAL COHOMOLOGY.

Let

G

be as above,

and consider the relationship between the cohomology of and that of its Probenius kernels (4.3.2.1)

THEOREM [14].

Let V

G

G

(3.1).

be a finite dimensional

rational G-module. a)

The natural restriction map

is an isomorphism for each b)

Hn(G,V) •> llm Hn(Br,V)

n.

The natural restriction map

H^G^V) + llm Hn(Gr,V)

is an isomorphism for n 2, be a bijection which,

together with its inverse, carries lines into lines. the PTPG says that if

m,n > 2, then m = n

exist a field isomorphism 31 1 1

11 1 1

p: k " " -•> k' " " with

(b)p(y) for

138

Homomorphisms of Algebraic Groups

a,b € k, x,y S kn+1, x € kn

, x £ 0.

such that

o(kx) = k'p(x)

for

(Recall that the above properties of

p

are called (p-semilinearlty . ) We can reformulate this result in a more invariant form. (1.2)

Assign (see [BT,l-7]) to an affine algebraic variety

V defined over

k

and to a field homomorphism

the algebraic variety the ring

k'l^V]

^V

defined over

k'

a

carry

Then there exist a field isomorphism

and a k' -isomorphism

varieties such that (1.4)

a: Pn(k)

$: ?Pn -*• Pm

of algebraic

a = "p o cp° .

Generalizations and interpretations of the FTPG are

closely related to different advances in and approaches to the problem of abstract homomorphisms of subgroups of algebraic groups.

The reasons for this are many.

But the

fact itself is not surprising at all if one remembers that the first result in the area, a description by 0. Schreier

B. Weisfeiler

139

and B.L. van der Waerden (Abh. Math. Sem. Univ. Hamburg 6(1928), 303-322) of the automorphisms of the projective special linear groups for any automorphism

PSL (k) was based on the FTPG: a

of PSL (k) with

there exist a field automorphism A € GLn(k)

such that

A cp (S*)""3^"1; PSL (k), if

S

here

S,

and a matrix

A cp (S)A""1

is the image of

is the transpose of

S = (s..).

q>: k -»• k

a(S) is either S

n £ 3,

or

S € SLn(k)

and

in

k

and an isomorphism

groups over

k

such that

£: ^PSL

-*• PSL

a(h) = pi o cj>°(h)

of algebraic

for

h € PSLn(k)

Since 1928 when this result was discovered the area has been developed by many a mathematician of renown: E. Cartan, H. Freudenthal, J. Dieudonne, Hua Lo-keng, Wan Zhe-xian, I. Reiner, C. Rickart, O.T. O'Meara, A. Hahn, D. James, B. McDonald, G. Mostow, A. Borel, J. Tits, G. Prasad, G. Margulis, M. Raghunathan and many others. attempt here to give a historical survey.

We will not

Instead we will

outline the major achievements in the subject and their interrelation. The ideal goal of the theory would be to obtain a theorem which includes all known results. goal is, of course, unrealistic.

This

But it is good to keep

it in mind for orientation and proper perspective. For this same purpose we discuss occasionally results from

140

Homomorphisms of Algebraic Groups

adjacent areas. 2.

Let

EVIDENT RESTRICTIONS

G and G' be connected algebraic groups defined

over fields k and k'. Let H be a subgroup of G(k) and a: H -*• G'(k') a group homomorphism. (2.1)

Suppose that G = G" is the group

e3L

(so that

<E8LQ(k) is the additive group of k). Let B be a basis of k

over its prime field.

Then any map

B -* k gives rise to

a homomorphism

GaQ(k) -»• a G'(k'), where H is a subgroup of G(k), then H should be Zariski-dense in G. This condition can sometimes be replaced by additional assumptions on a, H, and k' (say, k and k' are finite, H = G(k), and a is an isomorphism). assumption.

But in general, it is a very reasonable

B. Weisfeiler

141

(2.3) By a theorem of J. Tits (J. Algebra 20(1972), 250270, Theorems 3a^(vi)) G(k)

contains Zariski-dense free

subgroups H if either char k = 0 over its prime field. into

G'

For a free

or H

k

is not algebraic

no good map of

G

can of course be recovered from a general

homomorphism

H -*• G'(k'). This tells us that

well embedded in

G.

H

must be

To specify the meaning of this "well

embedded" is one of the problems of the theory. Examples of relatively small but well-embedded subgroups are (1)

full (of transvections or of rotations) subgroups of

O.T. O'Meara and (2) lattices in semi-simple groups over local fields (see §6 below).

In a number of cases it has

been shown that groups of type (1) and (2) are actually arithmetic (see (6.7) below and also [Va], [Se], and [VW]). 3.

ISOTROPIC SEMI-SIMPLE GROUPS OVER FIELDS.

(3-1) This case can be considered as a paradigm because the situation here is well-understood and can be completely described.

Let

G

be a connected semi-simple algebraic

group defined over k (or, shorter, k-group).

G is called

absolutely almost simple if all normal subgroups of G(k) are finite central; G is almost k-simple if only finite central algebraic normal subgroups of G are defined over k; G is k-isotropic if G contains a proper parabolic ksubgroup; G+(k) is the normal subgroup of G(k) generated by U(k) where U is the unipotent radical of a minimal parabolic

142

Homomorphisms of Algebraic Groups

k-subgroup. k-group.

For example,

SL (k) is a simply connected

Its subgroup U is the group of upper triangular

matrices with all diagonal entries equal to 1. Since SL (k) is generated by transvections, it is generated by the conjugates of U(k). Thus (SL^Ck) - SLn(k). Another example is SO (f), the special orthogonal group of a non-degenerate (and of defect at most 1 if char k - 2) quadratic form

f =

over k if a... € k

Z a..x.x.. l 3. There he also pointed out an application: (6.2) THEOREM.

(Weak arithmeticity, see [R, Proposition

6.6]).

be as in (6.1).

Let

G

lattice in G(E)°,

If

r

is a locally rigid

then there exist a number field

a structure of an algebraic group defined over and an element

g € GCR) such that

g r g"

k

k £ B, on

£ G(k).

(6.3)

REMARK. Actually, see e.g., [M2, Lemma 1],

field

(Q(tr Ad r)

generated over

G,

the

dj by the traces of all

154

Homomorphisms of Algebraic Groups

Ady* Y € r,

can be taken as

k.

Weil's theorem (6.1), and therefore its corollary (6.2), were extended to certain non-uniform latticesby H. Bass, A. Borel, H. Garland, and M. Raghunathan. A. Weil were cohomological —

The methods of

it was at the time when

deformations and their relation to cohomology were intensively explored.

The proof consisted: of two steps.

One of them was to show that r

H (J,Ad) = 0

with coefficients in Lie G(B),

Ccohomology of

and the second one was

to establish that the existence of deformations implies H1(r,Ad) # 0. In the same groundbreaking paper [S], A. Selberg conjectured that any uniform irreducible lattice group

G(E)°

PSLpCR) is arithmetic.

Recall that a lattice

arithmetic if there exist over

Q

in a

assumed adjoint, semi-simple, without compact

factors and different from (6.4)

r

T

in

G(B)°

is called

an algebraic group H defined

and an epimorphism of Lie groups IT: H(B)° •* GOO °

with compact kernel such that index in both

r

and

integral matrices in

Tr(.H(22)).

r fl rr(H(Z)) is of finite (H(Z)

0, cr. and a . *J > 0 for all i and all j > 0 . Then r is an arithmetic lattice in G(B) (take

H = RWQG>

then our

choice of the a,, implies that H(IR) = G(R) x (compact group)) This lattice is uniform if f (x) = 0 has no non-zero solutions, in particular, if

[k: Q] > 1.

uniform if k = Q and f(x) = 0 for some In the late

It is not x € Qn, x ^ 0.

sixties counter-examples to the conjecture

in [S] were found in the groups S0(n,l), 3 < n < 55 see §8.

New versions of the conjecture were proposed by

A. Selberg and I. Piatet ski-Shapiro.

The latterfs version

was very general, it included lattices in products of groups G1(k.), G. semi-simple over k.^ and k^ locally compact. It was essentially I. Piatet ski-Shapiro's conjecture that was subsequently established by G. Margulis (see (6.7)). However, a breakthrough was achieved by G. Mostow whose results are summed up by (6.5) THEOREM (Strong rigidity).

Let

G and

G' be

connected adjoint algebraic semi-simple groups over IR. Suppose that

G(B)° and G'(B)° have no compact factors and

are not isomorphic to PSLp(B). Let

a: r -»• I"

bean

156

Homomorphisms of Algebraic Groups

isomorphism between irreducible lattices of GCR)° and G'CR)°

respectively.

P: G -*• G'

Then there exists an B-isomorphism

such that

a(g) = p(.g) for

g € T.

[The absence of a field homomorphism by

AutJEl = {1}

2

and

Let

G

be a

B-group such that

G(B)° has no compact factors.

Let

G'

be

a connected k-simple adjoint algebraic group defined over a local field k of characteristic 0.

Let

a: r -*• G'(k)

be a homomorphism with Zariski-dense image of an irreducible lattice (i)

r c G(B)°

G'(k).

Then either

a(r) is relatively compact in the Hausdorff

topology of (ii)

into

G'(k), or

k =B

or

(D, G' = G£ x G£

(direct product of

algebraic k-groups), pr., o a: r -»• G£(k) has relatively

B. Weisfeiler

157

compact image, and there exists a homomorphism of algebraic k-groups

p: G -* G£

such that

pr2

« a(g) = p(g)

for

g € r. This theorem implies (6.5) if a

from (6.5) we have that

rk~G > 2.

I" = a(r)

therefore (6.6)is applicable with

is Zariski-dense and

k = B

must have by the assumption of (6.5) on But then (6.6) reduces to (6.5).

Indeed, for

if rk.^ > 2. We G'

that

G£ =

{!}.

The general version of

(6.6) (see [Ml] and [T2]) implies (see [T2] and [T4]) a perfect analogue of (3-7) for homomorphisms (with Zariski dense image) of lattices

r

such as in (6.6) into k-simple

k-groups over an arbitrary (infinite) field k. To see the relevance of the different conditions and implications of (6.6) one need only look (and we will in (6.12)) at Margulis1 proof of (6.7) THEOREM (Arithmeticity theorem) as in (6.6).

Then

r

Let

G

and

r

be

is arithmetic.

This theorem was first proven by G. Margulis in special cases; in one of these cases a similar result was obtained by M. Raghunathan. The counter-examples in S0(n,l) and SU(n,l) (see §8) show that the gap between (6.5) and (6.6) can not be closed without additional (as compared with (6.6)) assumptions on G or a or both. (6.8) Before proceeding further we mention several related developments.

G. Prasad has contributed very much,

158

Homomorphisms of Algebraic Groups

especially in the non-archimedean case, to the study of lattices, see e.g. [Pr], There is an ongoing investigation, led by R. Zimmer (see [Z] and [P]), of generalizations of Margulis1 results to ergodic actions. On the other hand, Y.-T. Siu has generalized the geometric version of (6.5)-

This version considers two

compact locally symmetric spaces X and X' of non-positive sectional curvature, of dimension ^2, totally geodesic factors.

and having no global

The claim then is that

any isomorphism of fundamental groups of X and X' extends, modulo normalizing factors, to an isometry X -*• X'. Y.-T. Siu [Sil,Si2] drops the assumption that X is locally symmetric but assumes that both X and X' are compact Kahlerian. of (6.5).

He also proves in CSil] other generalizations An example, by G. Mostow and Y.-T. Siu [MS],

shows that non-locally symmetric X exist for which the conditions of CSil] are satisfied. P. Parrell and W.-C. Hsiang, e.g.

[PH],studied

topological generalizations of the geometric version of (6.5). A class of compact manifolds M, dim M = n, whose universal covers are contractible but not homeomorphic to Bn is constructed in [Da] for n > 4. The fundamental groups

rr-CM) of such

M

are generated by

B. Weisfeiler

159

"reflections". In view of (8.7). it is improbable that these ir-jCM) are isomorphic to lattices in Lie groups for

n > 30.

However, such M defy the usual techniques to establish topological rigidity. (6.9) Margulis1 proof in [M2] of C6.6) splits naturally into two parts.

The first part constructs a measurable map co

between algebraic varieties, and the second part shows that co

is essentially algebraic.

The first part was somewhat

streamlined and conceptualized by R. Zimmer, see [Z]; we follow his exposition. In the notation of (6.6), let P(resp. P') be a minimal B - (resp., k-) parabolic subgroup of G (resp., G'). Then, by a theorem of C. Moore, r

acts ergodically on

G(H)/P(H).

Then a result of

H. Purstenberg ensures the existence of a measurable r-map co: G(E)/P(1R) •+ M(.G'(k)/P'(k)) 'from

G(B)/P(H) into the

space of probability (positive of total mass 1) measures on

G'(k)/P'(k).

The orbits of

are locally closed.

G'(k) on M(G'(k)/P'(k))

This implies that

co(.G(B)/P(B)) is

(up to measure 0) contained in an orbit, say G'(k).

G'(k)/D, of

It was shown by C. Moore and R. Zimmer that D is

either compact, or the Zariski closure H' of D is a proper algebraic k-subgroup of

G'.

In the first case

a(r) is

relatively compact. In the second case we combine the natural map

G'(k)/D -+ G'(k)/H'(k)

to obtain a

measurable r-map (also denoted by co) co: G(E)/P(E) •*

co with

160

Homomorphisms of Algebraic Groups

•* G'(k)/H'(.k). Assume for tlue rest of this outline that a(r)

is not relatively compact.

(6.10) Then one must recover from G -»• G'

co a rational map

of algebraic k-varieties Cor show that

essentially a map into a point if k ^ B

or

co is

C). To

succeed one must find a link connecting objects of absolutely different nature: measurable maps and rational maps.

Let

\|r: XQR)° -> Y(.k) be a measurable map of points

of algebraic varieties X and Y defined over B and k respectively with Call t

XCR) assumed to have a measure

M-y-rational if, up to measure 0,

into a point when

k #B

of a rational k-map

or

X -*• Y

(C and when

nx-

\|r is a map

\|r is the restriction

k =E

or

C.

To establish

a link between "measurable" and "rational1^ Margulis considers a measurable (with respect to the Lebesgue measure m+n

on E

m+n

) map

almost all n

f: K

x te E

-> Y(k). Then he proves that if for

x € Rm, y € Bn m

and B

^ y

the restrictions of

f to

are rational with respect to the

Lebesgue measures on Mn [im+n-rational.

M-m+n

and IRm, then

f is

To make the above theorem applicable,

Margulis proves that for any B-split subtorus T £ G the map

cpg: Z - ZQ(1R)o(T(B)) -> G'(k)/H'(k) given by

cp (z) = cp(gz) U

of

Z

is

Hy-rational for any unipotent subgroup

and for almost all g € G

Haar measure on

U).

(where

^

is the

The above statement is vacuous if

B. Weisfeiler

161

rkr,G = 1 — hence the restriction that

increases the unipotent subgroup of

q> (where 5

r f c G > 2.

Now one

IL- \ , the restriction

q> (u) = q>(gu)) to which is S

(in

-rational

U

for almost all g, by adding new root subgroups UQcL

one

after another and applying the Pubini theorem and the theorem about maps Bm x Bn -* Y(k) to the ^g1 U(r+l)= U(r)x Ua "* G'(k)/H'(k)-

Thls

completes

the proof in the case when k ? E,

a

not extend to

and therefore, by (6.6(11)),

tr(r)

p: G ->• G'

contradiction.

is relatively compact in

G' . Hence

Thus

a

does

H(B) ~ G(B)x

x (compact group). (6.12) The final step is simple. H(Q) -*H(