Algebraic Groups: Lecture Notes in Mathematics

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1271 A.M. Cohen W.H. Hesselink W. L.J. van der Kallen J.R. Strooker (Eds.)

Algebraic Groups Utrecht 1986 Proceedings of a Symposium in Honour of T.A. Springer

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Editors

Arjeh M. Cohen Stichting Mathematisch Centrum, Centrum for Wiskunde en Informatica Kruislaan 4 t3, 1098 SJ Amsterdam, The Netherlands Wim H. Hesselink Subfaculteit Wiskunde en Informatica, Rijksuniversiteit Groningen Postbus 800, 9700 AV Groningen, The Netherlands Wilberd L.J. van der Kallen Jan R. Strooker Rijksuniversiteit Utrecht, Mathematisch Instituut Budapestlaan 6, 3508 TA Utrecht, The Netherlands

Mathematics Subject Classification (1980, revised 1985): 11 FXX, 14LXX, 15A69, 17BXX, 18GXX, 20GXX, 22EXX, 35A27, 43A90, 57T10 ISBN 3-540-18234-9 Springer-Verlag Berlin Heidelberg New York tSBN 0-387-18234-9 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specificallythe rights of translation,reprinting, re-use of illustrations,recitation, broadcasting, reproductionon microfilmsor in other ways, and storage in data banks. Duplication of this publicationor parts thereof is only permitted underthe provisionsof the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violationsfall under the prosecutionact of the German Copyright Law. © Spdnger-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: DruckhausBeltz, Hemsbach/Bergstr. 2146/3140-543210

PREFACE A symposium on Algebraic Groups took place at the University of Utrecht from I - 4 April, 1986. It was organized to celebrate two birthdays: the 350th anniversary of the university and the 60th birthday of its distinguished member, Professor T.A. Springer. The university celebrated with a series of international scientific symposia and congresses, of which 'Algebraic Groups' was the first one. Our symposium was funded by the 'Stichting 350 jaar Rijksuniversiteit Utrecht', the 'Koninklijke Nederlandse Academie van Wetenschappen', and the Dutch research organisation ZWO. To the first of these bodies and to the convention bureau QLT employed by them, we are also indebted for help with the organization, as we are for secretarial help to the Mathematics Department of Utrecht University. To honour Professor Springer, we felt it would be appropriate to invite a number of leading experts in the field of algebraic groups to lecture on their current research; in this way a wide spectrum of topics would be covered and the central rSle of algebraic groups in mathematics emphasized. It is a tribute to the active part which Springer has played in the development of the subject, that all of the speakers have had close scholarly and personal contacts with him at one time or another. Of the fifteen invited speakers thirteen were able to come, while another mathematician graciously accepted to be a last-minute stand-in. Fourteen manuscripts were contributed to these Proceedings, which often, but not always, cover the subject of the talk delivered (cf. the list of talks and the table of contents below). They have been put in alphabetical order with respect to author's name(s) rather than in an order determined by subject. We briefly touch upon them here. As the reader will notice, there are contributions on various topics centered around algebraic groups. Now that algebraic groups have been with us for about three decades, much is known about their structure (nevertheless, Tits contributes new information on unipotent subgroups of reductive groups in positive characteristic). Thus, attention has gone to subsequent questions, such as a structure theory for finite-dimensional algebraic groups. Popov's contribution shows that there .is still little grip on a 'standard' example, such as Aut A n (he constructs

infinitely many nontriangular

l-dimensional

subgroups of ~

additive group • ). Nevertheless, a

(n~>3) isomorphic to the n many new insights have been obtained

for special classes of infinite-dimensional groups such as Kac-Moody groups. The contribution by Kac & Peterson illustrates this. It also reflects the interest regained in invariant theory, a classical aspect of algebraic groups. In their paper, Le Bruyn & Procesi pay attention to this subject by studying the GL(n)-orbit space of the affine space of m-tup]es of complex n by n matrices (on which the GL(n)action is componentwise by conjugation). Richardson uses modern techniques of

IV

invariant theory to derive an elementary necessary condition for normality of a closed subvariety of the Lie algebra of a semisimple group which is stable under the adjoint action of the group. Piatetski-Shapiro employs a Poincar~ series for split reductive groups to produce Langlands L-functions. Geometric invariant theory is concerned with a description of the quotient" space of a variety by a group acting on it. As a set on which this group acts, the variety can then be recovered from the quotient space and certain group data (the stabilizers). In the case where the group is an algebraic torus, Goresky & MacPherson indicate what kind of data suffice to reconstruct the variety as a topological space on which the group acts. Invariant theory usually starts with a known group action on a variety. Representation theory on the other hand tries to describe all linear representations of a given group. The (finite-dimensional) representation theory of algebraic groups where the characteristic of the representation space coincides with that of the group is reasonably well understood, especially in characteristic O. Part of the present interest in this area is directed to questions concerning special functions related to series of group representations. In his paper, Macdonald deals with a class of polynomial symmetric functions including the 'classical' Schur functions and zonal spherical functions related to various real forms of GL(n). In the cross characteristic representation theory, enormous progress has been made. In particular, for the finite subgroups of algebraic groups which are the fixed points of a (Frobenius type) automorphism, many class functions have been constructed which lead to characters. Lusztig, in his contribution, describes a Lie algebra variant of a special kind of class function he needed in his theory of character sheaves (in fact, functions vanishing outside the nilpotent variety, whose Fourier transforms have the same property). At the origin of the class functions in this theory are certain bundles on the flag variety. Borho exploits the theory of D-modules on the flag variety to study the orbits in the nilpotent variety and the classification of primitive ideals in the enveloping algebra of a semisimple Lie algebra. Relations with D-modules are also present in Brylinski's contribution, describing examples of cyclic homology of certain noncommutative algebras. The nilpotent variety, which turns up in so many contributions related to representation theory, also appears in Jantzen's survey of the determination of the cohomology of a restricted Lie algebra in positive characteristic. There are also contributions on Lie groups, the ancestors of algebraic groups. Borel proves a theorem in unitary representations theory of Lie groups. It concerns the vanishing o~ relative Lie algebra cohomology, and such theorems are of importance for the cohomology of cocompact discrete subgroups. The latter groups plays a central role in the contribution of Mostow & Yau. They use Morse theory to compute homological invariants of the quotient of the unit ball - viewed as the positive cone in natural PU(I,2) space - by a discrete subgroup arising from the monodromy of multivariate hypergeometric functions. As most of the lectures were excellent, they permitted the audience to gain a good impression of several areas of mathematics around the central theme of algebraic groups. The rather leisurely schedule of the symposium permitted many personal exchanges, and we hope the occasion stimulated and furthered the cause of research in the area. We wish to thank the invited speakers for also all the other mathematicians who attended into a success. Contributors and editors alike volume of Proceedings to Professor Springer as

The Editors: A.M. Cohen W.H. Hesselink W.L.J. van der Kallen J.R. Strooker

their talks and their manuscripts, and the symposium and helped to make it take pleasure in offering this a token of esteem and friendship.

•

LIST

OF

TALKS

Tuesday, April I, W. Borho, Nilpotent orbits, primitive ideals and characteristic classes R.W. Richardson, Invariant vector fields on a semisimple Lie algebra C. Procesi, Matrices and invariant theory

Wednesday, April 2, J.L. Tits, On rational unipotent elements of simple algebraic groups G.D. Mostow, Some surfaces covered by the ball and a problem in finite groups G. Harder, Cohomology and special values of L-functions I. Piatetski-Shapiro, L-functions and automorphic forms on classical groups

with Whittaker model Thursday, April 3, G. Lusztig, Fourier transforms on a semisimple Lie algebra over

q

I.G. Macdonald, Commuting differential operators and zonal functions J.C. Jantzen, Restricted Lie algebra cohomology

Friday, April 4, J.-L. Brylinski, Some examples of Hochschild and cyclic homology R.D. MacPherson, The variety of complete quadrics v. Kac, Unitary representations

of Diff S I, exceptional Lie algebras and

statistical mechanics A. Borel, A vanishing theorem in Lie algebra cohomology

TABLE

OF

CONTENTS

A. Borel, A vanishing theorem in relative Lie algebra cohomology

I

w. Borho, Nilpotent orbits, primitive ideals and characteristic classes

17

J.-L. Brylinski, Some examples of Hochschild and cyclic homology

33

M. Goresky & R. MacPherson, On the topology of algebraic torus actions

73

J.C. Jantzen, Restricted Lie algebra cohomology

91

V.G. Kac & D.H. Peterson, On geometric invariant theory for infinite-dimensional

groups

109

L. Le Bruyn & C. Procesi, Etale local structure of matrix invariants and

concomitants G. Lusztig, Fourier transforms on a semisimple Lie algebra over ~q I.G. Macdonald,

Con~nuting differential operators and zonal spherical functions

143 177 ~89

G.D. Mostow & S.S.T. Yau, Some surfaces covered by the ball and a problem in

finite groups I. Piatetski-Shapiro,

Invariant theory and Kloosterman sums

201 229

V.L. Popov, On actions of ~ a on ~n

237

R.W. Richardson, Normality of G-stabZe subvarieties of a semisimple Lie algebra

243

J.L. Tits, Unipotent elements and parabolic subgroups of reductive groups, II

265

AUTHORS" ADDRESSES A. BOREL, Institute for Advanced Study, Princeton, NJ 08540, USA W. BORHO, Gesamthochschule Wuppertal, Fachbereich 7 Mathematik, GauBstr. I, 4600 Wuppertal, Federal Republic of Germany J.-L. BRYLINSKI, Mathematics Department, Brown University, Providence, R.I. 02912, USA M. GORESKY, Department of Mathematics, Northeastern University, Boston, MA 02115, USA J.C. JANTZEN, Mmthematisches Seminar, Universit~t Hamburg, Bundesstr. 55 2000 Hamburg 13, Federal Republic of Germany V.G. KAC, M.I.T., Mathematics Department, Cambridge, MA 02139, USA L° LE BRUYN, Universitaire Instellingen Antwerpen, Departement Wiskunde Universiteitsplein I, 2610 Wilrijk, Belgium G. LUSZTIG, M.I.T., Mathematics Department, Cambridge, MA 02139, USA I.G. MACDONALD, School of Mathematical Sciences, Queen Mary College, Mile End Road, London El 4NS, England R.D. MACPHERSON, M.I.T., Mathematics Department, Cambridge, MA 02139, USA G.D. MOSTOW, Department of Mathematics, Yale University, New Haven, CT 06520, USA D.H. PETERSON, Department of Mathematics, University of British Columbia, Vancouver V6T IW5, British Columbia, Canada I. PIATETSKI-SHAPIRO, School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, 69978 Tel-Aviv, Israel and Department of Mathematics, Yale University, New Haven, CT 06520, USA V.L. POPOV, MehM~t, MGU, per. B. Vuzovskii 3/12, Moscow 109028, USSR C. PROCESI, Dipartimento di Matematica, Istituto "Guido Castelnuovo" Universit~ degli Studi di Roma "La Sapienza", Piazzale Aldo Moro 5, 00185 Roma, Italy R.W. RICHARDSON, Department of Mathematics, The Australian National University, GPO Box 4, Canberra A°C.T. 2601, Australia J.L. TITS, Coll~ge de France, t1, Place Marcelin Berthelot, 75231 Paris Cedex 05, France S.T.S. YAU, Department of Mathematics, University of Illinois at Chicago, Chicago, IL 60680, USA

To T. A. Spring~r, on hi6 60th anniversary A VANISHING THEOREM IN RELATIVE LIE ALGEBRA COHOMOLOGY Armand Borel

This paper is devoted to the proof of a vanishing

theorem for the relative Lie

algebra cohomology of a semisimple Lie group with respect to tensor products of irreducible

finite dimensional

and unitary representations.

role in the proofs of Zucker's conjecture in [B] and

[BC2] respectively.

the conjecture,

Since its context is somewhat broader than that of

and the techniques used to establish it do not occur in other steps

of the proof of Zucker's conjecture separate paper to it, although, Let algebra,

L

K(L)

of

space

[VZ].

For

the irreducible

L,

point) P

L

simple, connected

unitary representations

in our situation,

We consider

H

its Lie and

H

the relative Lie

in the highest degree

L

G

of various

and

E

as above,

of

L

for which the above cohomology

zero and a description of

they give a list of

H'.

[If

L

is a complex

this was done earlier by Enright

(which is not necessarily

is a factor of a Levi subgroup

of a simple Lie group

it starting from [VZ].

I

a finite dimensional

Our starting point to study this are the results of Vogan-

is not identically

the dimensions

E

and are interested

simple Lie group, viewed as real group, However,

L,

both irreducible.

H'(I,K(L);E@H)

in which it is non-zero.

H"

so far, it is of interest only through this application.

a maximal compact subgroup of

a unitary representation

space

in those cases, it seemed best to devote a

be a connected reductive Lie group with compact center,

algebra cohomology

Zuckerman

It plays an essential

in rational ranks 1 and 2 which are sketched

M.A

of a maximal proper parabolic

and the desired vanishing

symmetric spaces,

[E].]

simple, but this is a minor subgroup

condition involves

M,G

and

so that some work is needed to prove

The first proof, alluded to in [B], was a painful case by

case checking, which in fact could not be carried out in all cases involving the two exceptional

bounded domains.

of Q-simple groups Q-rank 1 case. vanishing

of Q-rank

However,

by a happy feature of the classification

I this was not needed to prove the conjecture

My goal here is to provide an essentially

theorem.

It is still not quite classification

in the

a priori proof of that free:

condition

(B) of

§i will have to be checked case by case, but this is of a much more elementary nature (and, I hope, will also be replaced later by a more conceptual argument). §i introduces

some notation and definitions

and (B) of the vanishing brief indications

the vanishing

§3 recalls some of the results of [VZ] and

stronger inequality

Finally §6 checks condition

alluded to above.

(A)

theorem in terms of Lie algebra data (3.7(7)).

is deduced from a slightly

is proved in §5. applications

and the two basic assumptions

The latter is stated in §2, followed by some

on its applications.

uses them to reformulate In §4, 3.7(7)

theorem.

4.4(i).

The latter

(B) in the cases of interest for the

§1. I.I. and

h

Let

L

a Cartan subalgebra

highest weight

~-@L

I.

~

H E L

of

G

For of

such that

G

~ E he,

L

@

Let

F

C(L,~)

Hq(I,K(L);H~F

L

denote an irreducible character

X~

(i.e.

be the greatest

) ~ O.

q

for q,

If there is no such

F

#

[V; VZ].

the associated Cartan involution, character

X%,

a Langlands decomposition

We fix a Cartan subalgebra

with respect to

(i)

h = a @ hM m.

hc

of

P

E

an irreducible

a proper maximal parabolic

P,

with

A,M

8,

where

invariant under

stable under hM

We choose an ordering on the set

is a

¢(gc )

of

such that

~+[a C # ( P , A ) u{0}

where

#(P,A)

representatives

denotes the weights of of

W(mceac)\W(gc).

a

in

1.3. M = L.Gp groups

We assume given a decomposition of e-invariant

K(L) = K ~ L

respectively.

and

(A)

K(Gp) = K N Gp

rk G = rk K

It follows first that

~0 @ h@p =

with

hG a0

WP a

be the usual set of is

>0

if it is a

¢(P,A).

of

M

into an almost direct product hM = hL

are maximal compact in

L

Zg(t)

of

(ao@a)

hGp.

and

The

Gp

is basic

and

rk Gp = rk K(Gp).

is a Cartan subalgebra of K(Gp). We may write P split and t O a Cartan subalgebra of K(L). Then

t

in

g

contains regular elements of

Cartan subalgebra split over

of

t is a Cartan subalgebra of a maximal compact subgroup

The centralizer t

We let v

closed subgroups and write accordingly

The following assumption

hL = a 0 @ to,

n.

An element

positive real multiple of some element in

since

let

with infinitesimal

with infinitesimal

P = N.A.M.

gc

a maximal compact subgroup of

is a connected real simple Lie group, with finite center,

fundc~nental Cartan subalgebra of roots of

A~sumptio~.

This is the case if and only if

In the sequel

subgroup and

K(L)

is dominant).

a maximal compact subgroup,

representation

8.

if

C(L,~) = -~.

1.2.

of

representation

which there exists then set

Notation.

be a reductive group,

finite dimensional

K

PreliminariLy.

can be written as ~.

t @ z I.

K(M) We have

From (A) it follows that

and a compact one.

Therefore

zI

zI

of

M.

Z 1 ~m = 0 has a split

is semi-simple

and

IR.

At the very end of the proof, I shall need to use the following assumption, which is easily checked in the cases of interest

(see §6) and which may well be true

whenever (A) is and

Gp

is the greatest factor of

M

satisfying the second equality

(A).

of

The intersection of

(B)

I

with

Zg(Z I)

is the Lie algebra of a compact

subgroup of ~ L.

Note that (e.g. if then

I

(B)

Z1

contains

a O.

is fulfilled.

XH

Notation.

For

is the quotient of

2.1.

hold.

is maximal ~R-split in

s ~ W P.

(i)

The vanishing theorem.

s E WP

H

we write

~(s%)

for

2.2.

By [C: Fs%,

H

reductive,

(A), (B)

of 1.3 to

Then

2.6] the condition

viewed as a

one, and nowhere else in

(2)

For

by a maximal compact subgroup.

C(L,~(s%)) + £(s) < (dim XG-dim XGp)/2,

By [K],

sllh L.

We keep the previous notation and assume

THEOREM.

Let

a0

It so happens that this stronger condition is often fulfilled.

§2. 2.0.

Therefore, if

is a complex semi-simple Lie algebra, viewed as a real Lie algebra),

M.A.

s%la > 0.

is equivalent to

module, occurs in

H'(n;E%).

Hq(I,K(L);Hi(n;E~)®H)

shla > 0

if

H%(S)(n;E%),

2£(s) < dim N. with multiplicity

We can therefore also write (i) as:

= 0

for

i < (dim N)/2

and

q + i ~ m,

where

(3)

m = (dim XG-dim XGp)/2.

• 2.3.

We now sketch two (related) applications of 2.1. (a)

Combined with the results of [BCI], 2.1 implies that if

arithmetic subgroup of

(I)

L,

Hq(I,K(L);Fv(s~) 8 L 2,~ (r\L))= 0

(b)

F

is an

for some given Q-structure, then

for

q ~ m-£(s)

and

£(s) < (dim N)/2.

We consider the setup of Zucker's conjecture [B; BC2; Z].

a Q-simple connected linear algebraic group, and

r

Let

an arithmetic subgroup.

that the symmetric space

X

symmetric domain and let

V

in [BB].

is the normalizer of a rational boundary component

Assume that

P

of maximal compact subgroups of

G = G(~)

be the minimal compactification of

~

be

Assume

is a bounded

V = F\X

constructed ~

of

maximal dimension. L

We have an almost direct product decomposition

is the greatest subgroup of

compact factor), Gp.

and

Xp

M

acting trivially on

is the symmetric

Modulo various reductions,

of the intersection with V * Vp of Xp in V vanishes

Xp,

M = L.Gp

where

(possibly up to a

space of maximal compact subgroups of

2.1 allows one to prove that the local L2-cohomology

of a suitable neighborhood

U

of a point of the image

from the complex codimension of

Xp

on.

This is the

* main point needed to show that the L2-cohomology to the middle intersection direct application more complicated on automorphic

cohomology

of 2.1.

sheaf on

sheaf along

For the next stratum

Vp.

V

is homology

(level 2), the situation is much

and 2.1 has to be used in conjunction with results of Casselman

forms

(cf.

[BC2]).

§3. Results of Vogan-Zucke~man. Reformulation of 3.1.

In the sequel,

assumption diagram to (cf.

of 1.3(A),

(see [BCI]).

ho, c

stable.

I write

-w G

h°

for

It leaves the set

Also

h L.

one, is given by

wG

whw G.

A(gp,hGp)

2.1.

I recall that under the first

gives the effect of complex conjugation onthe Dynkin

leaves

a

A(l c)

of simple roots of

stable.

[BCl]),which assigns to a representation

identity of

isomorphic

In this case, it is a rather

This is the same as in view of (A).

Moreover

Ic

with respect

the transformation

T1

its complex conjugate contragredient WMW G,

because

We write also

~

w

Gp

for

• wG

is the

WmWG~

(~A(Ic)).

We have then

(I)

C(L,~(s%))

3.2.

We have

assume that 8

# -~ ~>

(s%,a) = (sk,~*)

h° = to @ a°

~+(Ic)

is given by

and

by

rl"

8.

Its restriction

The algebra

subalgebra

q

of

subgroup of Let - PL

pair

is a real form of Xq

(q,w),

w

mq.

the symmetric

be dominant regular.

is the differential

and only if

h

is zero on

t.

~

skla ° = 0.

t ~ to .

K(L).

I.

We let also

It is indeed the

is 0-stable. h

A

c It is in particular

We let space

Mq, °

defined

standard O-stable parabolic

and has a Levi subalgebra O-invariant,

be the corresponding

m q and

which

analytic

Mq,o/K(Mq,o).

We say that

of a unitary character of

q

and Mq, o.

~

are

compatible

if

This is the case if

a

and on the derived algebra of m . Given a compatible o q [VZ] defines an irreducible representation Aq(~-pL) of L, which is

unitary by IV], such that

(I)

We may

is also the linear transformation

o

is one which contains

c of some element in

L *and

w ~ h°

to

for some regular

with respect to

hc = ho,c @a>0 Ic,a

I

is the centralizer mq, ° = mq ~ I

L

a E A(I c)

contains regular elements of

~(it) > 0

denote the Cartan involution of

restriction of

to

for all

H'(I,K(L);Aq(W-OL)~F ~) = H'(mq,o,K(Mq,o);C),

suitably translated.

"Suitably translated" means so that the left-hand side satisfies Poincar6 duality. It follows that the top cohomology occurs in dimension by [VZ], any irreducible unitary representation is so obtained.

with

2.C(L,v) = Maxq(dim ~ + d i m

q v.

Xq)

if

Moreover,

vla ° = 0,

The biggest possible M' q,o

M is the one which is generated by q,O whose roots are all the ~ ~ ¢(I c) such that

(3)

h

and a

O

(v,B) = (OL,B).

We can write

8 =

E

c e

(with

e~A(Ic))__

and the

c

integers all of the same

We have then

(4) But

Xq)/2.

H'(I,K(L);HfFv) # 0

runs through the standard 0-stable parabolic subalgebras which are compatible

semi-simple group

sign.

(dim ~ + d i m

such that

Therefore

(2)

where

H

~ ca(~,a) (OL,~) = (~,~)/2

and, since

v

= ~

ca(PL,a)-

is dominant regular,

(v,~) ~ (~,~)/2.

Equality

holds, therefore, if and only if

(5)

co. :~ 0 * (v,a) = (OL,OO.

The roots of

M' q,o

are therefore all the linear combinations of the elements of

(6)

A v = {a E A(Ic) l(v,e) = (pL,a)}.

Let us write

(7)

M ,M;,X v

If

for

M

q,o

vla ° = 0,

,M' ,X q,o q then

for this choice of

2.C(L,v) = dim ~

q.

We have

+ dim X v.

Moreover, it is clear that

(8)

where

dim X

X'

& dim X' + dim a , ~ o

is the symmetric space of maximal compact subgroups of

In all this, it was assumed that Tl-stable. (5).

However the assumption

TlV = v

Then

~(M~)

M;,M v

A(l c)

the corresponding groups.

M'

and

Av

are automatically

does not play a role in the implication

Generalizing the above slightly, for any dominant regular

the greatest ~l-Stable subset of and

TlV = ~.

v

we let

whose elements are orthogonal to Then

~(M;)

A

be v v - PL

is the greatest Tl-Stable subset

of

~(I c)

all of whose

elements

are orthogonal

to

w - PL

and

Av

is a basis of

~(M~).

8.3.

We now come back to 2.1.

We have

dim X G = dim X L + dim XGp + dim N + i.

Therefore

by 3.2(7),

(i)

2.1(i)

can be written

%(s) + (dim ~ + d i m

Xv(s%))/2

< (dim ~ + d i m

N+I)/2

or

(2)

£(s) + dim Xv(s%)/2

The map

s ~ s' = WMWGS

is an involution

(3)

of

W P.

We have

E(s') + E(s) = dim N

(4)

s'~la +

(5)

F~(s'~)

[This is the

sI

alent

< (dim N+I)/2.

is complex

occurring

s~la

conjugate

in the proof

= o

contragredient

of 2.6 in [C].]

to

Fv(s% ) "

The relation

(2) is equiv-

to

£(s) + dim Xv(sl)/2

~ (dim N)/2 = (i(s)+%(s'))/2,

hence to

(6)

£(s')

The left-hand regular

(7)

~.

- £(s) ~ dim Xv(sl )

side depends Therefore

Fix

s L@

where

~

only on

Let

s,

we are reduced

such that

Z(s)

£(s) < (dim N)/2

and we want

and

to prove

~,~

be regular

slla ° = 0.

2.1 for any dominant

to showing

< (dim N)/2.

Then

£(s')

runs through the regular dominant weights of §4.

4.1.

if

- Z(s) ~ max dim X

g

such that

~(s~)'

s ~ l a ° = O.

Further reductions.

dominant

such that

o = ~ - v

is dominant.

Then

(su,~) ~ (s~,~)

In fact

(so,~) = (o,s-la) >= 0

of

We also know that

W P.

(i)

since

~(s~)

for

o

~ ¢ A(Ic).

is dominant and

(s~,~) ~ (sw,~) ~ (pe,~)

Since

~ - p

is dominant,

(2)

s -I ~ > 0

is regular dominant (for

Ic),

by definition hence

(~A(Ic)).

this yields in particular

(s~,~) = (pL,~) ~ (sp,~) = (pe,~)

(~Eg(Ic)),

therefore

(3)

M (s~) ~ M (so),

dim Xv(s~ ) ~ dim X (so).

Note that we have not assumed Tl-Stability, the end of 3.2. X

t

!

~(sp) ,M~(s,p)

and have used the convention made at

In the sequel we replace the index by

s.

In view of

(3)

~(sp)

in

M (sp),M~(sp),X

(so),

we see that, in order to prove 3.3(7)

and hence 2.1, it suffices to establish

4.2.

PROPOSITION.

If

~(s) < (dim N)/2,

(i)

£hen

~(s') - %(s) ~ dim X . S

Remark.

It is not clear to me that

is equivalent to 3.3(7), because

~(sp)

dim X

is Tl-Stable.

If it is, then (I)

is then the maximum of the right-hand side S

of

3.3(7).

Otherwise,

it is conceivably stronger.

slightly stronger inequality,

(2)

£(s')

- £(s)

~ d i m X' + d i m a S

4.8.

We let

~n

In fact, we shall prove a still

namely

+ I. O

be the set of weights of

hc

in

n c.

Therefore

Let

(i)

As = {~ ~ ~+Is-l~ > 0 } ,

Bs = {~ ¢ #+Is-le < 0 } .

Then

(2)

¢+ = A s ~

Bs,

As =

~m'+

~(s) = Card Bs.

~+ = ~ + ~ m

~N"

(3)

sp = p - ,

The discussion that

Bs~

Bs,.

where

s

is the sum of the elements

in [C: 2.6] and standard

Let

Cs = Bs, - Bs.

facts about reduced

in

B . s

decompositions

show

Then

~(s') - ~(s) = Card C . s To prove

2.1, it suffices,

4.4.

in view of 4.2(i),

We have the inequality

PROPOSITION.

(i)

Card C

Notation.

to

8,

that

Let

i ~ but

5.1.

~ + @

We have

a ___~ B,

LEMMA

a &

then

(i)

~ dim X' + dim a + I. s o

~,B ~ ~. We write

i.e. if neither

(~,8) = 0.

s

Proof of Proposition

§5. 5.0.

(2), to show:

nor 8

~ + B

Let

~ ~

a - B

B

a - B

~

is strongly

This implies

[g± ,g±B] = 0.

orthogonal

in particular

Recall

that if

are roots.

Then

~ £ A(~e).

if

is a root.

if and only if and

4.4.

if and only if

(sp,a) = (pL,~)

-i s

is simple. (ii)

The assertion

A s = {a g A(Ic) IS -I a

(ii) follows

and

s

are simple}.

WmWGa

from (i) and the definition

of

A

(see 3.1(i)

and 3.2(6)).

s

Proof of (i):

Recall

(i)

that

2(O,B)

= (B,B)

if

B

is simple

and similarly

(2) If

2(PL,~)

s

a = B

is simple,

2(sp,~)

Assume now simple,

(3)

(sp,a)

cB C

~

= (B,~)

if

B ~ A(Ic).

then

= 2(p,s-l~)

= (pL,~).

= (B,B) = (s-la,s-l~)

Since

s

-I

~ > 0

we may write

and we have then

(sp,~)

= (~,~) = 2(PL,~).

= z cg(p,S).

s

-i

~ = E c86

with

8

or equivalently

(4)

(~,a) = z cS(~,S).

The possible values for the square norms of the roots, either

1 or I and 2 or i and 3.

In the first case,

suitably normalized, are -I s a is simple.

(4) shows that

In the last case, ~ h i c h

occurs only for the Lie type G2), a should be long, the -i s ~ should be a sum of three distinct simple P -I roots, which is absurd since g has rank 2. In the second case, if s ~ is not with

simple,

cQ # 0

should be short and

the only possibility

is

-I (5)

s

Then

BI,82

is a root, other),

are short and

leaves

~

(81,82 ) + (82,82) = (a,a).

long.

However

if the sum of two simple short roots

it is also short

a contradiction

5.2. T'

a = B I + 82,

(being the transform of one by the reflection -i since s ~ is also long.

We shall write ¢

n

stable.

T

for

Both

wMw a

~

and

(I)

and set T'

T' = -3.

are of order 2.

Then

to the

s' = T.s.

Also,

We claim

T'B s ~ Bs, = ¢.

In fact, if

a ~ B

s

then

S~--I.T~,a = S - I T T ~

Since

Card @n = £(s) + £(s') = Card B

= -S

+ Card B S

disjoint union of

Bs,

and

T'B s,

or also of

ag

T'B a

s,-la

(2)

s-ITia

5.3.

LEMMA.

(i)

For

a g

@+ We

it follows Bs'Cs'T'Bs

C

S

>0

>0

<0

>0

<0

<0

<0

>0

>0

have the equivalences

S

(iii)

+(z l) = cST' U (-c~'). S CT v s

@n

is the

contains a basis of

T'Cs = Cs-

in terms of the sign

B S

a ~ C T' (ii)

that

and that

of these subsets

S

s

a > 0.

S v'

We have then the following characterizations -I of s ~, s'-la or s-IT'a:

-i

-I

(ao~a)

,

.

~'~

=

~ ~

~It

=

0

10

Proof. The map identity Since

on

belongs

to

Bs~

T ',

@ a

o contains

b

since

a

-I

regular

~ .

which are zero on

B.

~,

therefore

eg,

with

B

go

5.5.

Similarly,

Let

LEMMA.

since

~+(m$)is

[By definition,

by

Let

Then

of

~ ~ ±As

a + gB

= s-la

out

~i"

C s,

(1.3) that

Hence

~i

(ii) ~ (iii).

acting by the

ms,

and

a E Cs.

is a root, where

This is clear s

is equal

s -I a

+ e . s-lB

is positive

= s-l~'a

we have

a ~ A

s-lr'~

> 0

s and

and

s-iB

~ + eB K C . s

(~/)

invariant ~/

since

- ~s-IB * > 0.

~ = {S e ~+(ms) IB __ 1 CTs }"

2 Card

generated

if

• '~ ~ C , s

imply that

.sense to speak of the set

pointed

to

those of

to C . It is the sum of a positive root ~ and s ~ = ±1. Therefore it is positive. Its transform

for the same reason:

(2) and 5.2(2)

then belong

are clearly

is invariant under

__[~,B~] C V s

s-lr'(a+eB)

(i),

We already

is a Cartan subalgebra

s-l(a+~B)

(2)

~i

zero on

s

s i m p l e and

is simple by 5.1.

root identically

such a root must

The roots of

Vs = ~ C

to show that

is also positive

Then

+ Card C r' ~ Card C . s s

under

T',

of orbits

on

hence ~

so is

~.

of the group

It therefore

makes

<~'> = {I,~'}

T'.]

~i" .... ~q

be the elements

of

C~ .

Set

E I = {B ~ ~I~ £

el }

E i = {B ~ ~i~ A

~I ..... 6 _ L ~i_ I, ~ £

~i }

(2%i~q).

~ = -ELi E..z If

belongs to

~',

(i).

of h = b @ a @ a, is the o the first equivalence in (i).

any positive

If not, we have to show that if

(i)

But

~,

(i) ~ (ii).

~o @ a

to i or to -i, then it belongs of

This proves of

This proves

and that

It suffices ~ ~

b.

elements

5.4. L ~ . The space adjoint representation,

if

as transformation

on

Being fixed under

~'Bs = ~"

is semisimple,

viewed

and

Cs

B ~ E i, to

C

s (because

then for some by 5.4. if

sB = ±i,

Moreover,

if

~i + sB ~ ~ Cs'

the element

B = B , then

~i + c B B

then both

T'(~i+sBB)

is a root, and

a. + B and a. - B belong i 1 = ~i - EBB G Cs. ) None of

11

these belongs

to zero on

therefore

to C T', since no root of m' restricts s s to show that these elements are all distinct,

i.e. we are reduced to

~.

It suffices

proving

(*)

Let

B ~ E i, 8' K E. 3

and

~. + ~ i

and assume

= a. + g'8'. 3

We may assume that

i ~ J.

(I)

e'~' - e8.

T',

the roots

But no root of

elements of

m).

m

Therefore

(2)

Then

i = j, e = e'

Assume first that

~. i

Being fixed under

ai + ~8, ~'3 + g'B' E C

that

-

~.

j

~i

=

e'B'

and

is zero on e'8 - gB

-

~ t

and

(~i,~j) < 0,

~8

and

i < j.

#

Then we have

0.

are zero on

J

(recall that a. - ~. i 3

(e'B',eB)

s

8 = 8'.

£, £

hence so is contains

regular

are not roots, and we have

< O.

But

(@i+eB,~j+e'8 ') = (~i+eS,~i+eS)

> 0,

therefore

(3)

(~i,~j) + (ei,e'8') + (eB,~j) + (eB,g'B')

By definition

8' --~ ~i"

hence a f o r t i o r i

(8',e i) = O.

> O.

In v i e w of (2), the relation

(3) implies therefore

(4)

(~6,~j) > O.

It follows

that

the definition If now ¢ = e'

and

5.6. therefore

(i)

of

~. - eB is a root, but then so is ~. - e'B', 3 I 8'. This proves (*) when i < j.

i = J.

Then

e8 = e'B'.

Since

~

and

8'

are

w h i c h contradicts

>0,

this implies

B = 8'.

By 5.3(iii),

Card C T' ~ dim a s

o

+ i.

In order to prove 4.4 it suffices

to show

2 Card

(~l)

~ dim X'. s

12

The dimension of

X'

is the complex dimension of the

s

(extended by linearity to of

gc

with respect to

gc ) g.

~ ~ (ms). (i)

ms, c.

We let

o

-i

eigenspace of

0

be the complex conjugation

We have

o8

Let

on

=

8

-

,

o8

=

-

8

(B~).

We distinguish three cases:

8 # 8 •

Then

g8 + gB* + g-6 + g-E* is 4-dimensional and (recall

that

01h c

~ is

dimension 2.

But

Therefore

belongs

8

stable. defined ~

8 ¢ 8 to

0

permutes

by * ) .

g8

Therefore

and its

implies that

81ao # 0,

and g i v e s

a contribution

~

gs* -1

(resp.

eigenspace

hence that 2

~ i

g-8

and

there v

CT s

g_8,)

has

by 5.3(iii).

to the left-hand

side

of

(1). (ii)

8 = 8 , ~ ~ ~.

contribution left-hand

to

side

(iii)

In this

the dimension of of

X' s

The root

B

Therefore

we m u s t show t h a t

is not in

By assumption

Then a g a i n

~,

this case

If

L

a t most two, b u t

gg + g - 8

is

a nd O - s t a b l e .

8 ~ ~

invariant

0

is

the

identity

on

The

a d d s two t o t h e

under

o

a nd

0.

g8 @ g - 8 '

B _--~ C T'.s Then 5.3(ii) shows that then yields

gS' g-8

centralize

z I.

the result.

The condition (B).

is compact, condition (B) of 1.3 is automatically fulfilled.

C(L,~) = 0

for all

~'s

which already follows from 2.6 in [C]. be checked case by case.

if

s~la > 0,

As pointed out in [B], it can very easily

A proof is also included in [Z].

As already remarked at the end of §I, condition (B) is satisfied if

is maximal ]R-split in

(i)

I,

a

o

i.e. if

rk K(L) + r ~ l

This equality holds in particular if a real Lie algebra,

In

and 2.1(i) amounts to the relation

2.~(s) < dim N + 1

6.2.

is ~-stable

hence does not contribute to the left-hand side of (I).

§6. 6. I.

is

gB ~ g - 8

(1).

B = 8 , B ~ ~.

Assumption 1.3(B)

case,

(in which case

I

= rk I.

is a complex simple Lie algebra, viewed as

rank K(L)

and

rklR(L)

are equal to half of

13

rk (L)), or if

L

K(L) = O(2m+l)

and

6.8.

is locally isomorphic

SO(2m+l,l)

symmetric space

X = G/K(G)

Underlying

pactification

Let

X

G

X.

§§1,2].

in which case

We now consider

be a simple non-compact

of

V

G

is a bounded symmetric

(see 2.3), there is first a Satake com-

Its boundary is the union of finitely many orbits of

each one of which is fibered in the so-called real boundary components. given moreover a Q-structure, real ones.

decomposition of

M

and

Xp

is characterized

subgroup of

G.

Let

P

the associated boundary component.

M0

stands for the present

(resp.

G(F)) stands for

domain,

therefore

6.4.

L

Xp,

(resp.

P = NAM

G,

G

is

N(F)

where

Xp

and

itself borrowed

is the normalizer

Gp).]

The space

which is

its Langlands

Then the identity component

M 0 = L.Gp,

acting trivially on

[The notation is the one used so far in this paper, F

by its normalizer,

be one,

has an almost direct product decomposition

greatest connected normal subgroup of

[BB],

If

then the rational boundary components are among the

A real boundary component

a proper maximal parabolic

M0

the irreducible

Lie group such that the

of maximal compact subgroups of

the construction

of

(m~IN),

= i, rk K(L) = m.

For the following, we refer to [BB:

bounded symmetric domains.

domain.

to

rk(L) = m+l, rk]R(h)

Xp

of

L ~

is the = Gp/K(Gp).

from [BC]. F,

and

In

Z(F) ~ M 0

is also a bounded symmetric

1.3(A) is satisfied.

In the sequel we are

only

concerned with types of Lie algebras.

between Lie groups are therefore meant to be only local isomorphisms

Equalities

of identity

components. Let of

G.

t

be the ]R-rank of

G

and

IRA = {~l,...,~t}

The Dynkin diagram for the ]R-roots is of type

canonical numbering of the vertices of [BB: of

X

is the one associated

subgroups, to

b.

up to conjugacy,

of

L,

are indexed by

L

L

one has first to determine

the roots in the Dynkin diagram L

(resp.

is then of type

b ~ [l,t]. Gp) are the

%-1"

the one of

Dyn gc

of

gc

I

c

is

I = g N I . c

BCt.

We use the

The proper maximal parabolic Let

Pb

~i (i
be the one assigned (resp.

i > b).

To find out the type I . This amounts essentially to finding c which restrict to the ]R-roots of

and is easily read off the tables of [T].

form of

or

The Satake compactification

to the last simple ]R-root.

The simple ]R-roots of

The ]R-diagram of

1.2].

the set of simple JR-roots Ct

Then one has to decide which real

These are simple computations,

which we do not give

in detail. 6.5. (see [H:

Up to local isomorphism,

the possible

G's

are, in E. Cartan's notation

p. 354])

A IIl, BD I (p=2), C I, D I I I ,

We now discuss each type separately.

E III, E VII .

14

Type A. III. SLb_I(C)

In Tits notation

[T:

p. 55] this is 2 & . n

In this case

L =

and 6.2 applies.

Type BD I. the case

Here

b = 2.

X = SO(p,2)/S(O(2)×O(p)),

Then

L = SO(p-l,l),

and

t = 2.

K(L) = O(p-l),

We have to consider

and the ]R-rank of

L

is

i. If

p

is even,

Let now hold.

then 6.2(1)

p = 2m + i

Some computation

Then

rk L = rk K(L) = m

x~

]Rn (n=2m+3) so chosen that G is the n 2 - x I + x22 - x~ + E 4 x i. As a maximal ]R-split torus,

group of the form

(el,e 2)

subgroup of

G

last

variables

n - 3

leaving

one of a maximal a 0 = 0.

eI

and and

BI

group

the orthogonal

group of the last

Here

the Siegel upper-half

assume that

L

c = [b/2].

e2

respectively.

fixed.

$O(1,2)

G

space of genus

and

L = SLb(]R).

n

Z =

I

0

"

t

of

variables

~

group

and

~o

is the l,

hence

is the Lie algebra of Its centralizer

is

and is indeed compact.

$P2b(]R),

t = n.

For

After permutation

As

group of the

of

is a block matrix with entries

Let

is the

It is a Cartan subalgebra

of the centralizer

is the symplectic

L

The Lie algebra

of the first three variables. n - 3

planes spanned

The group

It has the orthogonal

compact subgroup.

torus of the latter group.

The derived algebra

Gp ~ SP2n_2b(]R)

groups of the two hyperbolic

(e3,e 4)

as a maximal

the orthogonal

Type C I.

does not

in

we may take the product of the orthogonal by the basis vectors

and 6.2(1)

is needed.

We assume the coordinates orthogonal

is satisfied.

be odd.

K(G) = U(n), P = Pb'

b

is

we have

of the coordinates

A, tA-l,I2n_2b ,

we may take for

X

we may

(AGSLb(]R)).

Let

even the b l o c k matrices

with blocks

yl Z ..... yc Z, - yl Z ..... - yc Z, O2n_2 b

and

t

is the direct sum of

operating matrix c+l)

in

on the last (eb,e2b)

copies

D.

of

to

2n - 2b if

b

and of a compact Cartan subalgebra coordinates.

is odd.

SL2(]R)

(yiE~,i=l ..... c)

if

In b

Z

we have a direct product of

is even

(resp.

odd).

is the group of matrices

(I)

lal

b12]

~c12

d12~

of

To this we should add the

(ad-bc=l)

For

i •

$P2n_2bIR, 2 × 2 c

[l,c]

zero

(resp. D.

15

acting on the 4-plane Dc+ 1 of

Vi

is the unimodular ZI

with basis

is contained in the centralizer

show that the intersection obviously leaves the

c = ±I.

matrices

(i).

A

L

Di's.

is compact.

Vc+ I,

b

odd, The centralizer

It suffices to

This intersection

it centralizes

But the symplectic condition

SL2(]R),

hence -i

imposes that

c = c

V. (l~i~c) we have to find the centralizer of the group of l Writing an element of it in 2 × 2 blocks, we see easily that it A A0 ) (0

(AE~;L2(]R)).

But we must have in addition

A = tA-l,

is orthogonal.

Type D III. In this case, group

On

For

(eb,e2b).

of the product of the

invariant. cI 2.

spanned by

On

consists of matrices hence

Vc+ I

of the latter with

Vi's

consists of scalar matrices hence

(e2i_l,e2i,eb+2i_l,eb+2i).

group of the plane

L

Here

G = SO2n

L = SLb(H),

has rank

on quaternionic

2b-I

where

(notation of [H]), H

and ]R-rank

b-dimensional

K(G) = ~n

and

is the field of (Hamilton) b-l.

Moreover

K(L)

t = In/2].

quaternions.

The

is the unitary group

space and has therefore rank b.

This shows that 6.2(i)

holds.

Type E III. to

Here

SO(10) × SO(2),

Type E VII. isomorphic

to

G

is real form

and

Here

t = 2.

G

For

E6,_I 4

E6, K(G)

E7,_2 5

(where the compact

of

E6

b = 2,

we have

L = SO(9,1)

Let

b = 3.

Then

L

is the real form

~4"

It has real rank 2, hence 6.2(1)

6.6.

symmetric

boundary components for the statement,

unitary groups If

E6,_2 6

space is not necessarily

of

$p(p,q)

(l~p~q)

is locally

K(G) Moreover

t = 3.

E 6,

$O(p,q)

of Zucker's

conjecture,

(A).

See [Z]

Apart from some rank 2 p+q

quaternionic

the system of JR-roots is of type

in which

but where all the real

to be used satisfy

(l~p~_
of indefinite

with maximal compact

is again true.

hermitian,

of the Satake compactification

G = $O(p,q),

and

E7

where the new cases are also enumerated.

groups there are two new series:

is satisfied.

and we may again apply 6.2(i).

I have proposed a slight generalization

the underlying

and 6.2(i)

is meant).

For

subgroup of type

is locally isomorphism

b = 2, Z = SO(7.1),

is the real form

E 6 × $0(2),

of

~

odd, and the forms. and the relevant Satake

P compactification is different

is associated

from the one considered above.)

boundary components group

Gp

is

to the short simple ]R-root.

are again indexed by

SO(p-b,q-b)

the same as in the case

and

C I.

The ]R-rank is

b ~ If,p].

L = SLb(]R).

Given

(Thus, for p, b,

The computations

p = 2,

it

and the types of the corresponding to check (B) are

18

If root. by

G = $p(p,q),

the Satake compactification

Again the ]R-rank is

b E

[l,p].

For a given

p

is associated

to the long simple

and the types of boundary components are indexed

b,

we have

Gp = Sp(p-b,q-b)

and

L = SLb(]l~).

As

we saw above in the case D III, 6.2(1) is satisfied.

6.7. G

In all this, we have assumed

G

to be simple.

is the group of real points of a Q-simple

simple over

]R

and symmetric space of

of real simple groups.

G

In fact, in the applications,

algebraic group, hence is not always is then a product of symmetric spaces

The needed vanishing

conditions

follow from 2.1 and a suitable

KUnneth rule.

The Institute

for Advanced Study, Princeton,

NJ 08540 U.S.A.

References [BB]

W. Baily and A, Borel, Co~actifications of arithmetic symmetric domains, Ann. Math., 84, 1966, p. 442-528.

[B]

A. Borel,

varieties, [Bell

quotients of bounded

L2-cohomology and intersection cohomology of certain arithmetic in E. Noether in Bryn Mawr, Springer,

1983, p. 119-131.

A. Borel and W. Casselman, L2-cohomology of locally Duke Math. J. B0 (1983), p. 625-647.

symmetric manifolds of

finite volume, [BC2]

Cohomologie d'intersection et L2-cohomologie de vari~t&s arithm~tiques de rang rationnel 2. C. R. Acad. Sci. Paris 30!,

A. Borel et W. Casselman, (1985), p. 369-373.

[c]

W. Casselman, L2-cohomology for groups of real rank one, in Representation theory of reductive groups, Progress in Math., 40, Birkh~user, Boston, 1983, p. 69-82.

[E]

T. Enright, Relative Lie algebra cohomology and unitary representations of complex Lie groups, Duke Math. J. 47, (1980), p. 1-15.

[HI

S. Helgason,

[K]

B. Kostant, Lie algebra cohomology and Annals of Math. ?4, (1961), p. 329-387.

[T]

J. Tits, Olassification of algebraic semisimple groups, in Algebraic groups and discontinuous subgroups, Proc. Symp. Pure Math. A.M.S. IX, (1966), p. 33-62.

Iv]

D. Vogan, Unitarizability of certain Math., 120, (1984), p. 141-187.

[vz]

D. Vogan and G. Zuckerman, Unitary Comp. Math., 53, 1984, p. 51-90.

[z]

S. Zucker, L2-cohomology and intersection homology of locally symmetric varieties II, preprint, 1984.

Differential

Geometry and Symmetric Spaces, Adademic Press 1962.

the generalized Borel-Weil theorem,

series of representations,

Annals of

representations with non-zero cohomology,

NILPOTENT

ORBITS,

PRIMITIVE

CHARACTERISTIC

IDEALS,

AND

CLASSES

Waiter BORHO Bergische Universit~t - GH Wuppertal, GauBstr. 20, 5600 Wuppertal I , FR Germany,

and

Max-Planck-lnstitut for Mathematik, Gottfried-Claren-Str. 26, 5300 Bonn 3,

FR Germany.

CONTENTS I.

- Introduction

2.

- Review on S p r i n g e r ' s and Joseph's Weyl group r e p r e s e n t a t i o n s 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

3.

C h a r a c t e r i s t i c classes of cone bundles S p r i n g e r ' s r e s o l u t i o n of the n i l p o t e n t cone C o n s t r u c t i o n of c h a r a c t e r i s t i c classes from a n i l p o t e n t o r b i t Theorem H o t t a ' s t r a n s f o r m a t i o n formulae A l g e b r a i c c o n s t r u c t i o n of our c h a r a c t e r i s t i c classes

- C h a r a c t e r i s t i c class of a p r i m i t i v e 4.1 4.2 4.3 4.4

5.

G = SLn

- C h a r a c t e r i s t i c class approach t o n i l p o t e n t o r b i t s 3.1 3.2 3.3 3.4 3.5 3.6

4.

Basic n o t a t i o n Standard example Nilpotent orbits Primitive ideals Combinatoria| d e s c r i p t i o n Associated v a r i e t i e s Link t o Weyl group r e p r e s e n t a t i o n s , i 1 1 u s t r a t e d in case S p r i n g e r ' s correspondence Joseph's Goldie rank p o l y n o m i a l s Joseph's correspondence Comparison

ideal

C h a r a c t e r i s t i c v a r i e t y of a p r i m i t i v e Relation to nilpotent orbits Definition Theorem

ideal

- The e q u i v a r i a n t K-theory set up f o r proofs 5.1 5.2 5.3 5.4 5.5 5.6

Refinement t o the G - e q u i v a r i a n t l e v e l R e l a t i n g the e q u i v a r i a n t t o the geometric l e v e l Reduction t o the T - e q u i v a r i a n t l e v e l Formal c h a r a c t e r s A formula f o r c h a r a c t e r i s t i c classes in terms of formal characters Conclusive remarks on the p r o o f of theorems 3.4 and 4.4

- References

18

1. INTRODUCTION

~

Let

G be a semisimple complex Lie group, say l i n e a r algebraic, and connected.

The present report deals with the f o l l o w i n g three, a p r i o r i f a i r l y jects:

( i ) The geometry of unipotent conjugacy classes of

n i l p o t e n t o r b i t s in the Lie algebra

g

of

ideals in the universal enveloping algebra t e r f o r s i m p l i c i t y ; and ( i i i ) on the " f l a g v a r i e t y "

G; ( i i )

unrelated sub-

G, or e q u i v a l e n t l y , of

the c l a s s i f i c a t i o n of p r i m i t i v e

U(g), say with t r i v i a l

characteristzc classes in

H*(X)

central charac-

of certain bundles

X, that is the "universal" complete homogeneous space f o r

G. My main point w i l l be to report from recent j o i n t work with J . - L . B r y l i n s k i and R. MacPherson [BBMI,2,3] how ( i i i ) ( i ) and ( i i )

can be used as a tool to get i n s i g h t into both

simultaneously, and to understand t h e i r r e l a t i o n .

For some time, e s p e c i a l l y in the seventies and early e i g h t i e s , these subjects developed

more or less independently, each beeing studied f o r i t s own sake, by i t s

own s p e c i f i c methods. Extensive work has been done by many authors, and remarkable theories have been developed on the three subjects, making each of them i n d i v i d u a l l y into a h i g h l y c u l t i v a t e d area of mathematical research. Since they have been reviewed i n d i v i d u a l l y on various former occasions, I feel free here to focus a t t e n t i o n on some of the f a s c i n a t i n g r e l a t i o n s between these subjects. For more back-ground on the i n d i v i d u a l subjects, the reader may consult f o r instance the Lecture Notes by Steinberg [ S t ] , Slodowy [ S I ] , or Spaltenstein [Sp] on ( i ) , and Jantzen [Ja] on ( i i ) , ning ( i i i ) . 1.2

The f i r s t

and ( i i ) ,

the books by Dixmier [Di]

and say [ H i ] , [Fu] in combination with [BBMI,2,3] concer-

h i n t , suggesting that there must be some deep r e l a t i o n between ( i )

became apparent from the fundamental work of T.A. Springer [S] in 1976,

resp. A. Joseph [ J I , 2 ] a few years l a t e r , on ( i ) resp. ( i i ) : down to c l o s e l y r e l a t i n g ( i ) resp. ( i i )

Their r e s u l t s came

to the same kind of objects, namely i r r e -

ducible Weyl group representations. A careful comparison of Springer's and Joseph's correspondences, u l t i m a t e l y extended by D. Barbasch and D. Vogan [BVl,2] to exhaust i v e e x p l i c i t case by case c a l c u l a t i o n s , confirmed some s u p e r f i c i a l p a r a l l e l i t i e s on one hand, but also exhibited some i n t r i g u i n g discrepancies on the other hand, so t h a t the real r e l a t i o n remained a mystery f o r some years. This s i t u a t i o n was considered unsatisfactory by some people, including myself. S t r i c t l y speaking, in my case, t h i s challenge dates back already to my 1976 expos~ at the s~minaire Bourbaki [ B I ] , where I f i r s t

suggested to r e l a t e ( i ) to ( i i )

via

19

associated v a r i e t i e s , then reported Jantzen's conjectural p a r t i a l a n t i c i p a t i o n of Joseph's theory in case

G = SLn

(see loc. c i t . 2.9. resp. 5.9), and f i n a l l y

learnt from Springer about his new theory; consequently, i t was i n e v i t a b l e f o r me to wonder how these pieces may f i t

together into a common frame-work. The solution

of t h i s puzzle took me not only some time, but also some new advanced methods, as well as two good friends to teach me how to use them. Much of my j o i n t work with B r y l i n s k i or MacPherson was stimulated by the challenge of t h i s puzzle, and is f i n a l l y involved in the solution reported here. We use the intersection homology approach to ( i ) ,

as developed in j o i n t work with MacPherson [BMI,2], and the D-mo-

dule approach to ( i i ) ,

as developed in j o i n t work with B r y l i n s k i [BBI,2], and we

combine them using equivariant K-theory (on T'X) as a unifying concept, in order to obtain a common frame-work f o r the simultaneous study of ( i ) and ( i i ) of ( i i i ) ,

in terms

as elaborated in j o i n t work with both [BBMI,2,3]. Let me also r e f e r at

t h i s point to related work of V. Ginsburg [ G i ] . 1.3 The purpose of t h i s report is two-fold: F i r s t , to popularize the "puzzle" men___ tioned above (section 2), and second to state our solution (section 3 and 4). In section 2, I t r i e d to i l l u s t r a t e the problem in an i n t e l l i g i b l e way f o r the non-expert, using

G = SLn

as a standard example. Note that t h i s is not only a courtesy

to the reader not f a m i l i a r with semisimple Lie group theory, but i t also avoids here almost t o t a l l y those " i n t r i g u i n g discrepancies" mentioned above; t h i s w i l l make our problem (to explain the coincidences between Springer's and Joseph's correspondence) p a r t i c u l a r l y clear and persuasive. On the other hand, those "discrepancies" and t h e i r explanations f o r the other simple Lie groups are precisely what fascinates the real gourmet in Lie theory most of a l l . So in some sense, the experts may consider i t a loss of good taste, that I have s a c r i f i c e d such points r a d i c a l l y f o r the sake of p o p u l a r i t y ; I hope that they w i l l f o r g i v e me. In sections 3 and 4, I t r i e d to formulate an essential part of our results with as l i t t l e

e f f o r t as possible. I spent some care in defining important concepts, but

e s s e n t i a l l y no proofs are included here. However, I added here a f i n a l section with comments on the general strategy of proof (section 5), o f f e r i n g at least some f l a vour of equivariant K-theory and i t s use in t h i s context. Remark. This report is e s s e n t i a l l y i d e n t i c a l with my address to the I n t e r n a t i o nal Congress of Mathematicians at Berkeley, except f o r the augmentation by a f i f t h section, concerning methods.

20

2. REVIEWON SPRINGER'SAND JOSEPH'SWEYL GROUPREPRESENTATIONS

2:1

Basic notation. We f i x a Borel subgroup

B. We denote by

t ~ b :g

Weyl group, and by 2.2

B

the Lie algebras of

X = G/B

in

G, and a maximal torus

T ~ B ~G, by

T

W = NG(T)/T

in

the

the f l a g v a r i e t y .

Standard example. To s i m p l i f y the present review, I shall sometimes r e s t r i c t

(in the present chapter only) to

G = SLn

as a standard example. The reader not

f a m i l i a r with general semisimple Lie group theory may anyway prefer to think in terms of t h i s example through out t h i s paper: Then G is the group complex

n

by

n matrices

(n ~ 2) with determinant I, and

T

SL(n,$)

resp.

of

B are the

subgroups of i t s diagonal resp. upper t r i a n g u l a r matrices. The Lie algebra g = sl(n,$) bracket

consists then of a l l complex

[ x , y ] = xy - yx

n

by

n

matrices of trace

the commutator of matrices, and

t

resp.

b

a l l traceless diagonal resp. upper t r i a n g u l a r matrices. The Weyl group symmetric group

Sn

in t h i s case, acting on

T

values of a diagonal matrix. The f l a g v a r i e t y

. . . ~ Fn 2:3

sn; that is of

~n

F

is an ascending

such t h a t

Nilpotent o r b i t s . Let

Fi

consists of W is the

by permuting the

eigen-

F~ X may be thought of as real

chain of complex subspaces FI ~ F2 ~ . .

has dimension

i.

N denote the " n i l p o t e n t cone" in

subvariety of a l l a d - n i l p o t e n t elements, which is a cone in

g, that is the closed g. (A cone in a vector

space is a subset closed under homotheties.) Under the a d j o i n t action of N

n

X = G/B may in t h i s case be defined

a l t e r n a t i v e l y in the o r i g i n a l sense: Its points " f l a g s " in

and ~

O, with Lie

G on g,

decomposes into a f i n i t e number of o r b i t s , called " n i l p o t e n t o r b i t s " . In case

plex

n

by

G = SLn, these o r b i t s are j u s t the conjugacy classes of n i l p o t e n t comn

matrices. Recall t h a t the set

b i j e c t i o n to the set

P(n)

of p a r t i t i o n s of

N/G of n i l p o t e n t o r b i t s n

is here in

(theory of Jordan normal form), the

"parts" being j u s t the sizes of Jordan blocks. For example, the n i l p o t e n t o r b i t in

~6

generated by the n i l p o t e n t matrix

( a l l other entries zero)

21 corresponds to the p a r t i t i o n

~ = (3,2,1), which is also denoted

(notation of "Young diagrams"). 2__.4_ P r i m i t i v e ideals.

By d e f i n i t i o n , a p r i m i t i v e ideal

of some i r r e d u c i b l e representation some simple ( l e f t ) U(g)-module s a r i l y acts by a character on

of

L, notation

ideals. This set is f i n i t e . L

in U(~_)

is the kernel

J = Ann L. The center of

U(g_) neces-

L, which we assume here to be " t r i v i a l " ,

v a l e n t l y , t h i s means J c g U(g). Let as follows: The module

J

g, or in other words: the a n n i h i l a t o r of

×o

or equi-

denote the set of a l l such p r i m i t i v e

More precisely, one defines a surjection W~ × () above can always be chosen in a p a r t i c u l a r l y nice way,

as a so-called simple highest weight module (theorem of Duflo), and the highest weights in question here come from a single regular

W o r b i t in

t*

(Harish-

Chandra isomorphism), so the modules in question can be s u i t a b l y indexed by Weyl group elements

w £ W, and that surjection

wJ

P Ann Lw =: Jw"

~

Combinatorial description.

with t r i v i a l size

n

In case

W

> Xo

can then be described

G = SLn, the set

central character is in b i j e c t i o n to the set

Xo

of p r i m i t i v e ideals

T(n) of a l l tableaux of

(theory of Joseph's Goldie rank polynomials, see [ J a ] ) .

Here a "tableau" is the combinatorial object also f a m i l i a r as a "Young standard tableau" from the representation theory of symmetric groups. An example of a tableau of size

-g

m= 6

--

is

~

i t s "shape" is the p a r t i t i o n

k = (3,2,1)=

~

,

and there are exactly 16 tableaux of t h i s same shape. There is a canonical, e a s i l y calculable map T: W = Sn T(w)

> T ( n ) , producing from each permutation

(Robinson-Schensted algorithm). For example, the above tableau

duced as

T(w)

from the permutation

w a tableau •

is pro-

w = 216453, and from 15 other permutations.

Now the b i j e c t i o n between tableaux and p r i m i t i v e ideals, T(n)

P ×o'

22 :

~ J , may be described as follows (Joseph): J~ = Ann Lw f o r any permutation

w of tableau ~

• = T(w).

Associated v a r i e t i e s .

There is also a map Xo

to n i l p o t e n t o r b i t s . In case c r i p t i o n s of the two sets

>N/G from p r i m i t i v e ideals

G = SLn, in view of the preceding combinatorial des-

Xo,N/G, the reader w i l l f i n d i t not hard to guess what

t h i s map should do in combinatorial terms: I t sends the p r i m i t i v e ideal corresponding to a tableau shape

~

of

•

to the n i l p o t e n t o r b i t

~ = ~

J = J~

corresponding to the

~.

But how do we produce d i r e c t l y , and in general, a n i l p o t e n t o r b i t from a p r i m i t i v e ideal

J? For t h i s purpose, we i d e n t i f y the associated graded ring of

the symmetric algebra ~* = ~

(using f i r s t

with

the Poincar~-Birkhoff-Witt theorem, and second the K i l l i n g

form). Then we can define the associated v a r i e t y of of the associated graded ideal

J as the zero set g(grJ) :

gr J mS(g). I t turns out t h a t t h i s associated va-

r i e t y is i r r e d u c i b l e [BBI,2], see also [J3], and contained in a unique dense o r b i t

U(g)

S(g), and i n t e r p r e t i t as ring of polynomial functions on

~. So the desired map J i

• ~

N; hence i t contains

is given by

V(grJ) = ~. Let

me comment here that t h i s r e l a t i o n was suggested [ B I ] before, but completely proven [BBI] only a f t e r Joseph invented his c l a s s i f i c a t i o n theory of p r i m i t i v e ideals(2.9). ~

Link to Weyl group representations, i l l u s t r a t e d in case

G = SLn.

Young dia-

grams, and tableaux, the combinatorial data which we encountered above in the class i f i c a t i o n of n i l p o t e n t o r b i t s resp. p r i m i t i v e ideals, both have a well-known s i g n i ficance in the (Frobenius') theory of representations of the symmetric group: There is a b i j e c t i o n , denoted

~I

•

p~, P(n)

~ W^, from diagrams

(equivalence classes of) i r r e d u c i b l e complex representations of the tableaux

T of a given shape k

the representation

~

of size

n, to

W = Sn. Moreover,

correspond b i j e c t i v e l y to a l i n e a r basis f o r

pz. In view of t h i s , the results f o r

G = SLn

reviewed in 2.3,

2.5, may persuade you to make the f o l l o w i n g guess: To a n i l p o t e n t o r b i t ~

~, there should correspond an i r r e d u c i b l e representation

of the Weyl group, and to the c o l l e c t i o n of p r i m i t i v e ideals

variety

~

J

with associated

should correspond a basis of t h i s same representation. I t turns out

that t h i s is a c t u a l l y true, not only in case

G = SLn, but in general, and the main

purpose of my report is to explain how such correspondences can be constructed. I shall next introduce the correspondences of Springer and Joseph, which can be made to do t h i s job, although only with some major e f f o r t to v e r i f y that the two representations are a c t u a l l y the same. An a l t e r n a t i v e version, were t h i s d i f f i c u l t y appears, is then stated in theorems 3.4 and 4.4 below.

dis-

23

~m~ Springer's correspondence. Springer IS] attaches to each n i l p o t e n t o r b i t irreducible and l e t

representation

p~ of

W as f o l l o w s : Let

i r r e d u c i b l e on the homology group jection For

groups e t c . ) . For

~W^

u. Then W acts

G = SLn, t h i s action is

H2d(XU) of highest degree

p~; t h i s is Springer's e x p l i c i t ,

N/G

an

H,(xU). (For s i m p l i c i t y , here and below take com-

plex c o e f f i c i e n t s f o r (co-)homology t h i s defines

~

represent the o r b i t ,

Xu c X be the subvariety of a l l " f l a g s " respected by

l i n e a r l y on the homology groups

(2d = 2dim xU), and

geometrical r e a l i z a t i o n of the b i -

described only c o m b i n a t o r i a l l y in 2.7.

G a r b i t r a r y , one has to replace

action of the isotropy group

Gu c G of

p~, and one obtains only an i n j e c t i o n 2~9

u G~

H2d(XU)

by i t s i n v a r i a n t s under the

u to define Springer's representation

N/G

• W^

by mapping

~

to

p~.

Joseph's Goldie rank polynomials. Joseph [J1] attaches to each p r i m i t i v e ideal

J 6 ×o

a polynomial function

pj

proceeds in two steps. The f i r s t (Jp)~Z

on the Cartan subalgebra is, to replace

of "translated" p r i m i t i v e ideals

pending on a parameter

~

J

t.

His construction

by the whole i n f i n i t e family

Jp, with variable central character, de-

which varies in a Zariski dense subset

Z of

t*

" t r anslation p r i n c i p l e " of [BJ]). The second step is to consider the Goldie rank rk U ( # ) / J

as a function of

~, and to prove that t h i s extends to a polynomial

function on ~; t h i s polynomial, by d e f i n i t i o n ,

is

pj.

(For the reader not f a m i l i a r

with non-commutative Noetherian ring theory, l e t me also add the d e f i n i t i o n of Goldie rank: The quotient ring

U(g)/J

has a complete ring of f r a c t i o n s , which is simple

A r t i n i a n (Goldie's theorem), hence is a matrix ring over some skew fields (WedderburnA r t i n ) ; then ~Q the

rk U(g)/J~

is the rank of t h i s matrix r i n g . )

Joseph's correspondence. Next, Joseph attaches to the p r i m i t i v e ideal W -submodule generated by

representation, denoted

o(J),

pj

in

J ~ ×o

S ( t * ) , and proves t h a t the corresponding W -

is i r r e d u c i b l e . This defines Joseph's correspondence

X°

~W^, J~

~ o(J). Moreover, if

that

~ ( J ' ) = o(J), then the corresponding Goldie rank polynomialsprovide a basis

f o r the representation

J'

ranges over a l l p r i m i t i v e ideals such

o(J). The reader w i l l f i n d an e x c e l l e n t exposition of t h i s

b e a u t i f u l theory of Joseph in Jantzen's book [ J a ] .

~=!!

Comparison. The i n t r i g u i n g question, how the correspondences of Springer resp.

Joseph r e l a t e to each other, on which I commented several times before, can be made precise at t h i s point as follows: Does the correspondence from p r i m i t i v e ideals to n i l p o t e n t o r b i t s via associated v a r i e t i e s (2.5) combine with Springer's and Joseph's correspondences to a commutative t r i a n g l e ? Or in other words: Is to

p~, i f

J

corresponds to

~? In case

o(J)

equivalent

G = SLn, the e x p l i c i t combinatorial des-

c r i p t i o n s given above prove that t h i s is true. Barbasch and Vogan have v e r i f i e d t h i s

24 as a matter of f a c t f o r a l l cases, by an enormous amount of e x p l i c i t c a l c u l a t i o n s in [BVI,2]. More conceptual reasons are offered by Hotta and Kashiwara [HK] and below.

3. CHARACTERISTIC CLASSAPPROACHTO NILPOTENT ORBITS

In these next two sections, I shall now sketch the simultaneous, uniform approach to both subjects, ( i ) n i l p o t e n t o r b i t s , and ( i i ) c h a r a c t e r i s t i c classes on the f l a g v a r i e t y

p r i m i t i v e ideals, in terms of ( i i i )

X, as suggested in my recent j o i n t work

with B r y l i n s k i and MacPherson. For the elaboration of more d e t a i l s , I r e f e r to our o r i g i n a l papers [BBMI-3]. 3~

Characteristic classes of cone bundles.

The concept of a cone bundle

generalizes that of a vector bundle: The bundle map K locally trivial

f i b r a t i o n of

of Chern classes of

s(K)

( I ) f o r a vector-bundle s(K)

is assumed to be a

K by cones (in vector spaces). To extend the theory of a cone bundle. I t may be characterized by two axioms

K, s(K) = c(K) -I is the inverse of the t o t a l Chern class

is f u n c t o r i a l under proper push-forwards.

Now to define our c h a r a c t e r i s t i c class K of codimension

X

vector-bundles, Fulton and MacPherson [Fu] introduced the

notion of Segre class c(K), and (2)

~X

K on

d

Q(K)

in the cotangent bundle

the t o t a l Chern class of

in

H*(X), f o r any cone subbundle

T'X, we m u l t i p l y i t s Segre class by

T'X, and take the lowest (degree 2d) homogeneous term of

the product, n o t a t i o n a l l y : Q(K):= [c(T*X)s(K)] 2d 3.2

Springer's resolution of the n i l p o t e n t cone.

construction is the famous Springer map ~:T*X

Another key ingredient f o r our . ~ N, which w i l l allow us to pass

from n i l p o t e n t o r b i t s to the geometry of the f l a g v a r i e t y . So l e t me r e c a l l here that t h i s remarkable map has a very easy, elegant d e f i n i t i o n (as a Kostant-Souriau momentum map [BBI]):The natura| action of phism

g x X

bundle

T*X

> TX into

~

by v e c t o r - f i e l d s on

into the tangent bundle, and the map

g* = g

~

Construction of c h a r a c t e r i s t i c classes ~ : N , we f i r s t

taking the preimage •

~ of the cotangent

is then that i t s image in

N, and that i t resolves the s i n g u l a r i t i e s of

a nilpotent orbit

X defines a mor-

( K i l l i n g form) isthen.obtained by transposition and pro-

j e c t i o n . The remarkable point about potent cone

g

g

is the n i l -

N.

from.aiiiiini,!,,l,potent

orbit.

Starting from

produce a c o l l e c t i o n of cone bundles on

X by

under Springer's map ~ and then decomposing the closure

into i r r e d u c i b l e components KI . . . . . Kr. These are in f a c t cone bundles

25 Ki ----->X, c a l l e d o r b i t a l pends only on

for

~. We know t h a t t h e i r codimension

~, more p r e c i s e l y

2d = codimN~

Next,we take f o r these " o r b i t a l " classes 3.4

Q(KI) . . . . . Q(Kr)

d

in

T*X

de-

(work of Spaltenstein and Steinberg).

cone bundles

K 1 . . . . . Kr

the c h a r a c t e r i s t i c

as defined in 3.1.

Theorem.

a) The classes b) They form a c) This

Q(K I) . . . . . Q(Kr) W- submodule in

are l i n e a r l y H2d(x).

independent.

W- representation is e q u i v a l e n t to S p r i n g e r ' s

d) I t transforms the basis

Q(K I) . . . . . Q(Kr)

The f o l l o w i n g formulae were f i r s t context [ H I ] ,

pff.

according to the formulae below.

obtained by Hotta in a s l i g h t l y

different

[H2], see also [ J 4 ] .

3±5

Hotta's transformation formulae. For each simple r e f l e c t i o n

all

i = I . . . . . r, we have e i t h e r

sQ(K i ) = -Q(Ki), or else r sQ(Ki) = j=1 s nij(s)

s

of

W, and f o r

Q(Kj),

f o r some m a t r i x of non-negative integers

nil(s)

with diagonal e n t r i e s

nii(s)

= I.

Moreover, there is a geometrical i n t e r p r e t a t i o n f o r these ~ntegers, f o r which i ref e r to our resp. Hotta's o r i g i n a l papers. For example, t h i s says t h a t unless Kj ~6

intersects

Ki

nij(s)

= 0

in codimension ~ I.

&!gebraic c o n s t r u c t i o n of our c h a r a c t e r i s t i c

classes.

For the reader with less

i n c l i n a t i o n f o r geometrical elegance, but with a preference f o r abstract algebra, l e t me also o f f e r here an a l t e r n a t i v e ,

more algebraic d e f i n i t i o n

defined g e o m e t r i c a l l y in 3.1. This d e f i n i t i o n - B o r e l ' s d e s c r i p t i o n of t,

by a

W-equivariant

H*(X)

- The Chern character bundle on

X

in terms of polynomials on the Cartan subalgebra

isomorphism

W-harmonic polynomials on

B: H*(X)

~ • S(t_*) ~

onto the space of a l l

t.

ch: K(X)

~ H*(X), attaching to an ( a l g e b r a i c ) v e c t o r -

i t s t o t a l Chern class

- The Grothendieck group

of the classes Q(~

r e f e r s to the f o l l o w i n g i n g r e d i e n t s :

K(X)

(an isomorphism here).

of classes of ( a l g e b r a i c ) vector-bundles on

X, or

e q u i v a l e n t l y the Grothendieck group of the category of a l l coherent sheaves of Ox-modules. -

The analogous Grothendieck-group duced by the z e r o - s e c t i o n

o: X ~

K(T*X), and i t s isomorphism ~T*X

~*

with

K(X)

[Fu].

Given these i n g r e d i e n t s , we may attach to any closed subvariety

K of co-

in-

26 dimension d

in

T*X a cohomology class

Q(K) in

H2d(x), which we identify with

a degree d homogeneouspolynomial on t

via

proceeds as follows: Take the class

determined by the structure sheaf of

in

[OK]

B(H2d(x) = sd(t*)~). The construction K

K(T*X), and apply the composition of the maps o* K(T*X) "~, K(X)

ch 6 N , H*(X) "~ • S(~*)~;

f i n a l l y take the lowest degree term. So formally, this alterrlative, algebraic defin i t i o n of

Q(K) reads:

Q(K):= [B ch ~*[OK]] d. For a cone bundle K i t can be shown that this d e f i n i t i o n coincides with that in terms of Segre class as stated in 3.1.

4. CHARACTERISTIC CLASSOF A PRIMITIVE IDEAL

4~

Characteristic v a r i e t y of a p r i m i t i v e ideal.

J ~ Xo, we f i r s t

construct a cone-subbundle in

f i n e the c h a r a c t e r i s t i c class of

J

Starting from a p r i m i t i v e ideal

T*X

as f o l l o w s . Later, we shall de-

as the c h a r a c t e r i s t i c class of t h i s cone bundle

(4.3). Consider the quotient

U(g)/J

as a l e f t g-module

M, and take the correspon-

ding sheaf of modules

M:= DX~U(g) M over the sheaf

Dx

l o c a l i z a t i o n of

of rings of d i f f e r e n t i a l operators on

M on

X

(Beilinson-Bernstein

X); then the c h a r a c t e r i s t i c v a r i e t y resp. cycle of

well-defined notions from the general theory of D-modules, denoted Ch(MI

here. By d e f i n i t i o n , Ch(M)

is a closed subvariety in

M are

Ch(M) resp.

T'X, and

Ch[MI

is a

formal integer l i n e a r combination

Ch(M) = z mi [ V i ] , 1

in which each i r r e d u c i b l e component Vi

of the c h a r a c t e r i s t i c v a r i e t y occurs with

some well-defined p o s i t i v e m u l t i p l i c i t y

mi . Now c h a r a c t e r i s t i c v a r i e t i e s are (by

construction) f i b r e d by cones, and in the present case,this f i b r a t i o n is l o c a l l y t r i v i a l on

X

(as a consequence of the s t a b i l i t y of

and the G-homogeneity of cone bundles over 4.2

J

under the a d j o i n t G-action,

X). So Ch(M), and hence i t s components

Vi , are a c t u a l l y

X.

Relation to n i l p o t e n t o r b i t s .

By my work with B r y l i n s k i [BB2], these cone

bundles are a c t u a l l y o r b i t a l f o r some n i l p o t e n t o r b i t

~. In more d e t a i l , by l o c . c i t .

Springer's map ~ maps Ch(M) onto the associated v a r i e t y

V(grJ), which is the

27

closure of a n i l p o t e n t o r b i t , as reported already in 2.6. In the sequel t h i s n i l p o t e n t o r b i t determined by

J. I t follows that

Ch(~()

and now some t r i c k y dimension arguments show that each component tained in

~

cone bundles

O' denotes

is contained in Vi

-I~,

is even con-

as one of the i r r e d u c i b l e components, hence is one of the o r b i t a l KI . . . . . Kr (notation 3.3). - Hence we may w r i t e our c h a r a c t e r i s t i c cycle

as well as r Z z i [K i ] i=I

Oh(M) =

where we admit some of the " m u l t i p l i c i t i e s " zi

4_:_3 D e f i n i t i o n .

to be zero.

Now we define the c h a r a c t e r i s t i c class of

J

in H*(X) as the cha-

r a c t e r i s t i c class of i t s c h a r a c t e r i s t i c v a r i e t y by: r P(d):= Q(Ch(M)):= z z i q(Ki). i=I Here _..O(K i)

is the characteristic class of a cone bundle as defined in 3.1 (or 3.6).

4-:4 Theorem:

Let

J1 . . . . Jr ~ Xo be the c o l l e c t i o n of p r i m i t i v e ideals correspon-

ding to a n i l p o t e n t o r b i t

~

( t h a t is with associated v a r i e t y

a) The character~st__iic classes b) They span a c) This

P(JI ) . . . . . P(Jr )

W-submodule in

H2d(x) (Note

2d = codimNC.)

W-rePresentation is equivalent to Springer's

d) The class

P(Ji )

BP(Ji) = ~PJI

~ ) . Then:

are l i n e a r l y independent. p~.

is proportional tO Joseph's Goldie rank polynomialof Ji' that is

f o r some scalar

O ~ ~ ~ ~.

5. THE EQUIVARIANTK-THEORYSET UP FOR PROOFS

Let me conclude t h i s report with the f o l l o w i n g comments concerning the general s t r a t e gy of our uniform proofs of theorems 3.4 and 4.4. The purpose of these comments is to indicate the role played by the formalism of e q u i v a r i a n t K-theory, and to sketch a few ideas which are c r u c i a l in our approach. I r e f e r to my o r i g i n a l papers with B r y l i n s k i and MacPherson f o r detailed expositions of proofs, and also f o r more elaborated statements of these and f u r t h e r r e s u l t s . ~

Refinement to the

ses

Q(K)

bundle

G-equivariant level.

Our d e f i n i t i o n of c h a r a c t e r i s t i c clas-

as given in section 3 refers only to the geometrical structure of the cone

K. This is f i n e from the point of view of elegance of r e s u l t s , since i t

allows purely geometrical i n t e r p r e t a t i o n s . However, from the point of view of proof of some of our r e s u l t s , i t is more convenient to take also account of the additional structure on an o r b i t a l cone

bundle

K provided by the group action. The advantage

28 of our algebraic construction of the c h a r a c t e r i s t i c class

Q(K) given in 3.6 is

that i t applies almost word by word to the d e f i n i t i o n of a more refined notion of "equivariant c h a r a c t e r i s t i c class", denoted cise, t h i s is defined f o r any geneous polynomial on ~ ture sheaf on

K

QG(K), as w e l l . To be a l i t t l e

G - s t a b l e closed subvariety K

as follows: Take the class

in the Grothendieck group

KG(T*X)

[0 K]

in

T*X

more pre-

as a homo-

determined by the struc-

of the category of

G-equi-

v a r i a n t coherent sheaves on T'X, and then apply to i t the f o l l o w i n g chain of homomorphisms KG(T,X)

~

KG(X)

ch~Heven "G (X) BG ^S ( t * ) , which refines the completely analogous one in subsection 3.6 l e v e l . Here the "equivariant Chern character" the f l a g v a r i e t y of

X

to the

G-equivariant

chG maps the equivariant K-group of

ingo the even degree part of the equivariant cohomology group

X, which in turn by the "equivariant Borel picture" is i d e n t i f i e d with the ring

A

S(t_*)

of formal power series on t .

F i n a l l y , define

term of the power series manufactured from 5.2

Relating the e q u i v a r i a n t to the geometric l e v e l .

account of the a d d i t i o n a l

QG(K) as the lowest degree

K by t h i s procedure.

G - s t r u c t u r e on

Now that we haven taken

K we may look at the process of " f o r -

g e t t i n g " i t again; t h i s provides a canonical " f o r g e t f u l " homomorphism KG(X)--~ K(X), which corresponds to projecting a power series

F on ~

to i t s

F~, as is easy to see. I t turns out that the lowest degree term project to

W-harmonic part QG(K) does not

O, or has non-zero harmonic part. This fact r e f l e c t s an e q u a l i t y of the

codimension of support of a coherent sheaf on ding element in

in t h i s case, though a l i t t l e t e r i s t i c class by taking the

X and the degree of the correspon-

KG(X) with respect to Grothendieck's v - f i l t r a t i o n ,

which is true

d e l i c a t e to prove. We conclude then that our charac-

Q(K) can be reobtained from the

G - e q u i v a r i a n t version

QG(K) j u s t

W- harmonic part:

Q(K) = QG(K)~. 5~

Reduction

toiitihe

sists in switching from

T- equivariant l e v e l .

G - e q u i v a r i a n t K-theory on the f l a g manifold to

variant K-theory on a vector-space

E as follows. Starting from a

coherent sheaf on the cotangent bundle from

G to

A key technique of our approach con-

T'X, we f i r s t

r e s t r i c t the group action

T, and then r e s t r i c t the sheaf to the single f i b r e

base point, which is f i x e d by

T, to obtain a

other hand, we r e s t r i c t the sheaf on

T*X

T-equi-

G -equivariant

E of

T - e q u i v a r i a n t sheaf on

to the zero section

T*X

at the

E; on the

X, as we did a l -

ready in 5.1. These r e s t r i c t i o n processes give isomorphisms KG(X) ~ KG(T*X) ~ KT(E).

(*)

The point of these manipulations is now that equivariant K-theory of a l i n e a r torus action can be carried out very conveniently in terms of calculations with formal characters, as I shall explain a l i t t l e

more precisely below, and that we can re-

29 duce our problems from the context of

G- equivariant K-theory on

X

into t h i s more

convenient s e t t i n g . Furthermore, as explained in subsection 5.2, the l i n k to our previous geometrical considerations is made by means of the homomorphisms KG(X)

• K ( X ) / c h , H*(X).

In conclusion, t h i s e x h i b i t s our technique of t r a n s l a t i n g statements from a context most convenient f o r computational manipulations (formal characters) into a context most convenient f o r geometrical i n t e r p r e t a t i o n (cohomology of the f l a g v a r i e t y ) , and vice versa. This is one of the c r u c i a l ideas underlying

the strategy of proofs in

[BBM3]. 5.4

Formal characters.

Since

So the inclusion

L: {0}

into

KT(O)

KT(O). But

T

acts l i n e a r l y on

E, the zero point is

is nothing else but

of KT(E)

R(T), the representation ring of

which may also be considered as the group algebra of the character group T. lhe map ~*: KT(E)

T-stable.

~ E induces a f u n c t o r i a l ring homomorphism ~*

X(T)

T, of

~ R(T) turns out to be an isomorphism, which may be des-

cribed e x p l i c i t e l y as f o l l o w s . F be a T-equivariant coherent sheaf on

Let generated

S(E*)-module equipped with

an equivariant

poses into a d i r e c t sum of weight spaces the formal character of ah(M) =

E. Then M = P(E,F)

M×, where

T - a c t i o n , and so i t decomx ~ X(T). Now one can define

M as usual as a formal sum

~ (dim MX) [ × ] . xEX(T)

(Here one has to note that the weight m u l t i p l i c i t i e s of the p o s i t i v i t y of the weights of vious way by elements of a:=

is a f i n i t e l y

E.)

dim Mx

are a l l f i n i t e because

We may m u l t i p l y such expressions in an ob-

R(T), f o r instance by

n(l-x),

x where the product is extended over the p o s i t i v e roots (the weights of Then the desired formula f o r Proposition:

~*

T

in

E).

reads as follows:

~*[F] = A oh(M).

In p a r t i c u l a r , oh(M)

may be considered as an element in the f r a c t i o n f i e l d of

R(T). I t is easy to deduce from t h i s formula the Corollary: L~ 5:5 R(T)

is an isomorphism of

R(T).

A formula f o r c h a r a c t e r i s t i c classes in terms of formal characters. as a subring of the ring

by making a character

× of

A

S(t*)

of formal power series on

We consider

t . This is done

T correspond to the exponential of i t s d i f f e r e n t i a l :

~ ~I (dx) n = x. n>O PE ~(t ; ) is any power series, then we denote

edx If

KT(E) onto

=

term, which is of course a homogeneous polynomial.

[p]d

i t s degree

d

homogeneous

30 Theorem: Let

K be an o r b i t a l cone bundle. Let

M = Y(E,OK), the ring of the re-

gular functions on i t s f i b r e over the base point, considered as a

T- equivariant

S(E*)-module. Then a) As a formal power series on ~, A oh(M)

has i t s lowest nonzero homogeneous term

in degree d:= codimT, X Kb) The e q u i v a r i a n t c h a r a c t e r i s t i c class of

K

(as a polynomial on ~) is given by

the formula QG(K) = [A ah(M)] d. c) The c h a r a c t e r i s t i c class of Q(K) = ([~ ch(M)]d) ~

K

is given by the formula

as the harmonic part of the lowest degree term of 5~§

a oh(M).

Conclusive remarks on the proof of theorems 3.4 and 4.4.

This is the desired

e x p l i c i t expression f o r our c h a r a c t e r i s t i c classes in terms of the formal characters. Since t h i s expression relates our c h a r a c t e r i s t i c classes to the "character polynomials" as studied in the previous l i t e r a t u r e by Joseph, Jantzen, Vogan, and others, i t enables us to prove parts d) and c) of theorems 3.4 and 4.4. In case of theorem 4.4, one has to use the work on c h a r a c t e r i s t i c v a r i e t i e s of p r i m i t i v e ideals in [BB2] and some D-module theory as an additional ingredient. To make the i d e n t i f i c a t i o n with SpringerIs representations,that is to prove part c) of the theorems 3.4 and 4.4, we do not need the equivariant level but we work d i r e c t l y on the geometrical level, using as additional main ingredient the work on i n t e r s e c t i o n homology of closures of n i l p o t e n t o r b i t s in [BMI,2].

31

References

:

[BVI]

Barbasch, D. - Vogan, D.: Primitive ideals and orbital integrals in complex classical groups; Math. Ann. 259 (1982), 153-199.

[BV2]

Barbasch, D. - Vogan, D,: Primitive ideals and orbital integrals in complex exceptional groups; J. Algebra 80 (1983), 350-382.

[BI]

Borho, W.:

[BJ]

Borho, W. - Jantzen, J.C.: Ober p r i m i t i v e Ideale in der Einh~llenden einer halbeinfachen Lie-Algebra; Invent. Math. 39 (1977), 1-53.

[BBI]

Borho, W. - Brylinski, J.L.: Differentia] operators on homogeneous spaces I; Invent. Math. 69 (1982), 437-476.

[BB2]

Borho, W. - Brylinski, J.L.: Differential operators on homogeneous spaces I I I ; Invent. Math. 80 (1985), 1-68.

[BBMI]

Borho, W. - Brylinski, J.L. - MacPherson, R.: A note on primitive ideals and characteristic classes; in: Geometry Today, Birkh~user: Progress in Math. 60 (1985), 11-20.

[BBM2]

Borho, W. - Brylinski, J.L. - MacPherson, R.: Springer's Weyl group representations through characteristic classes of cone bundles; IHES preprint M/85/70, Dec. 1985.

[BBM3]

Borho, W. - B r y l i n s k i ,

[BMI]

Borho, W. - MacPherson, R.: Representations des groupes de Weyl et homologie d'intersection pour les vari~t6s nilpotentes; C.R. Acad. Sci. Paris (A) 292 (1981), 707-710.

[BM2]

Borho, W. - MacPherson, R.: Partial resolutions of nilpotent varieties; in: Analyse et Topologie sur les Espaces Singuliers, Soc. Math. de France, Ast~risque 101 (1983), 23-74.

[Di]

Dixmier, J . :

Alg@bres enveloppantes; Paris: Gauthier Vii1ars 1974.

[Fu]

Fulton, W.:

Intersection theory, Springer: Berlin-Heidelberg- New York Tokio 1984.

[Gi]

Ginsburg, V.:

g-modules, Springer's representations and b i v a r i a n t Chern classes; Advances Math. 59 (1986).

[H]

H i l l e r , H.:

Geometry of Coxeter groups; Res. Notes in Math. 54, Pitman: Boston-London-Melbourne 1982.

[HI]

Hotta, R.:

On Joseph's construction of Weyl group representations; Tohoku Math. J. 36 (1984), 49-74.

[H2]

Hotta, R.:

On Springer's representations, J. Fac. S c i . , Univ. of Tokyo, IA 28 (1982), 836-876.

[HK]

Hotta, R. - Kashiwara, M.: The i n v a r i a n t holonomic system on a semisimple Lie algebra; Invent. Math. 75 (1984), 327-358.

[Ja]

Jantzen, J.C.:

Recent advances in enveloping algebras of semisimple Lie algebras; S~minaire Bourbaki 1976, Springer LNM 67___77(1978),exDos#489.

J . L . - MacPherson, R.: Equivariant K-theory approach to nilpotent orbits; IHES preprint M/86/13, March 1986.

Einh~llende Algebren halbeinfacher Lie-Algebren; Springer: Berlin-Heidelberg-New York-Tokio 1983.

32 [J1]

Joseph, A.:

Go]die rank in the enveloping algebra of a semisimp]e Lie algebra I , I I ; J. of Algebra 65 (1980), 269-306

[J2]

Joseph, A.:

Kostant's problem, Goldie rank, and the Gelfand-Kirillov conjecture; Invent. Math. 56 (1980), 191-213.

[J3]

Joseph, A.:

On the associated variety of a primitive ideal; J. of Algebra 93 (1985), 509-523.

[J4]

Joseph, A.:

On the variety of a highest weight module; J. of Algebra 88 (1984), 238-278.

[Sl]

Slodowy, P.:

Simple singularities and simple algebraic groups; Springer 815 (1980).

[Sp]

Spaltenstein, N.: Classes unipotentes et sous-groupes de Borel; Springer 946 (1982).

[St]

Steinberg, R.: Conjugacy classes in algebraic groups; Springer LNM 366 (1974).

IS]

Springer, T.Ao: Trigonometric sums, Green functions of f i n i t e groups, and representations of Wey! groups; Invent. Math. 36 (1976), 173-207.

SOME EXAMPLES OF HOCHSCHILD AND CYCLIC HOMOLOGY

Jean-Luc Brylinski* Brown University Department of Mathematics Box 1917, Providence, Rhode Island 02912, U.S.A.

The theory of algebraic groups and their representations has made important progress in the last decade;

let

us

point out two

remarkable aspects of this progress. l) the use of sophisticated (co)homology theories like ~tate cohomotogy and intersection cohornology, in the work of Deligne, Kazhdan, Lusztig, Springer, and m a n y others. On the other hand, algebraic groups actions provide most interesting examples and much motivation to experts in intersection cohomololgy. 2)

the geometric importance of n o n - c o m m u t a t i v e algebras.

It

has proven important to consider the Sprin~er resolution, which is the cotangent

bundle

of

the

fla~

variety,

as

the

"shadow"

of

a

n o n - c o m m u t a t i v e object, the algebra of differential operators on the flag variety. In this article, we somehow combine both themes, by looking at the cyclic homology of some interesting n o n - c o m m u t a t i v e algebras.

We

mostly consider two sorts of algebras. One is the convolution algebra Cc (G) , where G is a (real or p-adic) Lie group. Together with P. Blanc, we show t h a t the Hochschild homology of that algebra is *partially supported by a National Science Foundat:on grant

34

equal to t h e d i f ~ r e n t i a b l e group homology Hdiff. (G , Cc* (G)) , where acts on

Cc (G)

This ties in, in a v e r y

via t h e adjoint action.

interesting w a y , w i t h t h e s t u d y of orbital integrals on course w i t h the orbit s t r u c t u r e of

G

G

itself.

G , a n d of

Even though cyclic

homology is in some sense d e t e r m i n e d by Hochschild homology, it is not clear w h a t the cyclic homology of Cc (G) is. One r e m a r k a b l e f e a t u r e though, is t h a t cyclic homology does m a k e a difference b e t w e e n the c o m p a c t c o n j u g a c y classes in

G and the others.

In particular, for

o-adic Lie Groups, P. Blanc and I prove an a b s t r a c t Selberg principle. w h i c h says t h e following: if e is an i d e m p o t e n t of C~ '~C (G) ' and if ~ is ' a regular e l e m e n t of G w h i c h is not compact, t h e n

ZG/G~ e(g ~ g-l) dg = 0 , where

G~

is t h e c e n t r a l i z e r of

~'

Such an "abstract Selberg

principle" w a s first proven, for G of split-rank [18

I, by Julg and Valette

], by v e r y different methods. The second t y p e of algebras is provided by relative differential

operators. The m a i n application is to obtain the Hod~e cohomology P

p,eqHq(Y , ~ y )

of a smooth projective algebraic v a r i e t y

Hochschild homology of an algebra

Y

as the

A , obtained as follows. According

to Jouanolou or Karoubi, t h e r e exists a fibration F • X

, Y w i t h fibre

a n affine space AN and total space X affine. Then A is the algebra of algebraic

differential

operators

on

X

which

only

involve

differentiations along the fibres of F . This result is joint work w i t h Jean-Benoit statement.

Bost and

Christophe Soul6;

see (2.26) for a precise

We c o n j e c t u r e t h a t the cyclic homology of A is equal to

the direct s u m of h y p e r c o h o m o l o g y groups of t r u n c a t e d de R h a m

35

complexes, and that the Hochschild to cyclic spectral sequence is the Hodge to de Rham spectral sequence for X (hence degenerates at El). I am

convinced that cyclic homology will prove extremely

valuable to geometers and to representation theorists. For example, it will be v e r y useful in the study of group actions on manifolds, where it is appropriate to introduce the "crossed product" algebra;

I have

computed the Hochschild and cyclic homology of such crossed products for a differentiable action of a compact Lie group on a manifold; the result involves an interesting auxiliary space associated to a group action, and will be described elsewhere. Finally, I will point out that the two sorts of algebras are rather similar in spirit, and hopefully the "equivariant" and "differential" themes m a y be combined in interesting ways. It is a pleasure to thank the organizers of this Symposium, which has been v e r y informative and stimulating.

In addition, I wish to

thank Philippe Blanc, dean-Benoit Bost and Christophe Soule, with w h o m I a m presently collaborating.

I a m also grateful to Joseph

Bernstein, Laurent Clozel, Alain Connes, Hetene Esnault, Pierre Julg, David Kazhdan and Alain Valette for useful discussions. In particular, A. Connes pointed out, in the Fall of 1985, that the algebra of relative differential operators on a Jouanolou-Karoubi fibration should have interesting cyclic homology.

51

Convolution ~rouD algebras All the work described below is joint with Philippe Blanc.

~1.I

Discrete ~rOUDS Let k be a c o m m u t a t i v e ring, with unit, G an abstract group,

k[G] the group algebra. For ~ ¢ fi, we denote bv g the corresponding

36

e l e m e n t of k[G] . A left (resp, right) k [ G ] - m o d u l e is t h e s a m e t h i n g as a

k - m o d u l e equipped w i t h a k - l i n e a r left (resp. right) a c t i o n of

Hence, t a k i n g

k

G .

as a base ring, a k[G]-bimodule is a k - m o d u l e

equipped w i t h a k - l i n e a r left a c t i o n of G , a n d a k - l i n e a r r i g h t a c t i o n of G , w h i c h c o m m u t e w i t h e a c h o t h e r .

If M is a k[G]-bimodute, w e

let Mad be t h e G - m o d u l e M , o n w h i c h

g e G acts by

m )

~ g m g-1

The following proposition is d u e to C a r t a n - E i l e n b e r g

[lO, C h a p t e r lO].

Proposition 1.1.!

For a n y k [ G ] - b i m o d u l e M , one has: H . ( k [ G ] , M) = H . ( G , M a d ) .

Here t h e first group is a Hochschild h o m o l o g y group, t h e second is g r o u p homology. Let us point t h a t t h e s t a t e m e n t is obvious, since b o t h sides a r e d e r i v e d f u n c t o r s of t h e f u n c t o r M

~H O ( k [ G ] , M ) = M / { g m - m g ;

= M/{gmg-l-m;

m c M,g

We a r e i n t e r e s t e d in t h e b i m o d u l e k[G]ad

m ~ M,g

¢ G} = H o ( G , M a d ) . M = k[G] .

is e q u a l to t h e d i r e c t s u m , o v e r a d j o i n t orbits

k[@]ad ; n o w if x ~ @ , t h e n

c G}

@ is i s o m o r p h i c to

The G - m o d u l e G

of

G , of

G/G x as a G-set,

h e n c e k[~]ad is i s o m o r p h i c to k[G/Gx] , By Shapiro's t e m m a [6, Prop. 6.2], H.(G , k[G/Gx]) ~ H,(G x , k) . Hence if X is a s y s t e m of r e p r e s e n t a t i v e s for t h e a d j o i n t a c t i o n of G on itself, w e o b t a i n

Corollary~$.l.2

H.(k[G]) ~

ff~ H.(G x , k ) . xeF,

This result, is d u e to B u r g h e t e a [9

]. However, his m e t h o d does

n o t s h o w t h e e l e m e n t a r y n a t u r e of t h e c o m p u t a t i o n . B u r g h e l e a also c o m p u t e s t h e cyclic h o m o l o g y ~ r o u p

H C~(k [G]) ,

37

for

k

a Q-algebra.

acts on

We wilt s t a t e his r e s u l t as follows. The group

Gx , t h e g e n e r a t o r of

Z

acting b y m u l t i p l i c a t i o n by

Z x ;

hence, up to h o m o t o p y , S 1 = BZ acts on BG×. The r e s u l t is S1 ® H. (BG x , k ) .

HC.(k[G])-

x~Z

S1 If x is of finite order, Hi (BG x , k) ~ H.(BG x , k) ® H . ( B S i) . If x h a s infinite order, setting N x -- Gx / x z , we have: 1 HS

(BG x , k )

= H.(BN x , k ) .

Up to h o m o t o p y , we h a v e a fibration S 1 = B 7/ ........~ B Gx

l B Nx .

The Gysirl e x a c t s e q u e n c e for this fibration: H . - I ( B N x , k)

, H . ( B Gx, k)

* H . ( B N x , k)

~ H.-2 (B N x , k) ~ - -

is a d i r e c t s u m m a n d of t h e exact s e q u e n c e of Connes [13 HC._I(k[G] )

B , H.(k[G], k[G])

I

, HC~(k[G])

], [24

]

S , HC._2(k[G] )

In p a r t i c u l a r , B a n d S h a v e a clean g e o m e t r i c i n t e r p r e t a t i o n . It is useful to c o n s t r u c t an explicit i s o m o r p h i s m b e t w e e n H.(k[G] , M) a n d H . ( G , Mad) , for M a k[G]-bimodule, a n d to use it to c o m p u t e

B

explicitly.

The first group is t h e h o m o l o g y of t h e

Hochschild complex M ® k [ G i÷l] k where b(m ® (gO''"'~;i))

~ M ® k [ G i] k

= (mgo) ® (g!'""gi)

)

38 1-1

(-1) J+l

7.

m

® (go

' "'''

j=O

(gi m) ® (go ' '"

+ ( -1)i+1

gj+l

gj

' "'''

gi

' gi-1) •

The second group is the homology of the s t a n d a r d complex

, M®k[Gi+l]-k

d ~ Mek[G i]~-"

k

d ( m (~ (go'''''gi)) = (go1 m g o ) ® ( g l ' . . . . ,gi)

where

i-I ' i + E (-i)J+ m ® ( g O ' " ' g j g j , l ' " ' g i j=O +

(-1) i+1

An isomorphism

m ® (go

'

....

gi-1) •

~0 f r o m the first complex to the second is

c o n s t r u c t e d as follows: ~ ( m ® (go . . . . .

gi)) = (go " gi m) ® (go

.

.

.

.

.

gi) •

Next, for M = k[G], one m a y c o m p u t e the operator B/= ~ o B o

~ - l . k [ G ] ® k [ G i] k

, k[G]®k[G i+11. k

One finds, working modulo d e g e n e r a t e cycles, i.e., those (go (D ( g l ' ' ' ' g i ) )

such t h a t gj = 1 for some ,j w i t h 1 <_j_< i

B/(go ® (gl '' =

gi))

i ij -1 -1 -1 -1 (-I) (gi_l...g I g0gl...gi_l) ~ (gj ..... gi' gi "gl go' gl .....gj-I) j=l

+ (g[~ .... g~l go g~ '

g~) ® (g;~ '

~: %' g: ' ....

F r o m this f o r m u l a , it is a p p a r e n t t h a t for orbit of x , B / m a p s

k[@]® k[G i] into k

' g~)

@ c G t h e adjoint

k[@]® k[G i+l] ; we denote k

/ again B x the induced m a p on group homology

39

/

Bx: H i ( G ,

k [ ~ ] ) ...... ; Hi+I(G, k [ ~ ] ) .

For each x ¢ Y., let 0 be its adjoint orbit, and define Bx" Hi(G x , k)

>

Hi+I(G x , k)

b y decreeing t h e following d i a g r a m

commutative

H i(G x , k ) ~ H

i(G,k[G]) /

Hi+I (Gx , k) ~ where a

Hi+I(G, k[O])

is Shapiro's isomorphism.

To describe explicitly subgroup, M

a , recall t h a t if

F

is a group, A

a (left) A - m o d u l e , the (produced) F - m o d u l e

t h e group of all m a p s F : F (i) F ( x . g ) = 8 -I .F(x)

PFA(M)

a is

~ M such t h a t for a n y x e F , g ¢

A

(ii) F has finite support modulo A . Then the m a p c( • M ® 7/[A i]

* PFi (M) ® :Z[Fi] is defined by

Z

a(m

® (gl ' ' ' ' g i ) )

Z

= a(m) ® (g1'''''

gi)

where

a(m)

is the

function f r o m F to M such t h a t (a) o¢(m)(g) = g - l m

for g :

A

(b) a ( m ) ( g ) = 0 for g c F \ A . Then

a

gives a m o r p h i s m of s t a n d a r d complexes;

the induced

m a p on homology is Shapiro's isomorphism. With the help of this c o n c r e t e o: , one easily verifies the following f o r m u l a for Bx :

Bx(gl,.--., gi) ÷

= ~ (_ 1) i j (gj ..... gi' gi- I "gl-1 x, gi' "' g.j-1 ) j=l ....

x.

gl

,.

....

a)

40 Example 1.1.3

For i = I , Bx(g) = (g-1 x, g) - (g, g-I x) .

For i= 0,Bx(~) : x E H I(G x). (Once again, all t h e s e f o r m u l a e a r e given m o d u l o d e g e n e r a t e elements.)

§ 1.2

Lie groups Let F be a non-discrete locally compact topological field, whose

topology is defined by an absolute value I

I . A Lie ~rouD over F (or

F-Lie group) is defined as in [6, ~i.i]. W e m a k e no assumption on the n u m b e r of connected components of a ~- or C-Lie group. In [3, 3°], P. Blanc introduced the category

~G

of differentiable

G-modules, for G

a real or corrlplex Lie group. Let us recall that an

object M

is a complete locally convex topological vector space

over

of ~ G

¢ , endowed with a group h o m o m o r p h i s m

p •G

~ Aut(M) ,

satisfying t h e following conditions: (D 1)

if C c G is c o m p a c t , p(C) is e q u i c o n t i n u o u s

(D2)

for a n y

m E M, the function g

~ p(g) . m

is

differentiable (J_D3)

the natural map

M

, C ~ (G, M) , sending m

to t h e

f u n c t i o n g ~ p (g) . m , is c o n t i n u o u s . The fact t h a t

J~G

is n o t a n abelian c a t e g o r y c o m p l i c a t e s t h e

algebraic h o m o l o g y of JDG . For

F

non-archimedean,

we let

J.DG be t h e c a t e g o r y of

admissible G-modules, in t h e sense of H a r i s h - C h a n d r a .

An o b j e c t of

JDG is a c o m p l e x v e c t o r space M , w i t h an action of G , w h i c h is s u c h t h a t t h e stabilizer of a n y v e c t o r of M is open in G. We will also talk of differentiable (instead of admissible) modules. JDG is n o w a n abelian category,

41

The definition of the differentiable homology, due to P. Blanc in t h e real case, m a y be extended to this m o r e general context as follows: Hidiff (G, M) is t h e homology in degree i of the complex of [loc. cit., 6"] ""

'Co (Gn+2

w h e r e (a 4) (go . . . . .

Cc (Gn÷l

M)G

>

gn) =

=n~ 1(_1) i I %b (gO' "''' gi-I ' g ' gi ' "'' gn ) d g. i=O O See also [12

] for another

approach

to differentiable homology.

Here d g is a fixed left Haar m e a s u r e on

G , C~-functions are

a s s u m e d to be locally c o n s t a n t in the n o n - a r c h i m e d e a n case, and

NG

is the space of c o - i n v a r i a n t s of the G-module N ; i.e., N G = H0(G, N) . One m a y shift, as usual, f r o m this "homogeneous" complex to the "inhomogeneous" complex. Consider the isomorphism OO

'C c (Gn+l , M) o

~

,C O (Gn , M )

defined by -i

(15 4) (gl,--.,gn)

= f~*(g,ggl,ggl

g2,.--,ggl-.,gn)

dg.

Using this isomorphism, the previous complex t r a n s f o r m s into ...

where

C oo c (Gn+l

(d F)(gl ,..., gn)

n

G

~ ...

= fG g-IF(g,gl,...,gn)

j

+ ~ (-1)J

j=l

, M ) ' cl ~ C c,~ (G n , M )

dg

-I F ( g l ..... g j - i '

g' g

gj' gj+l'

"" g n ) d g

+ (-l)n+l ~G F (gl ,- .... gn, g) d g.

42

Now let

complex-valued product

(£0 .

C~ (G)

be t h e algebra of c o m p a c t l y s u p p o r t e d

functions

on

G

, endowed

~(g/) q;) (g) = [G -

with

convolution

g) d g/

~(g/-1

n o n - a r c h i m e d e a n case, no topology need be p u t on a r c h i m e d e a n case,

the

C~

In

the

Cc (G) . In the

C0 (G)

is given its u s u a l locally c o n v e x topology,

for w h i c h it is complete.

Therefore, w h e n we speak of Hochschild

h o m o l o g y of t h a t algebra, we use c o m p l e t e d t e n s o r p r o d u c t s , as in [11, It, $5], in case F is a r c h i m e d e a n .

Proposition 1.2.1

For M an object of ~DGxG, one has: H.(C c (G)

Notice t h a t for

M) ~ Ndiff (G Mad)

M ¢ aOGxG , M

i n s t a n c e , t h e left action of C" c (G) is: F . m t h e discrete group case, studied in

oo

is a =

Cc (G)-bimodule.

I

F(g) g . m d g .

For As in

G

81.1, t h e proposition follows f r o m

a n i s o m o r p h i s m of complexes given by: c p ' M @ Cc (G i+1) given by

~(F) (S0 . . . . .

f o r m u l a as in ~1.1 (~

> M • C c (G1*1)

gi) = g O .. • g i F ( g 0 . . . . .

gi) ; i.e., t h e s a m e

is a m a p f r o m t h e Hochschild c o m p l e x to t h e

s t a n d a r d complex of g r o u p homology). Now w e c o n c e n t r a t e on t h e birnoduie M = C c (G) . The definition of B in [13, II, ~3], [24

] is not applicable, since o u r algebra h a s no

unit, a n d B involves a h o m o t o p y s for t h e cyclic c o m p l e x (C~ (G i+1) , b / ) , w h i c h uses a unit.

However, one m a y c o n s t r u c t

43 another map

with.[

q "C c (G)

C c (G2) , f r o m t h e choice of

u e Cc (C)

u(g) d g = I (u m a y be v i e w e d as a n " a p p r o x i m a t e unit") as G

follows:

(qF) (gl'g2) = u(gl ) F(gl g2) " (It should be possible to e x t e n d

o

to a homotopy, but we have not

w r i t t e n this down.) In a n y case, in degree 0 , t h e m a p B" H0 (C c~ (G), C c'~ (G))

~ Hi (C c (G), C c (G)) is t h e n given b y

B = (l-t) ~ o. So we find (BF) (gl ' g2 ) = u (gl) F (gl g2 )- u (g2)F (g2 gl ) " So t h e m o r p h i s m

¢J0

OCP

B / ' C c (G)

(*O

O¢

~C c ( G ) ~ C c (G) = Cc (GxG) defined b y

B/= q) o B o ~ - I = ~ o B , i s : (B/F) (gl ' g2 ) = u (g~l gl g2 )- u (g2)F (g~l) . Let

C~ (G)

o p e r a t e on

C c (GI+1) , for all

i , by m u l t i p l i c a t i o n , as

f u n c t i o n of t h e 1 s t variable; i.e.,

(F ~) (go ' "" ' gi) = F (go) . ~ (go ' " gi) "

Then t h e f o r m u l a for B / shows:

L e m m a 1.2.2

B / "C c (G)

' C~c (G2) is C c (G)-linear.

Let us now assume that

G

is (the group of

F-points of) a

connected reductive algebraic group over F. Let @ be the G-orbit of a re~_ular s e m i - s i m p l e e l e m e n t G x is a C a r t a n s u b g r o u p of G .

x c G , so t h a t

O is closed in G a n d

A p a r t f r o m a bit of n u i s a n c e c o m i n g

44

from

stable conjugacy

versus

conjugacy,

Lemma

1.2.2 i m p l i e s

the

e x i s t e n c e of a f a c t o r i z a t i o n B / H d iff (G, C c (G)) H d iff (G, C c~ (G)) ---*

H iff (G, C c (~)

~ H iff (G , C c (~)) .

As in 51.1, it is a p p r o p r i a t e to i n t r o d u c e S h a p i r d s i s o m o r p h i s m H diff (G , C c (0)) ~ B x • Hd iff (Gx , El) be t h e i m a g e in

H cliff i (G x , El) (cf [3, nO II]) . W e then define ~ ~" id+i fl f (Gx , El) as

Bx = oC1 o B/x o ~x . N o w let

HIdiff (G x , El) of t h e c a n o n i c a l e l e m e n t of

~,

H i ( Z , ¢) ;

let 11 be t h e c a n o n i c a l e l e m e n t of Hdiff (G x , El) .

Lemma

1.25

B x (11) = ~, e H cliff (G x , El) ,

This extension of Example 1.1.5 to Lie groups requires an explicit d e s c r i p t i o n of S h a p i r o ' s i s o m o r p h i s m .

For this, one

uses

t h e m e t h o d of

" e f f a c e m e n t " : for M e ~ G , one i n t r o d u c e s t h e e x a c t s e q u e n c e 0

, A

~ C c ( G , M)

subgroup of isomorphism

Hi-I(G

~M , A)

inductively.

measurable decomposition w e get a m e a s u r e on Y

such

C~-function

that

u

Iy

on

~ 0. ;

Then

this

The final

allows

H i ( G , M) to

construct

G = Y . Gx , a H a a r m e a s u r e on

X(y)dy

=

G such that %

and

i

Shapiro's

formulae i n v o l v e choices:

Y) , a c o m p a c t l y - s u p p o r t e d

u s e t h e s e choices a n d let

identifies a s a

and

a

a

Gx ( h e n c e

C~ - f u n c t i o n

X on

compactly-supported

~G u (g) d g = 1 . H o w e v e r , one m a y u

t e n d to '8-functions"

this in

45

some sense reduces the computation to a maximal discrete subgroup A

of C~ such that G x/A

averaging over

is compact (this reduction is obtained by

Gx/A)

So in fact no computation is ever really

necessary. This l e m m a implies t h e a b s t r a c t Selberg principle m e n t i o n e d in the

H

introduction.

iff (G , C c

explain

this,

we

first

need

to

relate

(G)) to orbital integrals,

L e m m a 1.2.4 Cc (G)

To

For a n y

x e G,•

~ H0d iff (G , Cc~ (G))

its a d j o i n t orbit, t h e composed m a p

, Hodiff(G , C{ (O)-- _--~ E is t h e "orbital o~

integral" f u n c t i o n a l

F ~

h /GxF(g x g - 1 ) d g .

This l e m m a is a tautology, since t h e Shapiro i s o m o r p h i s m is given b y

~ ~

m e a s u r e on

Gx

I@/G x %0(~)d~

<x-1

(note t h a t t h e choice of a Haar

a p p e a r s both in Shapiro i s o m o r p h i s m a n d in t h e

definition of t h e orbital integral). Now Connes h a s defined a m a p p i n g

K 0 (fi~) for a n y (topological) algebra

, H C2n ((2)

fl (see [13,

]) . If e is an i d e m p o t e n t

in Q, its i m a g e in H CO(fl) = H0 (fl, fl) belongs to t h e i m a g e of S "H C2 (Q) B "Ho((] , (1)

, HC 0 (@) w h i c h equals the k e r n e l of 'HI((~ , (D .

Here for

L e m m a 1.2.2, 1.2.3 a n d 1.2.4 t h a t for

(R = C~c (G) , it follows f r o m

x e G r e g u l a r s e m i - s i m p l e , one

has: [IG/G x e(g x g - l ) d g l ,

x should be 0 in H~iff(G x , E) . Now a s s u m e

46

x is not compact (i.e., x does not generate a compact subgroup of G). Let A be the group of l-parameter subgroups (in the algebraic sense) of Gx , defined over F. Then the absolute value morphism Gxt~g

I

I,A®~ Z

gives an isomorphism on differentiable homology.

compact,

Since

x

is not

Ixl = 0 . On the other hand, H1diff (A ® IR E) ~ A ® ~ and Z

m a p s to

Ixl u n d e r this isomorphism.

Hence

'

Z

x

is non-zero in

Hidiff(Gx, (t]) . tt follows t h a t J'G/Gx e(g x g-l) d g m u s t be 0 . Hence we h a v e proved the

Theorem 1.2.5 Let

(abstract Selberg principle)

G be a connected reductive Lie Group over

F , e

be an

i d e m p o t e n t in Cc (G) , x a regular semi-simple e l e m e n t of G w h i c h is not compact. Then

~GIG x e(gxg-t)dg = O.

The t e r m "abstract Selberg principle" was coined by Jutg and Valette, who established it for groups of F-split r a n k one [18

]. They

use analysis on t h e Bruhat-Tits tree, and Fredholm modules (in t h e sense of Connes), The "concrete" Selberg principle is deduced by taking e(g) = X< ~(g) v , v *

> where (~,V)

e

to be

i s a cuspidal representation of

G (here F i s n o n - a r c h i m e d e a n ) , ~ ¢ V , v* E V* w i t h ( v , v*) = O, and

X

is a suitable constant.

i d e m p o t e n t in

Indeed for

CC~ (G)(cf [11, T h e o r e m e 1.1])

t h e Selberg principle, see [16

].

% well-chosen, e

is

For a classical proof of

47 The e x a c t c o m p u t a t i o n of

H.

iff(G ,

Cc (G))

is a n open p r o b l e m .

For i = 0 , t h e following s t a t e m e n t : "F ¢ Cc (G) integrals

of

F

archimedean

[4

h a s z e r o i m a g e in are ];

H a r i s h - C h a n d r a [:1.5 For

zero",

H iff(G , Cc (G))

is c o n j e c t u r e d

t h e case

F

by

iff all orbital

P. Blanc

non archimedean

for

F

is d u e to

].

F n o n - a r c h i m e d e a n , let

U c G be t h e open set of r e g u l a r

s e m i - s i m p l e e l e m e n t s , Y = G- U . Let

S

C ~ (G) G

be t h e m u l t i p l i c a t i v e s u b s e t of

formed of functions

w h i c h v a n i s h n o w h e r e on U .

Lemma

1.2.6

The

C ¢~ (G) G - l i n e a r m a p

~ H ff(G , C c (U))

d~ff , c ~ , H i (G "c (G))

b e c o m e s a n i s o m o r p h i s m a f t e r localizing a t S . Indeed, t h e r e exists a f u n c t i o n - - t h e f u n c t i o n A - in w h i c h v a n i s h e s e x a c t l y on S.

Y.

Hence

C ~ (G) G - -

A kills C c (Y) a n d belongs to

The l e m m a follows t h e n f r o m t h e e x a c t s e q u e n c e 0

'C c(U)

~C c (G)

The interest of this l e m m a

, C c (Y)

~-0.

is that C c (U) is computable.

Indeed,

since U=

where

11 (G T modconj

N(T) T/) x

T is a C a r t a n s u b g r o u p of G, T / c T is t h e r e g u l a r subset, one

obtains H diff

i

(G, C c~ (U)) =

•

Tmodconj

H diff ]

(N(T)

'

C~ c

(T/))

48

@

[Hi (T, C) ® C~c (T/)]W (T)

T mod conj where W(T)= N(T)/ T acts diagonally on HI(T, ~ ) ® C c (Ti). The complete computation of H iff(G , C c (G)) will require at

least a clever use of a stratification of the Springer resolution of the nilpotent variety, It appears to be a quite challenging problem.

49

$2

Algebras of relative differential operators

2.1

Differential operators on a ~mooth affine v a r i e t y Let

D(X)

be t h e algebra of differential operators (of a r b i t r a r y

finite order) on a smooth affine algebraic v a r i e t y over a field characteristic

0

The Hochschild homology of

c o m p u t e d by Kassel a n d Mitschi [22

T h e o r e m 2.1.1

,

H i (D (X)

D(X)

k

of

has been

1.

2n-i D (X)) = t4 ~'DR (X) if X is of dimension n .

tt seems n a t u r a l to believe t h a t t h e Connes spectral sequence degenerates, so t h a t

2n-i (X) ~ HD2D2D-i÷2 2DR (X) ~ . . . HCi(D(X)) ---- HDR

but this appears not to h a v e been established as yet.* Because it is our purpose to generalize Theorem 2.i.l to algebras of relative differential operators, we will a d u m b r a t e a proof of 2.i.l. H i(D(X) , D(X)) is t h e homology in degree -i of t h e complex L

K" w h i c h belongs to t h e

=

D(X) D(X)®D(X )o D(X) ®

derived c a t e g o r y of bounded complexes of

k - v e c t o r spaces. Concretely, this is c o m p u t e d as

M"

®

D(X)

D(X)®D(X)" where M n = M- n , for . . .

, M i - - ~ Mi_ 1

~ ..

~ M0

a bounded projective resolution of the right D(X) ® D(X)°-module D (X) (recall D (X) ® D (X) ° has global homotogical dimension 2n , (*)I h a v e j u s t proved this, using t r a n s c e n d e n t a l methods.

cf[2

]).

50

We w a n t to sheafify this complex; i, e., to consider the complex of L sheaves ~X @ /)X' w h e r e ~X is the sheaf of germs of o

DX~D X algebraic differential operators on

X . A priori, this belongs to the

derived category of bounded complexes of sheaves of k-vector spaces on X x X. However, it m a y be described as ~"

®

~Dx , where ~" is a bounded projective resolution of

~x ~ ~x ~X as a right ~X [] ~X-module and ~" is defined by ~n = ~_n J as above. Each ~i

®

~X is a direct factor of (DX)k , hence is Q

D X []D X supported on the diagonal X % X x X. Therefore, we m a y view IT. X" = ~X ® DX as a complex of sheaves on X.

~X []DX

Lemma 2i.2

K" is isomorphic to ~ F(X, X') .

In fact, each

~X is acyclic for F (X --) since X is

® o

D x E]D x affine;; so ~ F(X, X') is computed from the complex ...--+ F(X, %i

®

,Dx ) - - ~ F ( x , % ~ + i o

DX m a3X Now let M" = F ( X x X , ~ I ' ) . m'

since

Mi

is a projective

® Dx )_~ Dx ~ DX

Then F ( X , ~ " ® ~X) is equal to

® D(X) D(X)®D(X) ° D(X) ® D(X)'-module, this complex is a

realization of K", q.e.d. Now let us recall some general concepts about

51

duality for complexes of modules over a non-necessarily c o m m u t a t i v e noetherian, unital ring A . We will a s s u m e t h a t homological dimension.

A has finite global

Our complexes of A-modules will be bounded

complexes of finitely-generated A-modules. If M" is such a complex of left (resp. right) A-modules, its dual a d e q u a t e derived category -- as

(M')*

is defined -- in t h e

~ HomA (M" , A) . It is a complex of

right (resp. left) A-modules, t h e action of A coming f r o m t h e action of A

on

may

A by right (resp. left) multiplication. To c o m p u t e it (M')*, w e resolve

M"

by

a

bounded

complex

P" of projective,

finitely-generated A-modules. Then (M')* is realized as the complex ..

* H o m A ( P i , A)

,HomA(p-i-l,A)

1

t

0 di

d o (i+l)

We h a v e a canonical isomorphism

" ',...

M" ~ ~((M')*)* . The following is

also obvious:

immmm2i2

Let M'(resp. N') be a bounded complex of

finitely-generated right (resp. left) A-modules.

There is a natural

isomorphism L N. ~ ~HOmA((M). * , N ' ) . M" ® A These considerations also hold for A a n o e t h e r i a n sheaf of rings (in t h e sense of [21

]), of finite global homological dimension. One

t h e n considers derived categories of bounded complexes of coherent (sheaves of) A-modules. Now in t h e case of the sheaf

~Dx, t h e r e is an equivalence of

52

categories between left 4~x-modules and right 4Jx-modules (see [20, $1] n,

for details). Here cox = 9X is t h e canonical sheaf, w h e r e n = d i m (X) . Since ~DX is a right JJ X ~ aJX-module (i.e., a Pl 1 ~ x - r i g h t module and a p2i ~ x - I e f t module), = P2 coX P2*®~X ~X is a 4JX ~ i~ X = 4DX x x - r i g h t - m ° d u l e .®

*

Similarly

~(X2)= P2(CO~-I) P2 @X °DX is a left aJXxX-module.

Lemma 2.1.4

L ®

¢Ux

~X identifies w i t h o

~x ~ ~x

• x( I )(9"-

~(2) JJx

~x,x

Proof:

Using (2.1.5), It

~Dx

.

~Dx ~ IR Horn

(9 o

~x ~ ~x

. ((JDX) , ~DX)

~Ox B D x *

-= ~ H°m~x ~ ~x ((P2

COX)®-I -(2) Pg~x2 (~x) ' 4DX ) *

_= [q Hom~xx X ((D(1)). ~2)) ~(1)

-= dOx

tL

4(2)

(9 ~)X ~Xx X

L e m m a 2.1.5 isomorphic to variety.

The left

~H xn (Ox xX)

~Dx x x - m o d u l e where

Ax

c

2) XxX

is canonically is the diagonal

53

The Grothendieck-Cousin fundamental class of A X c

Proof;

is a n e l e m e n t

~ of tt~X (p~ co x) , h e n c e w e h a v e a n

p2(co~ -I)

map

tt~ x (@X x X) , w h i c h e x t e n d s to a

XxX

@X x X - l i n e a r

~Dx x x - l i n e a r

morphism

(co1)xxx ®

where

, Hn

-ZXx (OX x X)

~Dx x X is v i e w e d as a r i g h t m o d u l e o v e r itself.

show that

this morphism

® Ico$11oxxx @Xxx,andif

~DX

xx-

factors through

But

(x I , . . , x

coordinates

It r e m a i n s to

the quotient

~(X2)

of

~f is killed b y t h e ideal of A x in

n) a r e local c o o r d i n a t e s on

(x I , . . , x n ; Yl , . - ,

point is that ~f =

/)X x X

Yn) on X x X n e a r

X,weget

local

A X , and the

8 and 8 have the same action on 8xi 8yi

dYl A - - - A dYn (Yl-Xl) . . . . (Yn-Xn)

__ ODDOSite of Lie d e r i v a t i o n ) .

simDle h o l o n o m i c

(notice

8 @y i

Since

2~X2)

a c t s on ~f b y t h e

and

Hn ,-,ZX x (@X x X)

are both

aDX x X - m o d u l e s , t h e m a p b e t w e e n t h e m , w h i c h is

n o n - z e r o , is a n i s o m o r p h i s m , q.e.d. The c r u c i a l l e m m a is t h e following

Lemma 2.1.6

The dual (~D~I))* of ~i~(1) is isomorphic to

Han X (O X x X) [-2n]

Proof:

Since locally

.D(1)

is isomorphic to

JD(2)

(the monkey

54

business of left and right m a y be a r r a n g e d locally), it is a simple holonomic right holonomic left

,DXx x - m o d u l e . .Dxx x - m o d u l e

Hence

( 4 1 ) ) * [-2n]

is a simple

(see [20] or [2] for the duality on

Its characteristic v a r i e t y is T* (X x X) AX

holonomic modules).

Hence

it is locally (on X ~ a X) isomorphic to Hna x (Ox x x) • The sheaf of germ of a u t o m o r p h i s m s

of the

.Dx x x - m o d u l e

ttRX_(Ox x X)

constant, equal to k* . The l e m m a t h e n follows from

is

Hl ( x , k*) = 0;

recall a constant sheaf is flasque, for the Zariski topology.

N o w use the Japanese notation

~BAx/XxX = HRx(Oxx x). W e

have obtained so far: IL

~x

_(1) =

® o ~Dx a ~Ux ® ~X [] ~Dx

= RHomh xx (

,

)[2n] .

Recall the following t e m m a , w h e r e for Y a smooth v a r i e t y ,

~

a left

~Dy-module, D R (Tll) denotes the de R h a m complex: ~l d Q~, ~y ~]t

~

"''

d,QdimY O Oy

! dO(- dim Y)

dO0

i

Lemma

2./.7

If Z ~ Y

over k , D R ( ~ Z / Y )

is a closed immersion of smooth varieties

is quasi-isomorphic to i . ( Q z ) [ d i m Z].

This l e m m a m a y be extracted from [20, ~4

].

55

We t h u s obtain a m o r p h i s m of complexes of sheaves: L "DX ® "DX -* ~ H ° m x x x ( i * f~X' i*f2X)[2n]" DX m DO Using t h e inclusion k % QX , w e get a m a p

~x

E ®

0

~X -~ I R H ° m x x x ( i .

k,

Qx)[2n) = Q x [ 2 n ] .

~x m ~X

Proposition 2.1.8

L ®

DX Dx

m

o

~X --* £2X [2n]

Dx

is a quasi-isomorphism. R e m a r k : This m a y be viewed as a r e f i n e m e n t of T h e o r e m 2.1.1.

Proof (sketch):

The question being local on

t h e r e exists a n etale m o r p h i s m x X is open and closed in

%o: X

X , one m a y a s s u m e

~ AI~. The diagonal A x

c

X

~0-1 (AAn) . In w h a t follows, we consider

sheaves w h i c h a r e supported on ~-I (AAn) , and w e r e s t r i c t t h e m to a neighborhood of X in X x X , so t h a t we m a y b e h a v e as if A x w a s equal to tp-I (AAn) . With these considerations in m i n d , w e let (x I , . . . ,

Xn) be t h e

s t a n d a r d coordinates on A n , h e n c e on X . On X x X , w e let (xl , . . . , Xn) (resp.(yl . . . . .

Y2)) be t h e coordinates on the first (resp.

second) copy of X . The ~ × [] ~ x - r i g h t m o d u l e ~X is t h e quotient of IDX [] ~X t h e right ideal g e n e r a t e d by

56 . ~ (xl-Yl .....' Xn-Yn ' ~Xl

8 ~Yl

a ' .... '

~Xn

a.). ~/n

These elements c o m m u t e with each other; to see this, n o t i c e t h a t if a and

b belong to

DX , w e h a v e :

their commutator

in

[a , b] 0 = - [ a , b] , w h e r e

~ X , [a , b] 0 t h e i r c o m m u t a t o r

in

[a , b] is

~ X ; hence,

e.g., 0 ]0 = ix I

[xl-Yl ' @xl =

[x I

~Yl

--~-~] - [ y l ' @Xl

__~_~] + [Yl "-~-~] ' @Xl ' ~Yl 0

---~-~ } = - 1 + 1

=

@Yl

'

O.

Hence one m a y introduce the Koszul complex K (i~X [] /~X" ; xl-Yl ' "" Xn-Yn ' @x I

@Yl

*..j

~Xn

~Yn

o

this is a complex of right D X [] ~DX-modules.

L e m m a 2.1,9

"

",

a

K (JDx [] ,Dx , x 1-Yl ,

....

_0)

' C)Xn C)Yn

is a r e s o l u t i o n of ~ X - Since this Koszul c o m p l e x consists of p r o j e c t i v e m o d u l e s , JDx

L ®

,Dx is realized b y t h e Koszul c o m p l e x o

DX ~] DX K'(D x

O ; Xl-Yl '"' Xn-Yn '. OXl

for t h e left

,DXl~,DX-module

@Yl

,..,

~0X .

8Xn

OYn

)

We will a n a l y z e t h i s Koszul

c o m p l e x in t w o steps; d i v i d i n g t h e s e q u e n c e @ @ ) into the subsequences ( x l - Y l , . ......, ~x n @Y (Xl-Y 1, "'"

Xn-Y n) a n d ( ~ - c~ c) 8Xn @Yl ' "'"" ~Xn

c~ .) . C)Yn

57

Let L" : K'(DX ; x l - Yl , . . . . quasi-isomorphic

to t h e

, Xn

-

Yn); t h e n o u r Koszul c o m p l e x is

simple c o m p l e x d e d u c e d f r o m

t h e double

complex •

8

K" (L", Ox1

~

a

Oy 1

Oxn

~.) 65Zn

So first w e s t u d y L" w h i c h m a y be d e s c r i b e d as t h e c o m p l e x

' An-i(kn) @ ~ X ''~ ' An-i+l(kn) ® ~ X k k

I

I

d0-i where

)

d o (-i+l)

8 is d e s c r i b e d as follows. Let

(e I , . . .

, e n) be t h e c a n o n i c a l

basis of Kn ; t h e n w e h a v e : n

8((e k A . . .

A el) ® P) = 7. ((e, A e k A . . . j=l

T h e r e is a m a p of c o m p l e x e s @x[n] Fe ~X

(note t h a t

L e m m a 2.1.10

[xj,F];

Ox[n]

A el) ® [ x j , P ] ) .

J

, L". w h i c h m a p s

(~'I ' " ' " ' ~n)

to

0 for F e 0×)

~ L" is a q u a s i - i s o m o r p h i s m .

Proof: On T* X , i n t r o d u c e c o o r d i n a t e s (x 1 . . . . where

F ¢OX

a r e d u a l to

(x I ,...,

' Xn ; ~i ' " " " ' ~n) , x n) . Filter

L°

by

subcomplexes L" (m) •

~

A n-i (k n) @ ~DX (m-i)

,

An-i+l(k n)

I dO-i

dO(-i+l)

~

~Dx(m-i+i)

the

58

Filter

OX in t h e stupid way:

@x[n] (0) - Ox[n] , @x[n] (-1) - 0 .

Then Ox[n] ........, L" is a m o r p h i s m of filtered complexes. We identify gr(L') = ~ L'(m) / L ' ( m - l ) differential forms on

w i t h t h e complex C2T.X/X[n] of relative

T*X , w h i c h is graded by t h e degree of

h o m o g e n e i t y along t h e fibers of

T*X

, X Precisely, w e m a p t h e

element (e 1 A . . .

A en_ i) ~) P

of

An-i(kn) ® JDx(m-i)

differential form ~m_i(P) d ~1 A . . . .

to the (relative)

A d ~n-i • To show t h a t this is

compatible w i t h the differentials, w e note t h a t (Ym-i-1 ([Xj , P]) =

Since 0 x

___a ~E,j

o m - i (p) "

' QT*X/X is a quasi-isomorphism, Ox[n]

; L" is

a filtered quasi-isomorphism. From L e m m a 2.1.10, w e deduce t h a t .

a

@Yl

, ..., @ @Xn

complex K (0 x ,

a

K" (L", axl

ax 1 Since ~xi

~ ) is quasi-isomorphic to the Koszul 8Yn ,,,-,

ay I

a ~x n

a.) ay n

, P] , it acts the same ~ acts on ~X by P ~ [__~_b aYi ~xi

w a y on (~xC ~ X ;

for F E 0 X [ c) ,F] = c3__F_F£ (~X" ' ax i ~x i o

Hence this complex is just QX [2n]. Hence we have found, locally on X , a quasi-isomorphism L J'DX ® J'DX --* QX (2n) ; it is still necessary, in order to prove

~X El DX

59

(2.18), to compare this quasi-isomorphism with the morphism defined before (2.1.8); we will neglect here to do that.

60

~2.2

Relative differential o p e r a t o r s In this section, we describe s o m e j o i n t w o r k w i t h J-B. Bost a n d

C. Soule. In t h e last section, we a n a l y z e d t h e c o m p l e x of s h e a v e s IL ® ~Dx , for X a s m o o t h algebraic v a r i e t y o v e r k .

~Dx

o

~X m ,Dx Here w e consider t h e algebra

~DX/Y of r e l a t i v e differential o p e r a t o r s

associated to a s m o o t h m o r p h i s m

F :X

k . We m a y define t h e sheaf of algebras

, Y of smooth varieties over ,DX/y c ~ X in either of the

following ways:

(i) ~DX/y is t h e s u b - a l g e b r a of b X , f o r m e d all e l e m e n t s w h i c h c o m m u t e with F -1(@Y) c O X .

(ii) ~DX/Y is the sub-algebra of dDX generated by @X and by TF C Tx , t h e sheaf of g e r m s of v e r t i c a l v e c t o r fields on X . ,Dx/Y is filtered as s u b - r i n g of ~Dx a n d we h a v e gr (~DX/Y) = @ T * ( X / Y ) ,

where

T*(X/Y)

is t h e r e l a t i v e

c o t a n g e n t space,

dDx / y

is a

].

n o e t h e r i a n sheaf of algebras, in t h e sense of [21 We n o w consider t h e c o m p l e x of s h e a v e s bx/Y

.0

X/Y

® []

°

bx/y;

~X/Y

'it is a c o m p l e x of s h e a v e s on X x X , c o n c e n t r a t e d on t h e diagonal A X . Hence, as in

A X ---= X

~2.1, w e m a y v i e w it as a complex of sheaves on

By t h e a n a l o g y of L e m m a 2.1.2, if

Hochschild h o m o l o g y of D(X/Y) = F(X , ~X/Y) complex

X

is affine, t h e

is c o m p u t e d f r o m t h e

61

RF(X, 9x/Y

We will set

(9 o 9X/Y) DX/Y m DX/Y

d = dim(Y) , n = dim(X) - dim(Y) . Following the

s a m e s t r a t e g y as in ~2.1, w e will produce a m o r p h i s m of complexes: TI.

9X/Y

®-1

@ 9 X / Y ~ ~Homg " DX/Y®DX/Y

(oos ,~S)@kg~X/S[2n+d] SxS

a n d t h e n j u s t i f y it is a q u a s i - i s o m o r p h i s m b y local computation. First, n

w e i n t r o d u c e t h e relative dualizing sheaf COX/Y = QX/Y ' °°X/Y m a y be used to t r a n s f o r m left

~X / y -modules into right

~X / y -modules.

In particular:

* 9(i) X/Y = P2 °°X/Y

~ 9 X / Y is a right 9(X x X)/(Y x Y ) - m ° d u l e ; p"2 0 x

and * ®-i 9(2) X / y = p2(COX/y)

®

9 X / Y is a l e f t

9(XxX)/(yxy)-module.

p"2 0x

L e m m a 2.2.1

9 X / y (2) is canonically isomorphic, as a left

Hn (OX~ X) 9(X x X) / (Y x Y) -module, to _AX This

is proven

just

like L e m m a

f u n d a m e n t a l class of Angeniol-Elzein [1 A x c_~ X x X of s c h e m e s over Y 46

~ X (X~X, P2 COX/y)"

2.1.5,

usin Z the

relative

], for the inclusion

Y . This f u n d a m e n t a l class belongs to

62

Lemma 9.9..2

The dual (¢DX/y(1))* of ¢DX/y(l) is (maybe

® co~- t [-d - 2 n ] . non-canonically) isomorphic to I-I<(@x x X Oy Y Idea of Proof:

Locally on X , (~X/y(1))*[+d + 2hi is isomorphic, as a

Hn ((gX~ X) left ~D(XxX) / (YxY) -module, to ,_,AX automorphisms of t h a t module is equal to

The sheaf of 8erms of

Oy

(recall t h a t

central in JDX/Y) , or more precisely to the inverse image

Oy

is

1~ - 1 0 y .

Therefore we have: n

®

(~x/y(1)) =-_H~ (0x xY x ) F. ~ (~y for some invertible sheaf

£

F -I ~[-d - 2n]

on Y .

It remains to identify £ .

For

this purpose, we endow the two sides of this isomorphism with their natural filtrations; as a simple holonomic ~DX ~ x-module, I-I~x(OX~.X)

has a natural "depth of layer" filtration (under the

isomorphism 2.2.1), this corresponds to the order filtration on ~ X / y (2) . Hence

(~X/y(1))*[d + 2n] also has a filtration.

We will

show t h a t

£ --- co~ "I by comparing the graded modules for these

filtrations.

There is a n a t u r a l filtration on the dual of a filtered

complex. For ~Dx/y (1) , the "duality spectral sequence" of [2, Chapter 2] degenerates, since gr ~Dx/y (I) is Cohen-Macaulay, and we obtain gr (~DXZy(1)*) --- gr (JUX/y(1)) * (on the right-hand side, ~ denotes duality for complexes of @T*(X/Y) x T* (X /y) -modules).

Now, let

co-normal bundle for the immersion

A

be the (relative)

X ¢-~ Xx X of

Yx Y-schemes;

63

i.e., A is a vector bundle over X , w h i c h is t h e quotient of t h e usual c o - n o r m a l bundle TX (X x X) by its horizontal part (in t h e case w h e r e F is a trivial fibration so t h a t X = Y x Z , w e h a v e

A = YxT~(ZxZ)) .

Then g r ~ x / y (1) ~ O^ @ COX/Y . Now recall t h e following simple case of Grothendieck duality (see [141). i

imnmm2.2/

Let

S % T

be a closed i m m e r s i o n of smooth

k-varieties, then the dual (@S)~ of the @T-module @S

is equal to

cos ® i* (COT e ' l ) [-d], w h e r e d is the codimension of Y in X .

W e apply this l e m m a to the embedding

A c T* (X IV) x T * (X IV) W e get (@^)~ = COA (~ COT*(X/Y) x T*(X/Y) ~(-I) [-2n - d]..We have a smooth morphism

T~(X/Y)

symplectic manifolds

T~(Xy)

, Y , the fibers of which ;

are the

since the canonical sheaf of a

symplectic manifold is trivial, w e have

coT. (X/Y) ~ coy ; hence w e

get (@^)~ = coA ® 6o~-2[-2n - d]. Similarly, the fibers of the smooth morphism A ... " Y are the symplectic manifolds

TXy(XyxXy) ~ T*(Xy) ; therefore co^ = coy and (@^)* = co~-i[-2n- d], so w e get: _(1)

,*

[gr~x/YJ

®-1

*

®-1

®-1

= coX/Y (~ (OA) = C°X/Y ~ coY

On t h e other hand, using L e m m a 2.2.1, we find:

[-2n - d].

64

gr

( ~ X (OX x X)) = gr(.Dx/v (2)) = ooX/y ® g r ( d J X / y )

It follows that

Y

/~ must be equal to co~-1 . This includes the proof of

Lemma 2.2.2. Now, just in 82.1, we have:

~X/Y

® o ~X/Y= ~X/Y ~] ~X/Y

= R

IRHorn .-, ((IJ Cl~) , & / y ( 2 ) ) '~(XxX)/(YxY) X/Y' '

®-1 H°ms(x xX)t(YxY) (~BAx/X x X ® COX ' ~SAx/X xX )[2n +d]" ¥

Now we need the following: For ~" a complex of left DX / y -module (where F :X over

, Y is a smooth morphism between smooth algebraic varieties k), we let

DR(~') = DRX/y(~')

be the simple complex

associated to the double complex i ®71, J is in degree i + j - n ) (where QX/Y Ox

i l~ 71],j+l d _i+l @~[lj+l -~ QX/Y Ox QX/Y OX

T

T

i ® ~llj d ni+l ~ qn j -+ QX/Y ~ ~'X/Y~~'~ . . . . . 0X ~x

where the horizontal differential uses the action on ~" of the vertical vector fields on X . Notice DR(~IF) is a complex of F -103Y)-modules . We apply

65

this construction to F x F ' X x X ~ Y x Y , a n d

~ = ~ax/XxX. Y

Lemma

2.2.5

DRF(~Sax/X x X) is isomorphic to i.(QX/y)[n], Y

where i • A X % X x X is the inclusion. (This is a generalization of Lemma 2.i.7.) Hence we obtain a morphism of complexes 'DX/Y

® ~)X/Y -* IRH°mOyxy(i*(Qx/Y) ® DX/Y ~ DX/Y

® COy-1 , i.(nX/V))[2n + dl. There is a

Oyx y - l i n e a r injection i . O y % QX/Yof complexes, hence

we obtain a morphism of complexes E :OX/Y @ :OX/Y ~ IRHom@yxy(i.(Qx/Y) ® DX/Y ~] ~XtY ® CO~-1 , i.(QX/y))[2n + d]. Now notice that the action of O y x y on i.~2X/Y factors through the quotient @y = @ay of @YxY. It follows that the last complex is IL . equal to IRHom@yxy(OOy-1 , @y) ® ~X/y[2n + d]. @y Now we want to prove t h a t this morphism is a quasi-isomorphism. Just as in §2.1, we m a y assume that X is etale over A n x y ; then essentially we are in a product situation and we have: ~DX/Y = ~D~n [] @ y . Then our m o r p h i s m is a tensored product of the m o r p h i s m

66

E

®

An ~

[] ~

An

~ o

An

-~ Q" [2n] . I~

An

which we know by §2.1 to be a quasi-isomorp ~ ~d of a morphism n. @y ® @y-~ NHomoyx (COy-I , @ y ) [ d ] Oyx Y Y which is a special case of the isomorphism (2.i.3).

Hence we have

found a quasi-isomorphism between E

a~x/Y

®

o

a~x/Y

~X/Y [o ~X/Y

and L NHomoy,~ (COy-1 , @y) ® ~2X/y[2n + d] .

which is the same as

(Oy

g_ n ® @y) ® Q x / y [ 2 n ] . @YxY

Now Loday and Quillen [24 IL

oy

% z -,

is

ayEil i

Oyx Y

(this

] prove an isomorphism

a

little

stronger

than

Hochschild-Kostant-Rosenberg [17 following

the

classical

computation

of

]). Hence we have proved the

67

quasi-isomorphic to ®(Qy @ QX/y)[i

Theorem 2.2.6

+ 2n].

The complex of sheaves lk ~DX/Y ® ~X/Y on X ~X/Y [] ~X/Y o

i

is quasi-isomorphic to ~ (Qy (D Qx/y)[i + 2n]. I ~y The operator B has a nice interpretationon the direct image

( 9 ~ F . ( Q y (~ Q ; / y ) [ i + 2 n ] ; then N F . ( Q ; / y ) i s a i Oy quasi-coherent complex of Oy-modules, endowed with a connection V, called the 6auss-Manin connection [23

Proposition 2.2.7

i O~ N NF (Qx/y)[ i + 2n] B maps Qy

].

to

Qi+l @ ~F.(f2X/y)[i + i + 2n], Y O~ and is induced by the 6auss-Manin connection. Details will be presented elsewhere. Now for X affine, the Hochschild homology of D(X/Y) is computed E from the complex ~ F(X, IDX/Y JDX/Y) . o

~X/Y ®~X/Y We want to explicit this for

F : X

, Y

a Jo~anolou-Karoubi

68

resolution of a smooth projective v a r i e t y

Y , i.e., F

is supposed to

satisfy

(i) X affine; (ii) F is a locally trivial fibration with fibre A n . For a n y smooth projective Y , the existence of such F :X proved by Jouanolou and Karoubi (see e.g., [25,

> Y was ]).

We first apply [RF. to the Hochschild complex of J_DX / y , and w e obtain i@.(Q

~F.(Qx/Y)

@v@~ F ( Q x / y ) ) [i + 2n] . Because of property (ii),

is quasi-isomorphic to @y, w e obtain 6B Qy[i + 2n]. i

Hence we have proved

Theorem 2.2.8

If F : X

> Y is a Jouanolou-Karoubi resolution,

then Hk(D(X/Y ),D(X/Y)) = ~H2n+i-k(Y, QX) . i

We

have not verified the following description,of the cyclic

homology of D(X/Y) , which in any case is extremely likely (as the Connes spectral sequence should degenerate). W e should have: ~2n-k+2i (y i i HCk(D(X/Y)) ~ @ , Qy -~ . . . -* Qy) a direct s u m of i

hypercohomology groups of t r u n c a t e d de R h a m complexes. The Connes

spectral sequence then would become, essentially the Hodge to de R h a m spectral sequence, and then degenerate at E l . Conversely, of course, if the Connes spectral sequence was shown to degenerate by some cyclic homology wizzard, the degeneration of the Hodge to de R h a m spectral sequence would follow. The higher algebraic K-theory of D (X/Y) is easily determined

Proposition 2.2.9

Kj (D (X/Y)) = Kj (Y) .

69

Proof: D(X/Y) is filtered by order, hence by [25, Kj(D(X/Y)) = Kj(f~(X)) = Kj(X) and by [25,

], ], Kj(X) = Kj(Y).

Karoubi defines [19, $3.25] Chern characters Ch~ n 'Kj(A)

, HCj+2m(A)

for any algebra A, with the property that c

n-1

-

s° T

In our case, it appears t h a t the interesting Chern character is Ch~ (for j

variable).

Indeed this wilt give for each

character Kj(Y) = Kj(D(X/Y)) a conjectural description of

, ~t21-J(Y, (~y

i and ,..

j

a Chern

> Q~f) (using

H C~ (D (X/Y)) . Presumably this Chern

character factors through Hi-J (Y , Q~,) ; this would follow from the vanishing of Cb.jn-1

Of course, this sort of Chern character should

coincide with those obtained by Karoubi, in de Rham cohomology. We hope t h a t Some modification of the algebra

D(X/Y) ,

probably some sort of algebra of Toeplitz operators, will have closer relations with the Beilinson-Deligne cohomology theory.

70 BlblloaraDhv

[t] Angeniol, B. and Elzein, F.; La classe fondamentale relative d'un cycle; Bull. Soc. Math. Memoire n'58 (1978); pp. 67-93.

[2] Bjork, E.; Rings of dJfferential operators; North Holland (1982).

[3]

Blanc, P.; Cohomologie diffe'rentiable et changement de groupes; Asterisque 124-125 (1985); pp. 113-30.

[4]

Blanc, P.; Sur les fonct~bns d'integrales orbitales nulles sur un groupe re'duct~f; preprint Ecole Polytechnique (1985).

[5] Blanc, P. and Wigner, D.; Homology of Lie groups and Poincare" duality; Letters in Math. Physics 7 (198S); pp. 259-270.

[6]

Bourbaki, N.; Groupes et alg'ebres de Lib; chapitre III; Diffusion C.C.L.S., Paris.

[7] Brown, K.S.; Cohomolo8~" of groups; Graduate texts in mathematics n'87, Springer Verlag (1982).

[8]

Brylinski, J.-L.; A differential complex for Poisson manifolds; preprint I.H.E.S./M /86 /12 (1986).

71

[9]

Burghelea, D.; The cych'c homology of the group rings; preprint Ohio State University (1984).

[1o] Caftan, H. and Eilenberg, S.; Homological algebra; Annals of math. studies; Princeton University Press n°lgn (1956).

[11] Cartier, P.; Representations ofp-adic groups; Proc of Syrup. in Pure Math. vol. SS (1979); pp. 111-155.

[12] Casselman, W.; A n e w non-unitarity argument for p-adic representations; Journal of the Faculty of Science, University of

Tokyo 28 (1982); pp. 907-928.

[15] Connes, A.; Non-commutative d]fferentialgeorneto,; Publ. Math. I.H.E.S. 62 (1986); pp. 257-$60.

[14] Grothendieck, A.; Cohomologie locale des laiseau,v cohe'rents et theoremes de Letchetz locaux et glol~ux; (SGA); North Holland.

[15] Harish-Chandra;

Admisszble distributions on reduct]ve p-ad]b

groups; Queen's papers 48 (1978); pp. 281-348.

[16] Harish-Chandra and van Dijk, G.; Harmomc anal.ysis on reductivep-adic, groups; Lecture Notes in Math.

[171 Hochschild, G., Kostant, B. and Rosenberg, A.; DzYferentialforms on reKular affine algebras; Trans. Amer. Math. Soc. 102 (1962);

pp. 383-408.

72

[18] Julg, P. and Valette, A.; T~,,isted coboundary operators and the Selbergprinc~)~le;

preprint (1986); to appear in J. Oper. Theory.

[19] Karoubi, M.; Ilornologie cych'que et K-theorie I; preprint;

Paris

(1985). [20] Kashiwara, M.;

On the holonornic systems of linear differential

equations II; Invent. Math. 49 (1978); pp. 121-155.

[21] Kashiwara, M. and Kawai, T.; On the holonornic systems of llnear differential equations ('s.ystems with regular singularities) III; PuN. R.I.M.S./Kyoto University 17 (1981); pp. 815-979.

[22] Kassel, C. and Mitschi, C.; Algebres: dbl~@ateurs d]ffe'rentiels et ' cohornologie de de Rharn; in preparation.

[23] Katz, N. and Oda, T.; On the differentiaObn of de Rharn cohornology classes with respect to parameters;

J. Math. Kyoto

University 8-2 (1968); pp. 199-215.

[24] Loday, J.-L. and Ouillen, D.; Cychc homology and the Lie algebra homology of matrices; Comment. Math. Helv. 59 (1984); pp. 565-591.

25] Ouillen, D.; Higher algebraic K-theory; Springer Lecture Notes in Math 541 (1975); pp. 85-147.

Qn the T o p o l o g y

of A l g e b r a i c

Torus

Actions

by M. Goresky*

and R. M a c P h e r s o n z

To T. A. S p r i n g e r On his s i x t i e t h

I.

Introduction.

Suppose

a compact

algebraic

complex

torus

(c

two

of

(Sl) n c

(C*) n

reals.

In this note,

topological

and

1.

the t o p o l o g y

2.

the t o p o l o g i c a l the

orbits Knowledge

of

reconstructing

structure structure

X

of the of

the ~+

following

(R+) n

the

topologically,

goes

compact is

the

on

as e x p l a i n e d

torus positive

B.

action,

we

Stab ( + ) n ( b ) a

is the

, and

action

(~+)n

torus

of an

information:

B = X/(SI) n

subgroups

information

has an a c t i o n

the a l g e b r a i c

where

the

space

stabilizer

this X

(e* )n,

we d e t e r m i n e

of the orbit

the

variety

subgroups:

(R+) n c

topological and

algebraic

As a Lie group,

product

By

b~rthday

long in

way ~ 8.

i.

Partially

supported

by N.S.F.

grant

# DMS

860-1161

2.

Partially

supported

by N.S.F.

grant

# DMS 850-2422

mean c

the

(R+) n • towards

74

We express which data

this

c a n be a s s o c i a t e d is

of

two

varieties them.

types:

ZF

and

If

X

effective,

map

various image

of

([All,

the

geometric

points

ingredient

here

~ ZG

quotient

This

and

space

is

a

what

picture

are of

of

ordered

definitions

~

(with

If F ~ G

(b)

u

(c)

If X G N X F

{X F

of

(£)

of

action

is

is

of

done

by

the

are

the

in

the

with

the

subvarieties

of

ZF points

standard.

algebraic

play

in

is

k-n.

identified

various

this

between

at m o s t

to d i f f e r e n t be

in

algebraic

n

varieties

also

action

polyhedra

, ZG

action

The

they

now

The

maps

new ~FG:

reconstructing

of

a

section be u s e d

ordering

"piece")

then G < F

results

are

techniques.

presentation

topological

) = X

~ @

standard

will

then X F N X G =

I F ~ ~

The

structure

this

which

(or

paper.

efficient

partial

e a c h F ~ ~ of a s u b s e t (a)

an

In

pi~cification set

by

the o r b i t

Definitions.

A

that

of

dimension

collection

expository

consists

topological

of

All

role

largely

contribution

2.

may

Torus

ZF

the

torus

[K]).

X.

data"

the

B.

of

%

the

action

collection

and

quotients

X.

the

consequences

complicated

to

on

{FG:

have

These

the

a

maps

ZF

"torus

collection

associated

theory

is

a

k

[MW],

in

action

is

dimension

map.

invariant

is

algebraic

quotients

moment

semistable

ZF

second

[GS],

symplectic

of

The

polyhedra

of c e r t a i n

*)n

first

then the varieties

association

moment

(c

The

some

had

in t e r m s

to the

n-space.

Euclidean

The

information

we

give

denoted

X F c X such

X

~)

the

that

rather

elementary

the p a p e r .

is and

main

action.

some

throughout space

Our

of

of a t o r u s

easy

a a

partially choice

for

75

Remarks.

stratification

A

piecification

is m o r e

general:

piecificatlon

does

frontier

the

closure

allow

the

(c)

implies

(i.e.

pieces).

We

ordering

axiom

Definition.

together

of

with

~GH~FG

G ~ F,

and ~FF

that

Definition. space

suppose

with

on ~.

pieces

The

topological

piecificatlon,

the

may

satisfy

however

be

singular,

the

axiom

is not n e c e s s a r i l y that

XF

each piece

cofunGtor

continuous

is the

pieces

possibility

topological

with

the

of a p i e c e

~

spaces

=

@.

(indexed

a

the

a union

of

partial

closed.

on a p a r t i a l l y ZF

and of

The

is l o c a l l y

a

ordered

set

by the e l e m e n t s

maps

~FG whenever

a

necessarily

A space-valued

is a c o l l e c t i o n of ~)

not

is

: ZF

' ZG

property

that

if H

~

G

~

F

then

~FH

=

identity.

~

is a p a r t i a l l y

indexed

realization

by ~, R(~)

and

over

ordered ~

is

X of

set,

X is a p i e c i f i e d

a space-valued

the

triple

cofunctor

(~, X, ~)

is the

space R(~)

=

J_L z F × XF / ~ F~

where

~

identifies

whenever

Of

partially

over ~FG:

ZF

cylinder

Remarks.

a

(z,x)

realization:

ordered

with ~

point

~

ZF

x

XF

with

(~FG(Z),

x)

x ~ X 8 A V.

Example

[O,1]

a

set

9

piecifJcation

is a p a i r , Z G.

of

The

consists XG =

spaces The

mapping

Z F,

of

{0)

two

and

Z G,

realization

cylinder. elements

XF =

(0,1].

together R(~)

with

over

X

Suppose

G < F. A

Let

the

realization

is c a n o n i c a l l y

piecified

with

map

mapping

of gFG'

The

X =

cofunctor

a continuous is

the

pieces

76 F

R(~) The

realization

which

is

R(~)

is

uses

the

proper

3.

In

algebraic

Recall finite subspace

A

interior

of

itself.

is

obtained

through

span

a

is

~:

spaces

R(~)

ZF

is

.

and

compact

each

~FG

X,

compact.

Z F is H a u s d o r f f

locally

compact

collection

variety

we

will

from

(this

if

X

is

is p r o p e r .

of

a

is

how

is

hull

is

interior

the

with

C c ~n

subspace

C

C°

an to

which

can

be

of

an

action

reconstruct

of A.

affine

the

of

convex

the C

The

Euclidean

the

data

the

data.

affine

The

of

X

show

this

polyhedron

translating

smallest

is

the

A

so

of

a

affine

topological

interior

subspace

hull

hull

of a p o i n t

span(C) that

it

which passes

origin.

Data

TAD3,

consists

TAD4

finite

such

that

(a)

If C ~ w

then

each

(b)

Each

C ~ •

a basis

possibly

face

is r a t i o n a l , consisting

obtain

of

the

following

four

ingredients:

:

collection

in ~ n

We

5

as

points)

Remarks:

define

C.

viewed

by

R(~)

locally

Its

dimensions,

has

is

%

various

~n

each

Hausdorff

X/(Sl) n

TA

a

and

points.

C,

Definition.

is

the

is

In

The

TAD2,

of

a convex

of

the

each

ZF

containing

is

if

projective

space

set

only

we

any

that

projection

Data

torus.

topological

obvious

relations).

section

from

an

X

each

Action

this

obtained

TADI

if

compact,

with

and

commutation

Torus

TAD1,

if

Hausdorff

locally

%

comes

xF

= ZF x

of

a partial D ~ C

~

of

(closed)

overlapping,

D of

C

i.e.

integral

order

~=~ D is a

possibly

is a l s o

the

convex

an

Euclidean points

sharing

element

b I .....

of C

(of

interior

of ~.

subspace

on w by defining face

polyhedra

span(C)

b r ~ Z n.

c

77 Define to

P = d

be

the

compact

polyhedra.

There

topological

space

pieces: if

two

they

Thus

are

~

subset is

a

is

the

inclusion.

set

for e a c h s u b s e t We

remark

this and

pF A pG ~ ~

TAD

a

is

algebraic

varieties

TAD

3

for

each

RC

4

and

is

then given

by

--

U

{C

all ~

these of

the of

of P if a n d o n l y polyhedra

partially

C ~ 4.

ordered

by

I C ~ ~-F)

of

particular ~:~

pF c V

~

over

it

of

in

satisfies

complex

R c, a n d

is

fact the

a

Whitney

axiom

of

the

c=, F c G

the p a r t i a l l y

variety

P

(not

necessarily

ordered

for e a c h

s e t ~,

compact)

i.e.

for e a c h

f a c e D < C an a l g e b r a i c

~ RD"

is a c o f u n c t o r F ~ ~

an algebraic

TAD

piece

convex

•

of

C E ~ is a u n i o n

same

plecification

cofunctor

C ~ T an a l g e b r a i c map PCD:

that e a c h

the

I C ~ F}

in

frontier:

2

union

piecification

in the s a m e

of

of P a r e

{C

the

F c ~.

that

stratification

~ P are

subsets

The pieces

is

(coarsest)

in e x a c t l y

of

pF = N

which

the p r o p e r t y

x,y

contained

Rn

natural

P with

points

of

an

map

~ of c o m p l e x

algebraic ~FG:

is a c h o i c e ,

algebraic

variety

ZF

ZF and

varieties for e a c h

o v e r ~,

relation

i.e. G < F

' ZG"

for e a c h

F ~ ~ and

for e a c h

C E ~ such

that

pF

c C °, of a n i n c l u s i o n .C

IF: We

shall

denote

furthermore Axiom

i.

is a l l o w e d

assumed

Each

ZF

to v a r y

the

image

to s a t i s f y is

plecified

over

RC

~ ZF

i (R c) the by

( C ~ ~

C ZF

following the

the partially

~F =

by

images ordered

I cO D pF}

These

data

are

axioms: ZC F set

=

i ~ ( R C)

where

C

78

(which

is p a r t i a l l y

ordered

by c o n t a i n m e n t ,

i.e.

D ~ C ~=~ D c C)

Axiom

2.

If G < F e ~,

and

if C e ~, w i t h

.C o ZF ~FG

Axiom then

3.

If

the

G ~ F e y and

following

diagram

if

=

~

D ~ C e ~ with

Suppose of

the

to

a

Action

algebraic

linear

torus

action

Kaehler

metric

(sl) n c

(E*) n a n d

associated

on

z,

PFG

Rise

~o TA D a t a .

complex n

(C)

on

the

pN

which

let p:

moment

IG

G~ves

X is a p r o j e c t i v e

map

pF c C °

, RD

F

A Torus

pG c D ° a n d

commutes: RC

% 4:

pF c C ° a n d pG c C ° t h e n

algebraic

We

assume

ambient is

X

zG

'

the

projective

invariant

, R n be

the

[All,

[MS],

([K],

variety

with

an

torus

action

space

pN.

under

the

(restriction [A2],

action extends

Choose

a

compact

torus

to X

the)

[GS]).

of This

map

factors X through TAD

the q u o t i e n t

I:

We

fol l o w s :

define

a

space a

the c l o s u r e

J

orbit

torus

itself

orbit (c)

n

Rn

B = X / ( S I ) n.

collect~on

~

of

convex

polyhedra

in

Rn

as

in X =

of e a c h

p......

B

projects

(c* )

n

.x

to a p o l y h e d r o n

.x p r o j e c t s

to

the

C = p(T)

interior

and

the

torus

C ° of C ([A1],

[GS],

[K]) Proposition. 1, i.e.

The p o l y h e d r a

they satisfy

the

obtained

following

in

this

hypotheses:

manner

constitute

TAD

79

I.

Each C = g(T)

is a r a t i o n a l

2.

Only

many

3.

If C

finitely = p(T)

of C is a l s o

is

polyhedra

collection

X as

follows :

Definition.

L e t C ~ ~.

only

convex

if

the

is e q u a l

to C,

2:

point

a

(x • X

< C

then

characterized suppose

ye

*

( c )

follows:

as

.x

let

above

P

and

x

face

closure.

a canonical

~ X

then each

is

in

piecification

the

piece

XC

to the o r b i t

of

if

and

through

x

is

cofunctor

for e a c h =

R

C) of

convex

complex

algebraic

polyhedron

C e ~ let

X c / (c*) n

a unique

follows:

=

map

suppose

: RC

PCD

x ~ XC

of y ~ R D.

is

~ RD which a

Then PcD(x)

llft

of

can

x ~

be

RC and

= y if a n d o n l y

if

.

ProPosition.

Now

there

y ~ X D is a l i f t n

orbit

corresponding

space-valued

o v e r ~ as

_

D

collection

I P((£~)n'x)

RC If

this

i.e.

Define

varieties

A

in the c o l l e c t i o n

indexes

polyhedron

XC = TAD

in

of a t o r u s

~ of p o l y h e d r a

polyhedron

ppear

a polyhedron

the g - i m a g e

This

convex

Each map PCD

= g(X)

denote

let • be

is w e l l

the

union

the n a t u r a l

defined

and algebraic.

of

convex

the

piecification

polyhedra

defined

of P as d e s c r i b e d

in %

2.

TAD

3.

Define

F ~ ~ choose

a

space-valued

cofunctor

is

particular [K],

[M],

follows:

given

any p e pF and set ZF = ~ - l ( p )

(This

~ o v e r ~ as

the

"symplectic

"geometric [A2].

It

is

= -1(p)/(sl)n

quotient"

invariant known

that

which

theory" the

is

identified

algebro-geometrlc

symplectic

quotient

with

a

quotient does

not

80 depend

on the point

p ~ pF).

If G < F G ~ then we obtain

a map

~FG: which

iS c h a r a c t e r i z e d

means

that

pG.

Then

in the

p = p(p')

E pF.

the c l o s u r e

in a s i n g l e

ZF

of

following

Choose

the

(sl) n orbit,

~ ZG

~FG(P')

Proposition. and

variety TAD

torus

orbit

thus d e t e r m i n i n g

the

preceding

c

[K])

C°

natural

identification

RC

for

=

it is clear

then any

that

the axioms

The AXI,

of s e m i s t a b l e

=

For

each

topological a

of

XC

quotient

categorical

This

Choose

-1 (q)

intersects

a single

q E

point

Z F and gFG are well

is

of

~

an

point

(p-1(p)

each

TAD2,

/

is

inclusion P

~

pF

an

algebraic

iC: F c

C°

RC

, ZF

there

is

a

N X C) / (SI) n

description

p-1(p)

ZF

defined

of R C is a subset

of

(s1)n.

TAD3,

TAD4 d e f i n e d

here s a t i s f y

F ~ ~

there

is a

~n (c)

- invariant

set

points,

of the u n i o n

p-image

Z F.

AX3.

ss

the

choice

TAD1,

xF

consisting

there

this s e c o n d

data

AX2,

of Drool.

TAD

X C / (c*)n

ZF

Proposition.

of

of

map.

(by

Sketch

choices

and each ~FG is an a l g e b r a i c pF

E

= Z G.

i.e.

If

criteria

because

and

(c*)n.p

2,

4.

p'

= q'.

The

satisfy

Let

a lift p ~ a-l(p').

q' e M - l ( q ) / ( s 1 ) n We d e f i n e

way:

=

u (x c I P F c

of those contains

pieces the

c) X c such

stratum

that pF

the closure A/though

)n m a y not e v e n be Hausdorff, X F /(c*

quotient

ss

([M]),

i.e.

an algebraic

variety

of the

there

is

(which

we

81

still

denote

SS

by X F

t /~c*

, Y is an a l g e b r a i c point,

then

f factors

)n ) w i t h the p r o p e r t y map which

through

takes

an a l g e b r a i c

ss

g: X F I ( C * By

[K],

the c a t e g o r i c a l

quotient ZF

for

any

that

p

X ss

choice

and

of

must

~

be

pF

lift of made,

equivalently,

an embedding

of b a s e p o i n t

choice

of X ss.)

ss

c

ss

XG

agrees

categorical

with

the

quotient

(E

*)n

chooses

a

ss X F /(C

necessarily

map is

~FG

*

with

emphasize

language, to the

the

[M]

invertible

basepoint

p

correspond

an a l g e b r a i c

fact

p.148,

=

sheaf {0},

I02 of G i n t o P G L ( n + I ) .

so we o b t a i n

CFG: This

not

to a s i n g l e

map

Kirwan

Mumford's

[K]p.

p does

orbit

XFS

lln

nor

of S =

Kirwan

torus

f:

, Y.

(p)/(S

In

the a c t i o n

choice

-I

with

F.

whenever

c a n be i d e n t i f i e d

Mumford

while

If S < F t h e n X F

= ~

)n

(Neither

Z vary

each

that

a 5 or

Kirwan's

to M u m f o r d ' s

map

n x~s/(c*)n ) ........,

as

defined

homeomorphlc

to

above the

because

universal

Mumford's Hausdorff

quotient.

%

5:

Construction

suppose TADI:

of

we are given a finite

= u ~ which TAD2: over • ,

a

the s p a p~ ~ a collection

collection

is p i e c i f i e d

space-valued

from T A D a t a of TA data,

• of c o n v e x

polyhedra

by the decomposition cofunctor

~

of

i.e. in ~ n w i t h

union

P

~,

algebraic

varieites

defined

82

a space-valued

TAD3:

cofunctor

~

of

algebraic

varieties

defined

o v e r ~, TAD4:

a system

pieces

indexed

of

c

inclusions

: RC

iF

by the p a r t i a l l y

ordered

~F = ( C ~ •

Construction (over

I.

Define

P) B = R(~)

Construction follows:

of

2.

the c o f u n c t o r

for e a c h

set

indexes

cofunctor

of

algebraic

=

space

triple

B to be the r e a l i z a t i o n

(~,

P, ~). of

a partially

B

indexed

ordered

by

•

as

set

{F e ~ I pF c cO}.

the p i e c e s

spaces,

Z F into

set

piecificatlon

C e • define

~C This

a

piecify

I pF c C ° )

a topological

Construct

....,. Z F w h i c h

in the p i e c i f i c a t i o n

~C

which

associates

of C ° a n d a d m i t s

to

any

a

F

E

~C

the

of

B

is

the

since

C°

is a

has

been

subvariety C ZF c ZF

Definition. realization Remark.

The R(~C)

Since

cell, t h e r e

piece of

BC

in

the c o f u n c t o r C ZF

each

is

is a c a n o n i c a l

so B C is f o l i a t e d

ConstFuCtion

3.

For

St C = exp as

follows:

identified its

Span(C) with

annihilator

multiplication of

the

(~+)n

by

dual

lies

with

Co,

R C,

~C )

and

C°

x

RC

C° x

(point).

a subgroup

(~ A n n ( s p a n ( C ) ) )

(~+)n

of

~ = J(il)

This

(~C'

C ~ • we a s s o c i a t e

in

The e x p o n e n t i a l

isomorphism.

~

is a s u b s p a c e

exp iS a n

triple

homeomorphism

by s u b s e t s

each

piecification

identified

BC and

the

the

of R n Lie

the

c

which

algebra

Lie

[AI], of

algebra

identifies

this

with

(sl) n. of the

map

: Lie

(R+) n

is s u m m a r i z e d

[GS]

, (R+) n in the d i a g r a m

Therefore (SI) n , Lie

and

algebra

83 (Rn)* ~ Lie u

(sl) n

=

, Lie

(m+) n

~= ~ (R+) n exp U

Ann(span(C) )

Theorem. action Data, as

Suppose of

the

to 2

construction

3.

there

that,

a

algebraic

construction

(1)

is

X

TAD1...TAD4.

applied

St C

projective

torus

(c*) n

wlth

an

the c o r r e s p o n d i n g

TA

obtained

this

TAD,

let

BC

be

and

let

St C

be

the

the

variety

from c o n s t r u c t i o n

pieces

subgroups

1

obtained

from

obtained

from

Then:

is a c a n o n i c a l

homeomorphism

h : B

, X /

(Sl) n s u c h

for e a c h C E ~ we have: h takes

BC homeomorphically

(3)

h takes

each

(~+)n orbit for

to xC/

leaf C°x(point}

($I) n

c B C homeomorphically

to a single

isotropy

(~+)n (x)

in X/(S1) n

each

x G xC/(sI) n,

precisely

the s u b g r o u p

Example.

See

[GGMS]

are e x p l i c i t l y of

manifold

G_ ,.(£n) w i t h

nonempty

pieces

to m a t r o i d s

k

on

and are n

of e x a m p l e s ~n one

elements,

are

Stab

the

usual

action

representable

where

to one

and

in the p i e c i f l c a t i o n

which

subgroup

is

St C.

described

rank

the

for a family

matrolds

t~6

Extract

Let B be the space

(2)

(4)

algebraic

where of

the p o l y h e d r a

correspondance X

the

is

the

torus

over

the c o m p l e x

with

Grassmann

(c)

of the G r a s s m a n n ~ a n

C

n

.

The

correspond

numbers.

An e x a m p l e .

Suppose space

that with

(c)

2

acts

homogeneous

X = CP 3,

on

coordinates

(zl:

complex

projective

z2:

z3:

(Zl: sz2:

tz3:

z4),

formula

(S, The moment

t) • (Zl: map

Z2:

Z3:

is then g i v e n

Z4) by

=

stz 4)

three by

the

84

=

M([Zl:Z2:Z3:Z4])

(}z212

+ Iz412

, Iz312 + 1=412)

Izll 2 + Iz212 + Iz312 + Iz412 The

image

P

is

the

square

{(x,y) in

R2.

The

with

the

part

make

the

convention

TYPE

I:

various of

• ~21 o < x < i, polygons

X

which that

no

C

in

projects

o ~ y ~l } ~ to

coordinate

are

listed

their

typed

below,

interior

z.

1

along

(where

is z e r o ) :

:-:-i'bi':':'i'Z'ZO:':'FZ'D:':';-bF:':

il!iiiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiiii

:::::::::::::::::::::::::::::::::::::::::::::

::::::::::::::::::::::::::::::::::::::::::::: i:!:i:!:i:i:i:i:!:i:i:i:i:i:i:i:i:i:i:i:i:i:!

- ........,.............

::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::

i::iiiiiiii::iii::i i iiiil;ililii::i::i::i::i: ( I:

O: O: O)

iiiiiiiiiiiiiiiii!iii!i!i!ii!ii!iii!ii!iiiiii

:-:. F:G:

.,......................,............. ... ,........

.,.................,.,.,..............

.-.-...........u........,......,./......... •. ....,' .. .' .. .' .. '.. ' / . .•. . .•....,.... . v • , • . • / /. ./. . . .../.......... . ..........................

• ........,.....

iiiiiiiii!i!iiii!i!i!iiiiiiii!iiii!iiiiiiiiii ::::::::::::::::::::::::::::::::::::::::::::: "'"""'"'~

.....

(Zl:Z2:

TYPE

I'1'1'1 . . . . . . .

O:

O)

.......................,..,...

:::::::::::::::::::::::::::::::::::::::::::::

: :G:-:-'-'.'.'.'.'.'.'.-.-.'.

....,.,.........................,.....,..

( O: I :

O: O)

( O: O:

1:

O)

( O: O: O: i )

liiiiiliiiiiiiii

iii?i!iiiiii!iliiiiii?iiiii!iliiiii?ii!ili!.

:.F:-:.:.Z.Z.Z.:.:.:-Z.Z-:,:.Z.Z.:.:-:-:-:.:

)iiiiiiiiiiii!iii;i!iiiiiii;i!ii)iiii;i i{iii

iiiliiiiiiliili!iiiiiiliiiiliiiiiil

II . . . . . . . . . . . . . . . . . . . . .................................

-.-...,......... • • .............., ,......... •.............///

J

• ...............

:';-:-F:':':':':-:-:.:-FI.:-;.:.:.:.:.:-:

...................,.,...............

i:!ii:i iiiiiiiiiiii!:i i i:i:i:i:iiiii!i:i iiiii~iiiiiii!i[iiiii!iiii!iii[~i!i!iiiiii: :::::::::::::::::::::::::::::::::::::::::::: ( O: O : z 3 : z 4 )

(zl:

O:z3:

O)

( O: z2:

O:z 4)

II

(Zl:Z2:Z3:0)

(O:z2:z3:0)

( O : z 2 " z 3 : z 4)

we

85

TYPE

IIl

i

TYPE

( Z l : O : z 3 : z 4)

( O : z 2 : O : z 4)

( Z l : Z 2 : O : z 4)

IV

m (ZI:

The

RC

is a p o i n t

TYPE

IV.

Over

each

piece piece

ZF

complex

a

standard

It

1

for

is e a s y

S 4, so

the

RC

(0

: I)

see

it f o l l o w s

that from

of

edge

line,

P

each

if

C

of

the in

the

maps as

of

ZF

£

is

of

for

a point.

the

identify

homogeneous CFG

is

this:

may

their TYPE

and

interior

we

(with

have

or III,

square,

which

the

is

II,

is like

of

line

, ZF

of T Y P E

z 4)

I,

is c o n t a i n e d

take

.C IF:

C

to

•

projective may

z3:

of T Y P E S

projective

we

II,

the

on

F which

inclusions

is of T Y P E cF

F

complex

[yl:Y2 ]) and The

C

The p i e c i f i c a t i o n

For e a c h is

for

Z2:

to

III,

with

the

coordinates

be

image

square,

the (1

and

identity.

: O) the

if

C

rest

of

IV.

the the

realization theorem

that

of

s

is

the o r b i t

the space

four B

sphere is

S 4.

86

%7

Sketch

topology. 1.

a

Of

the

Suppose

compact

prQof.

we

consider

Hausdorff

of R n,

2.

piecification

~: B

space

B

mapping

of

P

(indexed

by

a

linear

the

the c l o s u r e

of

the

lemma

in

pure

to

a piecewise

linear

~ P

into f i n i t e l y m a n y p i e c e w i s e axiom

a

that we have:

subset a

First

frontier:

partially

subsets

pF,

ordered

set

~)

and w h i c h s a t i s f i e s

pF of a n y p i e c e

is a u n i o n

of pieces, 3.

a

many)

disjoint

decomposition

topological

("open")

of

balls

B

into

of v a r i o u s

(possibly

uncountably

dimensions,

s u c h that: a.

the m a p

of p i e c e s b.

~

F°

takes e a c h o p e n ball h o m e o m o r p h i c a l / y of

the c l o s u r e

For

each

the

fiber

~FG:

pF

of

~-l(p)

ZF

cofunctor

of

on

•

open

ball

these

ball

p E pF G

is a

by the c o n d i t i o n through

@

takes

z.

and let face

of

that ~FG(Z)

This

forms

space

ZF F

be let

lles

in

valued

~.

hypotheses,

to the r e a l i z a t i o n

which

of P.

Whenever

w h i c h we call

Under

homeomorphic

p.

be d e f i n e d

the

"closed"

P, c h o o s e a point

over

~ ZG

the c l o s u r e

is a

to a u n i o n of p i e c e s

piece

Lemma

P

of e a c h o p e n ball

homeomorphlcally

onto a union

R(~)

the

space

B

is

canonically

o v e r P of the c o f u n c t o r

triple

(~, p, ~).

For

example,

plane,

P

in

the

is a s u b s e t

the o p e n balls

in

B

following

picture,

of the llne, and pieces

of

B

is

a

~

is v e r t i c a l

P

are s k e t c h e d

subset

of

the

projection,

and

in.

87

In this example,

the

ZF

and

-

The

proof

and

a

the maps

CFG

|

of

the

point

p

lemma

G

are as

@

J

is s t r a i g h t f o r w a r d :

pF.

follows:

By

(3a)

and

(3c)

(p)

........,. @

fix

a stratum

there

exists

pF of P a

unique

homeomorphism pF

hF: which

commutes

(point))

is a

Furthermore,

the p r o j e c t i o n

leaf

of

by

it is easy

(3b),

the

V

that

the r e a l i z a t i o n

To a p p l y

this

(R+) n

orblts.

B We

lemma

into claim

c

B

lies

~

l(pF)

in a s i n g l e map

c

is c o m p a t i b l e

ball).

B with

the

relations

of ~.

topological of

(i.e.

~

this

(pF)

to a c o n t i n u o u s

x ~-l(p)

to check

-i

to pF and s u c h that e a c h hF(P F ×

foliation

h F extends

defining

decomposition

1

with

hF: and

x @-

to

open

that

the

balls

these

theorem to

be

satisfy

of

the the

~5,

we

take

the

decomposition

by

conditions

of

the

88

topological

lemma.

This

follows

from

the

following

facts

about

the

composition X

I.

A

single

open disk

(C

which

*)n

, B

........

orbit

O

is a s i n g l e

~ Rn

#

in

X

(R+) n

homeomorphically to the i n t e r i o r

projects

orbit C°

0

to

in

a

topological

B, w h i c h

of a c o n v e x

projects

polyhedron

C

in

P. 2.

The

closure

orbits.

It

topological each

To

closure moment and

% 8. In

closed

of

facts,

we

interior

observe

are

standard

for

toric

subgroups

any point

x e X

is

the

itself

is that

be

the

projecting

because

of

TAD,

to

many

the

the of

O

which

is

a

(~+)n

orbits,

C

P.

in

moment

closure

* n (c)

map

for

the

0

of

the

of

is a t o r i c

variety,

varieties.

topology

Stab(~+)n(b )

X

(x) = S t a b

B,

many

X.

constructed

remark

in

of a face of

facts

of

O

that

restriction

the

finitely

finitely

closure

first

integral

to the

of

the

The

space

consisting

of

of

However

topology

the

closure

X.

the

In terms

the

for

we

(R+)n

consists

is

stabilizer

Stab

to

disk

Reconstructing

This

X

0

map

~5,

in

projects

these

these

0

projects

of w h i c h

prove

of

(b)

the g r o u p

of

linear

values

on

stabilizer to

b

TAD.

To

from

since

is *n (e)

the

(SI) n on

points.

by

Rn If

with

modulo x

can

(sl) n

Stab

(x)

of

(~+)n (b).

determined

action

identifies

extent

the

information?

Stab

is d e t e r m i n e d

and

what

this

subgroup

, n(X) (c)

functionals

integral

from

B = X/(SI) n

reconstructed

Stab

(~+)n

of

is a l g e b r a i c . (R n)

those

projects

by

to

(z n)

that

,

take

b ~ B c,

89

then

Stab

(s 1 )

n(X)

Stab

is the s u b t o r u s

n(X)

=

Ann(span(C))/Ann(span(C))

(z n)

(s I )

We

call

each

X

a

piece

torus

"piecified

Bc

in

torus

B

(sl)n/stab

bundle"

fibers

over

over

BC

(x) = s p a n ( C )

B.

with

/ AC

The

preimage

fiber

where

the

a

of

quotient

covector

in

(sl)n span(C)

is

integral

values

the L e r a y above

A

remarks

Z n.

to the p u r e l y bundles". stabilizer

subgroups

bundle

over

class.

It

B,

and

would

classifying

In

that

twisting

in the

determined X

the

by

is a t o r i c

X

the p r o j e c t i o n

the

E2

term

of

reconstructing

which

from

X

this

Leray

question

of

X

to

torus

identity,

its

topology

is

determined

interesting

to

have

"bundles"

from

"piecified

X

to

torus

alone.

from

classifying

the

TAD

~n

takes from

B.

The

spectral

from TAD alone.

are

such

to

c a n be c o m p u t e d

the

map

the

of

if

be

classes

extension

cohomology

topologically

example,

some

for

that

topological

For

has

sequence

imply

of

it The

c a n be c o m p u t e d

question

case

iff

on

spectral

sequence

The

in

is

a

X

a

reduces

"piecified

circle

then

TAD

and

all

torus of

the

a principal

by

the

first

Chern

of

first

Chern

a section,

there

is no

theory

S

1

is

in g e n e r a l .

B

admits

bundle"

This

is

and the

the

topology

case,

for

of

example,

X

is when

variety.

Bibliography

[All

M.

F.

London

Math.

Atiyah, Soc.

14,

Convexity 1-15

(1982)

and

commuting

Hamiltonians,

Bull.

90

[A2]

M.

F.

Atiyah,

algebraic geometry.

[BBS]

Angular

Proc.

torus

Carrell,

ed).

in ~rOUD

Springer New York

SL(2,E)

actions.

V.

Nauk.

I. Danilov,

33

(1978),

[GS]

V.

moment

mapping,

[GGMS]

[K]

F.

Press,

manifolds

[M]

Math.

S.

in

quotients

Vectorflelds mathematics

Quotients

Soc.,

1982.

toric

varieites,

of

Sternberg, 67

(1982),

Goresky,

Cohomology

of

Mathematical

N.J.

Marsden

by

(J.B. #

956,

by £

Uspekhi

and

Mat.

properties

of

the

491-513 MacPherson,

and

V.

and Schubert

Serganova, cells,

to

Quotients

Notes

# 31,

in

SvmDlec~ic

Princeton

and

University

(1984)

and

with symmetry,

Springer

R.

Convexity

convex polyhedra,

A.

WeJnstein,

Reports

D. Mumford and J. Fogarty,

edition)

and

121-138

in Mathematics.

Geometry.

J.

M.

Kirwan,

Princeton

[MW]

and

geometries,

in Advances

Alqebraic

Amer.

Inv. Math.

appear

C.

Complete

and

85-134

Gelfand,

Combinatorial

26 (1983),

and A. J. Sommese,

The geometry

Sulllemin

I.M.

notes

polyhedra,

(1982)

A. Bialynicki-Birula

[D]

Actions

lecture

[BBSo]

Trans.

Soc.

and J. Swiecicka,

actions,

Springer-Verlag,

convex

Edinburgh Math.

A. Bialynicki-Birula

algebraic

momentum,

Verlag,

on Math.

Geometric

New York

(1982)

Reduction Phys.

of

5 (1974),

Invarlant

symplectic 121-130

Theory,

(second

RESTRICTED

LIE A L G E B R A

COHOMOLOGY

J.C. J a n t z e n Mathematisches Seminar Universit~t Hamburg Bundesstr. 55 D - 2 0 0 0 H a m b u r g 13

The c o h o m o l o g y Hochschild

of r e s t r i c t e d

in 1954,

could get more non-trivial

cf.[11].

precise

cases.

The most

regular

functions

purpose

of this article

fascinating

on the n i l p o t e n t

result

char(k)

General

in

the t h e o r e m the c o h o m o l o g y

group

is the ring of

in this Lie algebra.

to give a survey of r e c e n t

this p a p e r

let

k

It is the

developments

be an a l g e b r a i c a l l y

in

Lie(G)

of an a l g e b r a i c

structure:

the p-th power map.

set of all

(k-linear)

derivations

functions on

on

k[G].

(because of

G) w h i c h commute

char(k)

with

We get thus a map denoted

by

enveloping

as in 1.1 has

(also called

over

k

is a Lie a l g e b r a

x

from

g

to itself w h i c h

xp

field

ag a i n w i t h Lie(G)

of

has to satisfy

Lie(G)

as the

of all regular

G, hence

belongs

to itself w h i c h

of

to

is

it from the p-th

Lie(G).

Lie algebra).

k

has an

representation

to d i s t i n g u i s h

algebra

k

is a d e r i v a t i o n

lead to the general

over

over

(the algebra

also

from

restricted ~

G

the left regular

x~-~ x [p] in order

in the u n i v e r s a l situation

x~-~ x p

group

We can r e g a r d

k[G]

= p) and commutes

usually

p-Lie a l g e b r a

of

For any such d e r i v a t i o n

Lie(G).

!=~-The

closed

Theory

additional

power

p

groups

= p # O.

1.1 The Lie algebra

G

is still

for large

by

that one

cohomology

algebraic

cone

first d e f i n e d

only r e c e n t l y

these

that

of a r e d u c t i v e

was

theory. Throughout

I.

about

and Parshall)

ring of the Lie a l g e b r a

with

It was h o w e v e r

information

(proved by F r i e d l a n d e r

this

Lie algebras

to g e t h e r

with

definition

of a

A p-Lie a l g e b r a a map

some c o n d i t i o n s

x~

which

x [p] one can

92

look

up e.g.

to k n o w

in

the

If one

[12],V.7

following

takes

G = GL(V)

then

one

just

the p - t h

gets

Lie(G) power

bra

of a p - L i e

Lie

algebra.

a suitable

or

[4],II,§

facts.

as

for some

vector

space

x

algebra finite

g£(V)

with

= Endk(V)

is c l o s e d

have

a p-Lie

algebra

~I

by t a k i n g

get

from

the m u l t i p l i c a t i v e

group.

On the o t h e r

h a n d we

taking

x [p] = 0

the a d d i t i v e ~

From

~

algebra

a homomorphism gZ(V)

call

as a p - L i e

In c a s e

(i.e.

of

G

lated

over

~

(taking

be

is just

algebra

algebra

of

~O

algebra

by

of

k.

is c a l l e d

the

a representation

x 6 g,

standard

i.e.

p-th

if

p

power

is

map

ideal

of all

g-module

if we

algebraic group

representation are d e a l i n g

of

with

g

a repre-

with by all

envelopin~ restricted

G

(i.e.

any

"G-module")

of

~

as a p - L i e

of

~.

Then

with

any

representati-

x £ g.

~-modules

of all

Set

the

~. By

leads al-

This

with

category

annihi-

where

algebra

I

is c a l l e d

above,

that

cate-

subcatego-

U(~)-modules

the r e m a r k s

is i d e n t i f i e d

the

the

full

u[P] (g) = U ( ~ ) / I

x p - x [p].

of

way with

identifies

that

these

al~ebra

algebra

in a n a t u r a l

construction

~-modules

x p - x [p]

group

~ ~ g£(V)

enveloping

is i d e n t i f i e d This

generated

the r e s t r i c t e d

a fixed

~-module) .

the u n i v e r s a l

restricted

by all

with

as an a l g e b r a i c

U(g)-modules.

tegory

a p-Lie the p - L i e

for all

to a r e p r e s e n t a t i o n

of all g - m o d u l e s

is the

space

for some

to a r e s t r i c t e d

U(~)

ry of all

x 6 k. T h i s

into

algebra.

g = Lie(G)

G ~ GL(V)

of all

into

g = k

as the L i e

into

algebra

= p(x) p

algebras

it a r e s t r i c t e d

by d i f f e r e n t i a t i n g

gory

p(x [p])

is a v e c t o r

sentation

1.5 L e t

a p-

as in 1.2).

on it. We

gebra

is a g a i n

can m a k e

k

is just

p : g ~ g_~Z(V) of

if

of p - L i e

A ~-module

on

subalge-

subalgebra.

for all

g = k

x 6 k. This

be a p - L i e

A representation

on

can m a k e

Lie

c a n be e m b e d d e d

One

regarding

)

group.

n o w on let

as a p - L i e

x [p] = x p

1.1 w h e n

for all

one.

algebra.

(dim V < ~

(for any x 6g_~£(V))

x [p]

algebra

dimension

k

V. A n y

x~

as a p - L i e

examples

is

of

dim(V)

< ~

over

x [p]

under

be e n o u g h

is a p - L i e

V

p-Lie

simplest

we

and

us it w i l l

1.1

dimensional

!z~ T h e

the s t r u c t u r e

in

as an e n d o m o r p h i s m

which

Any

For

Lie(G)

= g_~(V)

of

7,3.3.

Any

the ca-

of all

u[P](~)-modules. Let In b o t h in one

us look cases

at this

U(~)

indeterminate

construction

can be i d e n t i f i e d X.

in the with

(The e m b e d d i n g

two s i m p l e

cases

the p o l y n o m i a l

~ = k ~ U(~)

maps

ring I

from

1.3.

k[X] to

X.)

93

One sees easily U [p] (~Q ) = k [ X ] / ( X p) and U [p] H k[X]/(X-a) So U [p] (~I) = k [ X ] / ( X P - X ) ~ a6~p . (KI) is isomorphic to a direct p r o d u c t of copies of

k, it is a semi-simple algebra a d m i t t i n g

simple modules

(all of d i m e n s i o n one)

p

p

different

and each r e s t r i c t e d ~ 1 - m o d u l e

is semi-simple. On the other side, restricted ~O-mOdule

there is

(up to isomorphism)

(the trivial one). For each

is an i n d e c o m p o s a b l e module of d i m e n s i o n restricted

~O-mOdule

only one simple

i (I S i S p) there

i , namely

k[X]/(xi).

Any

is isomorphic to a direct sum of such k[X]/(xi).

It is a p r o j e c t i v e

(or injective)

all i n d e c o m p o s a b l e

summands have d i m e n s i o n

restricted ~O-mOdule

if and only if

p, i.e. are isomorphic to

U [p] (~O) • 1.6 Both categories, ~-Modules,

that of all ~ - m o d u l e s and that of all r e s t r i c t e d

have been identified w i t h categories of all modules over

some ring. So we have injective and p r o j e c t i v e r e s o l u t i o n s

in these ca-

tegories and can use them to compute d e r i v e d functors. Let us look at the fixed point functor M ~ - ~ M ~ = {m 6 M l x m = O

for all

x 6 K}.

It is left exact. W h e n r e g a r d i n g it as a functor on all ~ - m o d u l e s (reap. on all r e s t r i c t e d K-modules)

we get d e r i v e d functors w h i c h we i H i (K,M) are called shall denote by Hi(K,?) reap. by H~(~,?). The i the Lie algebra c o h o m o l o g y groups of M, and the H~(K,M) for restricted

M

are the r e s t r i c t e d Lie al@ebra c o h o m o l o ~ y groups of M. The i notation H~ has been taken from [11] where these groups were introduced for the first time. One can interpret the the of

M

Hi(K,M)

also as

Exti(k,M)

as the e x t e n s i o n groups of the trivial K - m o d u l e

in the c a t e g o r y of all r e s t r i c t e d K-modules.

compute the

Hi(~,M)

The cup p r o d u c t makes

H:(~,k)

= ~ H~(~,k) iaO

all a u t o m o r p h i s m s of

as a p-Lie algebra acts on

~

tion of

M G

is a G-module, on

the action of

H:(~,M) H:(K,k)

If then

into an

H~(K,k)-module.

~ = Lie(G) G

The group of

H:(~,k)

through

for some a l g e b r a i c group

acts on each

is c o m p a t i b l e w i t h that on on

M, but

into a graded asso-

H:(g,M)

graded algebra automorphisms.

and

k.

ciative algebra and any

and if

k

We can t h e r e f o r e

not only using an injective r e s o l u t i o n of

also using a p r o j e c t i v e r e s o l u t i o n of

G

and s i m i l a r l y

K

H~(K,M)

H~(~,M). H:(g,k)

The acand w i t h

H:(g,M). (There are similar results

for the

o r d i n a r y Lie algebra c o h o m o l o g y groups w h i c h we do not mention.)

94

1.7 The functors One can observe

Hi(~,?)

use the notation Obviously trivial

U(~O)

plication by

and

this already

H~(~,?) ~

have quite different

in the simple examples

properties.

from 1.3. Let us

from 1.5. 0 ~ k ~ ] ~ k[X] ~ k ~ 0

= U(~1)-module

k.

is a free resolution

(The map

k[X] ~ k[X]

of the

is the multi-

X.) One gets easily

(I) H i ( ~ o ' k ) =

Hi(~1'k)~

~

for

Any restricted ~1-module

ii = O,I,> I.

is semi-simple.

Therefore

the fixed

point functor is exact on the category of all restricted

~1-modules.

This implies: i (2) H~(~1,k)

~'k

for

i = O,

0

for

i > O.

\

A minimal projective

resolution

of

u[P](~O)

~

k

as a restricted ~O-mOdule

has the form .... where

the maps

o[P](go) from

~

u[P](~O)

induced by the m u l t i p l i c a t i o n

u[P](~O)

~

k

~

O,

= k[X]/(X p) to itself are a l t e r n a t i n g l y by

X

and by

X p-I. Then an easy compu-

tation yields: i H~(~o,k)

(3)

~ k

for all

Let us generalize dimensional

vector

i ~ O.

the last result

space

V

over

make it into a p-Lie algebra with just a direct product of

dim(V)

k

as follows:

Consider

as a commutative

x [p] = O copies of

for all

a finite

Lie algebra and x £ V. So

V is i ~O" One can get the H~(V,k)

from (3) using a KHnneth formula. When formulating the result below we want to take into account the algebra structure on H:(~,k) and the operation of

GL(V)

all automorphisms

on this algebra.

(Of course,

of the p-Lie algebra

GL(V)

V.) One gets

is the group of

(cf.[I],I.6.

or

[14],I.4.27): I (4)

S(V ~)

for

p = 2,

for

p > 2.

H:(V,k) A(V ~) ~

S'(V~) (I)

Here we use the usual grading on the symmetric gebra when w r i t i n g we take gree but

2i. V~.)

we write

S(V ~) rasp. A(V~).

The notation

S(V ~) but with a grading such that each (We shall use this convention Furthermore

as a group and where any

space over

a £ k

acts as

exterior al-

S'(V ~) means that

si(v ~) appears

in de-

also for other vector spaces

for any vector space

M (r) for the vector

resp.

M k

over

k

and any

r 6~

which coincides with M p-r a does on M. (Note that

95

we do not change the action of some S' (V ~) (I) .)

g £ GL(V)

1.8 Let us assume from now on that The map u[P](~)

~

x ~-~ I ~

u[P](~) ~

x + x ~

I

dim(~)

on

< ~

S ' ( V ~) w h e n taking

.

induces an algebra h o m o m o r p h i s m

u[P](~).

It makes

Hopf algebra and by d u a l i s i n g

u[P]( ~)

u[P](~)

bra. T h e r e f o r e one can regard

U [p] (~ )~ ) as the ring of regular functi-

ons on some i n f i n i t e s i m a l group scheme T h e n the c a t e g o r y of all

into a (co-commutative)

into a (commutative) ~

over

H o p f alge-

k, c f . [ 4 ] , I I , § 7,n°3.

~ - m o d u l e s is the same as that of all r e s t r i c -

ted ~ - m o d u l e s . C o n s i d e r the special case

~ = Lie(G)

d e f i n e d over the prime field. Let e n d o m o r p h i s m on Then

~

M

F

f £ Fn[G]

to

induces an i s o m o r p h i s m

M

(~ = Lie(G) with

and any

r 6N

fP.

G/ker(F)

ker(F),

r > O. Conversely,

hence also

because of

acts t r i v i a l l y has the form MI

F

~

M (r) for all

there are

M r £ ~.

the case of similarly

g

is

M

does on

M [r] with M

on w h i c h MI.

and denoted by

M [-I]

~p,

is i s o m o r p h i c

g 6 G

It is also a basis of r (fP.(g))13 in the case of F~(f.

M (-I) ~ M [-1]

Fr(g)

then

M [r]

Indeed, one can choose a basis over such that any

M jr]. As

acts as

M = M~ I] for some other G-module

respect to this basis. the matrix of

g £ G

~ G: any G - m o d u l e

is d e f i n e d over

fij 6 ~ p [ G ]

M [r] for

acts t r i v i a l l y on each

G/ker(F)

is u n i q u e l y d e t e r m i n e d b y

If a G - m o d u l e

as above) we set

equal to the G - m o d u l e which c o i n c i d e s

as a vector space and w h e r e any

M. O b v i o u s l y

to

maps any

(taken in the c a t e g o r y of g r o u p schemes),

Let me m e n t i o n that

In this s i t u a t i o n

Then

G

cf.[14],I.9.5.

any G - m o d u l e with

F

for some a l g e b r a i c group

be the c o r r e s p o n d i n g F r o b e n i u s

F ~ : k[G] ~ k[G]

is the kernel of

cf.[14],I.9.7. ~G,

G. So

F

) = fP.

inl~ase t ~

has m a t r i x M (r) and of M (r) and

~ . Then P (f~&(g)) w i t h ~ M [r] and

(fij(Fr(g))

these m a t r i c e s coincide.

in

One has

latter module is defined.

1.9 The c a t e g o r y of r e s t r i c t e d ~ - m o d u l e s

is equal to that of all ~-modu-

les and the functor M ~ - ~ M ~ coincides w i t h taking ~ - f i x e d points. i H~(~,M) are equal to the H o c h s c h i l d c o h o m o l o g y groups

T h e r e f o r e the Hi(~,M). or

These can be c o m p u t e d the " H o c h s c h i l d complex",

[14],I.4.14 - 4 . 1 6 .

cf.[4],II,

§ 3

One can c o n s t r u c t a natural filtration of this

complex. The a s s o c i a t e d graded c o m p l e x turns out to be the H o c h s c h i l d c o m p l e x w h i c h we get w h e n we regard the v e c t o r space

~

as a c o m m u t a t i -

ve p-Lie algebra w i t h trivial p-th p o w e r map and w h e n we r e g a r d

M

as a

trivial m o d u l e over this p-Lie algebra. So the c o h o m o l o g y of the graded

96

complex

is g i v e n by

c u l t to c o m p u t e quence which

converging enables

In c a s e complex (1)

1.7(4)

to

V

replaced

by

~.

It is not d i f f i -

W e get thus the E l - t e r m

H:(~,M).

T h e r e are m a n y terms

y o u to s i m p l i f y

the r e s u l t s .

p = 2

finally

one gets

H:(~,M)

of a s p e c t r a l equal

se-

to z e r o

as the c o h o m o l o g y

of a

of the f o r m

O ~ M ~ M ~ ~* ~ M ~

The differentials In c a s e E(p_2)r+1

terms

E~'J(M)

$ 2 ~ * ~ M ~ $ 3 ~ * ~ ...

c a n be w r i t t e n

p # 2

then Eo-terms (2)

with

the g r a d i n g .

into

equal

E r terms

the s p e c t r a l

sequence

for the n e w s p e c t r a l

making

sequence.

old

O n e gets

to

= si(~*) (I) ~

O n e has e n o u g h

down explicitly.

one r e - i n d e x e s

AJ-i(~*)~

information

about

M.

the d i f f e r e n t i a l s

to be a b l e

to c o m p u -

te the E l - t e r m s : E i,J(M) I

(3)

= si(~,)(I) ~HJ-i(~,M).

We get o b v i o u s l y :

If

E~'J(M)

% O, t h e n

i ~ j ~ i + dim(~).

If

d r : E i'j ~ E i + r ' j + 1 - r ( M ) is n o n - z e r o , then a l s o i +r < j +1 - r < r r i +r + dim(~), hence 2r -I ~ j - i ~ d i m ( ~ ) . So for 2r ~ dim(~) all differentials (4)

E

are zero.

This

(M) = Er(M)

for all

H e r e w e use the n o t a t i o n If dule, phisms type

~ = Lie(G)

then each

= ~ Ei'J(M). r i,j for some a l g e b r a i c g r o u p G is a G - m o d u l e ,

r

is c o m p a t i b l e

1.11 A s s u m e

for the m o m e n t bilinear

put together we get thus

proved

with

is a G - m o are h o m o m o r -

factors

of the

G.

in

[5],5.1/2,

resp. H , ( ~ , M )

Er(k)-module. One gets

structure

which has with

p % 2. The cup p r o d u c t y i e l d s

as an a l g e b r a

as an

property.

t a k i n g the g r a d e d

So

H:(~,M)

Ei'J(k) x E£'m(M) r r maps E r ( k ) x Er(M)

to b i l i n e a r

a structure

dr

maps

a structure

a derivation

Any

of

w i t h the a c t i o n of above were

M

[I],1.8,

[14],I,9.10 -9.19.

r,i,j,l,m

Er(M)

and if

all d i f f e r e n t i a l s

a n d the f i l t r a t i o n

The results mentioned [7],I.1,

r ~ dim(~)/2.

Er(M)

Ei'J(M)

of G - m o d u l e s , Ei'J(M)

implies:

r ~ 1

Er(k),

in g e n e r a l we get on

The d i f f e r e n t i a l s

the s t r u c t u r e s

associated

the

on

for all

~ Ei+i'J+m(M) w h i c h can be r ' ~ Er(M). In the case M = k

on

E

to the f i l t r a t i o n

of

have E

then

(M) by

H:(~,k)

E -terms vanishes

Ei'i'k" r+1 ~ ) is a h o m o m o r p h i c

as factors. Ei'i(k) as on r

dr

(k) a n d

image of

Ei'i(k) r

E ir + r ' i + 1 - r ( k )

By i n d u c t i o n

= O.

97 i,i (I) O EI (k) = S(g ~) " On the i~O is a s u b a l g e b r a of H:(~,k) in fact of the com-

Ei'i(k)r

is a h o m o m o r p h i c

i~O other hand

~ Ei'i(k) iaO ev subalgebra H~ (g,k)

mutative

a homomorphism

K : S(~)

For any r e s t r i c t e d Er(k)-module S(~)

If

yields

(1)-module.

modules

and

a finitely

that

E K

If

M

E

In the case any

f £ si(~)

plex 1.9(I) plies

for

that

S(~*) ~

M

K : S(~)

n e d by the l i n e a r m a p ~(I) 2 H~(~,k) as set of e q u i v a l e n c e

(x,a) [p] =

~ £ ~

2.1 over

The R e d u c t i v e

let

S(~)

graded

(1)-mo-

one.

If the

so is the o r i g i n a l

~-module,

in

one.

[7],I.4:

then H:(~,M)

i__{s

commutative

k-algebra.

(I) ~ H~e v (~,k) by m a p p i n g

property

module

of the d i f f e r e n t i a l im2i in H~ (~,k). As

a class

over

{f21f 6 S ( ~ ) } ~ S ( ~ )

(I) H~ev (~,k) 2 H~(~,k). Using

also is

(for any

extensions

one can give a d i r e c t

for all

p) d e t e r m i -

the i n t e r p r e t a t i o n

of c e n t r a l

the d i r e c t

(I),

for p = 2.

product x 6 ~

of

~ ~O ~

construction

~ = ~ xk and

the same m a p as b e f o r e ,

O

with

of p-th

a 6 k = ~O"

cf.[9],I.1.

Case

In this p a r t let k,

as an

(I)

first proved

follow easily

(x[P],~(x) p)

One can s h o w t h a t this gives

2.

S(~)

El(M)

and

is the 2 i - t h t e r m in the com-

defines

classes

~ O, c f . [ 4 ] , I I I , § 6 , 8 . 5 ,

this map: One associates to power map

This

hence

hence

immediately:

K : S(~)

generated

S(~)~-module.

(I)

generated

a n d its c o r o l l a r y

The homomorphism

~ ~

w h a t was

yields

one gets

is a f i n i t e l y

as an

induction

over

a structure

S(~)

f2 6 s 2 i ( ~ ) .

is a c o c y c l e ,

the p r o p o s i t i o n

Now

generated,

M = k. The d e r i v a t i o n

f2

as an

1.9(3).

dimensional

is a f i n i t e l y

p = 2

Er(M)

generated

p % 2

over

M = k

(I) to

of

H:(~,M)

for

Er(M)

is f i n i t e d i m e n s i o n a l ,

is f i n i t e l y

module

case

on

algebras.

~ Ei'i(k) a s t r u c t u r e as an r ihO are t h e n h o m o m o r p h i s m s of S ( ~ ) (1)-

(M) is just the a s s o c i a t e d

is a f i n i t e

qenerated

ev H~ (~,k)

the s t r u c t u r e

S(g~) (I) by

a l s o on on

of c o m m u t a t i v e

to

(M) is f i n i t e l y

graded module

The s p e c i a l Corollary:

M

Hi(~,M)

over

So we h a v e s h o w n at l e a s t Proposition:

i~O H~ev (~,k)

~-module

then each

The s t r u c t u r e

associated

(I)

is a s u b q u o t i e n t

generated

We h a v e via dule.

2i H~ (~,k) . So we h a v e c o n s t r u c t e d

~

The d i f f e r e n t i a l s

dim(M) < ~ ,

imply

=

of

by r e s t r i c t i o n

Er+I(M)

is f i n i t e l y 1.9(4)

image

G

T c B c G

be a c o n n e c t e d be a m a x i m a l

and r e d u c t i v e

torus

algebraic

and a B o r e l

subgroup

group of

G.

98

Denote the u n i p o t e n t radical of

B

by

U. Let us assume

G # T. Set

= Lie(G), b = Lie(B), ~ = Lie(T), ~ = Lie(U). We w a n t to assume that all these groups are d e f i n e d and split over sponding F r o b e n i u s endomorphism. racters of

T

ponents of

G.

2z2 As

u

and by

h

the m a x i m u m of the Coxeter numbers of all com-

is an ideal in

for any b - m o d u l e

~1

so

(I)

with

b = u ~ t

~

T

M~

one has

M ~ = (M~) ~

is a direct p r o d u c t of

is a d i r e c t product of p-Lie algebras

(as in 1.3). T h e r e f o r e

dule is s e m i - s i m p l e and implies e a s i l y

b

M. The a l g e b r a i c group

m u l t i p l i c a t i v e groups, i s o m o r p h i c to

Denote by

~ . Let F be the correP X(T) the group of all cha-

M~

(cf.I.5)

any r e s t r i c t e d ~-mo-

is exact on r e s t r i c t e d t-modules.

(for all r e s t r i c t e d b - m o d u l e s

M

and all

This

i £ ~):

H~i (~,M)~

H~ (b,M)

We can apply the c o n s t r u c t i o n of a spectral sequence as in 1.10(2) the Lie algebra

u

and a r e s t r i c t e d b - m o d u l e

to

M. Then all terms will

be r e s t r i c t e d t-modules and all d i f f e r e n t i a l s will commute w i t h the o p e r a t i o n of

~. T h e r e f o r e taking ~ - f i x e d points yields a spectral se-

quence c o n v e r g i n g to (2)

H:(~,M)

E 'j = (si(~ ~) (1) ~

2z3 Any T - m o d u l e

M

because of

A 3"- 1"( ~ )

~

(I):

M) ~ ~ H,i+j (b,S) .

is the direct sum of its w e i g h t spaces,

and

M~

is the direct sum of all w e i g h t spaces c o r r e s p o n d i n g to a w e i g h t in pX(T). L o o k i n g at the weights of 2.2 or [14],II,12.10) (I)

If

p > h, then

Ai(u~) ~ = O

On the other hand,

and

~

an e l e m e n t a r y argument

for all

(cf.[1],

i > O.

as all groups are d e f i n e d over

a d j o i n t r e p r e s e n t a t i o n of Si(u~) [I]

A(~*)

yields:

T

on

~

and on

S(~).

acts trivially on this module,

the spectral sequence in 2.2(2)

degenerates

for

F ~ - so is the

So ~ i s ( ~ ) (I)

cf.

1.8. T h e r e f o r e

p > h

and

M = k. We

get, using the same c o n v e n t i o n as in 1.7(4): (2)

If

p > h, then

H:(b,k)

~ S' ( ~ ) ( I )

This was first proved in [I],2.3. There are in [I],2.9(2) on

H~(b,M)

for other r e s t r i c t e d b-modules

2=4 For any B - m o d u l e

(I) with

ind~ M = G

M

{f : G ~ M

M

the induced G - m o d u l e

with ind~ M

r e g u l a r l f ( g b ) = b-lf(g)

acting by left t r a n s l a t i o n . ( H e r e

a finite d i m e n s i o n a l subspace

M' c M

also results

dim(M)

= I.

is d e f i n e d as

for all g £ G , b £ B }

"regular" means that there is

d e p e n d i n g on

f

such that

99 f(G) c M'

and such that

G ~ M'.) Then les and

is regular in the usual sense as a map

~M : ind~ M ~ M, f~-~f(1)

is a h o m o m o r p h i s m of B-moduG _~ induces an i s o m o r p h i s m HomG(V,indBM) HOmB(V,M)

~'~ ~M o ~

for any G-module

V. The induction

right adjoint to the dules.

f

(exact)

It has right derived

functor

forgetful

One can associate __~C~/B-m°dules on

Hn(G/B,~(M))

for all

trivial B-module

k

n G Rind B k

More generally

n 6~,

theorem implies k

for

n = O,

[

O

for

n > O.

~ M

for

n = O,

t

for

n > O,

O

2.5 For all G-modules in

V,V' V~

tural isomorphisms,

as

any h o m o m o r p h i s m ~

that

V~

(V~) [-I]

This construction

forgetful

functor

tor

M~

and

functors map injective

(I)

~(k).

of G-modu-

V '[I]. So one has na-

to

obviously

functor ~ o m functor

V~-~ V [I]

to B-modules

Therefore

to all algebraic

is obviously

isomorphic

functor and then the func-

also the adjoint

in [I],3.1.)

E i,j 2 = H~i(~,RJind ~ M) [-I]

groups

on G-modules with the

functors

are isomorphic.

converging

These

being right ad-

Therefore we get two Grothendieck

E2-terms

V [I]

B.

M ~ - ~ ind~((M~) [-I])-

(first constructed

{G-modu-

V~--~

H~(~,?) [-I]

objects to injective objects,

joint to exact functors. abutment with

~ :V '[I] ~ V

to the exact

of at first this forgetful

M [I] on B-modules.

sequences

as

~ HOmG(V, ' (V~) [-I])

of the functor

from G-modules

M ~ - ~ ( ( i n d ~ M)~) [-I]

~G/B

4.6):

cf.e.g.[14],I.4o8.

is a (left exact)

generalizes

Fp, especially

The composition to the composition

~(M)

cf.I.8:

functors can be identified with

defined over

free sheaf

R n ind BG M In the case of the

sheaf

acts trivially on

les} to itself which is right adjoint Its derived

(All

I 3.)

M:

tensor identity",

HOmG(V, [I],v ) ~ HOmG(V, [II,v~) This implies

that

(cf.e.g.[14],II

~

because of the "generalized

les takes values

a locally

cf.e.g.[14],I,5.12.

one has for any G-module

n G Rind B M

(3)

M

It has the property

to B-mo-

R n ind,.

[14],chapter

one gets just the structure

So Kempf's vanishing (2)

cf.e.~.

to any B-module

G/B.

is left exact and

functors which we denote by

this is true in greater generality, of

ind~

functor from G-modules

spectral

to the "same"

100

respectively Ei,J i G j = R indB(H~(b,M) 2

(2)

2.6 Because of 2.4(2)

[-I]).

the spectral sequence 2.5(I)

degenerates

for

M = k. As 2.5(2) has the same abutment as 2.5(I) we get therefore a spectral sequence w i t h E~, j

(I)

i S j [-I]) .i+j [-I] = R indB(H~(b,k) ~ H~ (~,k)

Suppose now

p > h. Then

compute all

Rlind~(S3u~).

and all

G R indB(SJ~)

C o m b i n i n g this w i t h

= O

and we have to

(cf.[I],3.4 -3.6)

for all

i > O.

(I) we get for

p > h:

i [-I] H~(~,k) ~

(3)

is known by 2.3(2)

One can show

for any

. _G_i/2 ln~B~ (u ~)

for

i

even,

O

for

i

odd.

There is a natural h o m o m o r p h i s m

S(~ ~) ~ ind~S(~ ~) = ~ i n d ~ S i ( ~ ~) iaO f 6 S(~ ~) to the function G ~ S ( ~ ) , g~-~ f o Ad(q)-11u.

Map any

kernel consists of all functions v a n i s h i n g on of n i l p o t e n t elements in k[N]

(4)

IWI of the Weyl group k[~] ~

cf.[I],3.9.

If

A d ( G ) ~ = ~, ~ e

~

into

ind~S(~).

The

variety

one gets,

if

W

of

p

does not divide

C o m p a r i n g the dip

does not divide the

G:

ind~S(~), p > h, then

IWI. Therefore:

ev [-I] Theorem: If p > h, then H~ (~,k) is isomorphic to h2i+1 (~,k) = O for all i. This was

:

~. We get thus an e m b e d d i n g of the algebra

of regular functions on

m e n s i o n s of the h o m o g e n e o u s pieces order

p

j 6N : i

(2)

H:(b,k)

first proved

(for

k[N]

and

p ~ 3h - I) in [6]. The a p p r o a c h des-

cribed here is taken from [I]. 2.7 Some results in 2.6 can be i n t e r p r e t e d in a d i f f e r e n t way. Set Y = {(gB,x)

6 G/B × ~ I A d ( g ) - I x

be the two projections. isomorphic to

6 ~}. Let

The map

~

Hi(y,~y). rates

(~

for all

i. As

~

and

T : Y ~

is locally trivial w i t h all fibres

~, it is e s p e c i a l l y affine. The sheaf

in 2.4 can be checked to be equal to Hi(G/B,~y)

n : Y ~ G/B

~

~y

is affine,

~ ( S ~ ~) m e n t i o n e d

. Hence

G R i indB(S£~).~ =

one has

H i ( G / B , ~ c~¢/.~)

The spectral sequence

Hi(N,RJT,~y)

b e i n g an affine variety)

and yields

~ Hi+J ( Y , ~ y ) ~degeneisomorphisms

101

Hi(y,~y)

(1)

~ (Ri~,~y)

RIT,~y

(~).

SO

= O

2.6(2), (4) i m p l y

for all

i > O,

and

(2)

If

(p, IWl

= I, t h e n

I*~Y

~N" =

The m a p (from

T

is S p r i n g e r ' s

[17]).

It is p r o p e r

(Any

p

not

dividing

that

~

is n o r m a l

gous

theorems

Hesselink

over C

It is i m p o s s i b l e theorem from

gets

due

For

p = h

There

Y

good"

p

variety birational.

is smooth,

results

(2)

implies

generalise

(the n o r m a l i t y

of

N)

analo-

resp.

to

(I)),cf.[16],[IO]. a bound

false

on

for

are a l s o

the r e s u l t s

"very

good.)As

to K o s t a n t

to a v o i d

of the n i l p o t e n t

for

= I. T h e s e

as in

definitely

3.3(2).

at l e a s t

(p, IWi)

(the v a n i s h i n g

the tely

and

IWi is v e r y

for

2.8

resolution

some

depend

p

in T h e o r e m

p < h

as w i l l

examples

on the

in

2.6.

Indeed,

follow

immedia-

[I],6.5 - 6 . 2 0 .

isogeny

class

of

G. For

G = PGL

and a l s o for G = GL one has s t i l l H:(g,k) [-I] P P indUS' (~*), but this is no l o n g e r true for G = SLp, cf. [I],6.3. For

sults

G-modules

of the G H: (~,indBM) [ - 1 ]

on

information One

in case

can a l s o

Hitker(Fr),k)" " " 1.8/9.)

Here

In g e n e r a l generated

3.

3.1

called

are

algebra

For H:(~,M ~ X (M) of =S

complicated

case

dimensional to h a v e

r = 2

p-Lie

finite

is f i n i t e l y

in

results,

dealt

algebra

~.

is a f i n i t e l y in

[7],I.11.

k. All

over

variety

~-module M the a n n i h i l a t o r ev (-I) H~ (~,k) defines a closed support

cf.

cf.[I],2.4.

with

over

generated

variety

the

restric-

variety

of

k,

over

so its k.

It is

of subvariety

M. O b v i o u s l y

~(k).=2 Such

varieties

have

been

studied

re-

dimension.

dimensional

. It is c a l l e d

are

complete

type.

Hev(ker(Fr),k)

is a f i n i t e of

there

rather

cohomology groups i H,(~,?) = H ,i( k e r ( F ) , ? ) ,

that

more

= I

restricted

M~) (-I) X

of c l a s s i c a l

know whether

H~V(~,k) (-I)

X =S cohomology

any

G

for the

assumed

spectrum the

even

except

be a f i n i t e

The

=

to e x p e c t not

Let

maximal

~

does

Varieties

~

for

r > I. (Recall

Support

ted ~ - m o d u l e s

ind2M with dim(M) D [I],3.7,5.5 They give

for the H o c h s c h i l d

one has

algebra,

in

p > h

ask

with

one

type

extensively

for

finite

groups

102

(instead of p-Lie algebras),

cf.the

survey in [2],2.24 -2.27.

sults proved there carry over to our situation

Some re-

(cf.[8],I.5,3.2),

for

example: (I) dim ~g(M)

= 0 ~ M

is an in~eqtive

restricted module.

(One has to know that in the category of all restricted ~-modules jective objects

are projective

and vice versa.

ral theorem about finite dimensional complexity

of

M

constant

b

and

of

M

projective

dim =Xg (M)

defines

like

u[P](~).)

The

such that there are a

resolution

complexity

0 - M

~ Po ~ PI " P~[1"'" dim(Pn)

~ bn

S(g~)-module.

Let

V (M) be the variety S(T).

Vg(k)

H~" (g,M®

any

in

g

Proposition

and that

f(Xg(M))

M~ )(-I) is made into

defined by the anni-

1.11 implies that = Vg(M)

f

for any restric-

= dim Xg(M)

is the complexity

and

= {O} ~ M

is an in~ective

So one may hopeDot tO with the

<

M. So also

dim Vg(M)

Vg(M)

M.

f : Xg ~ g. Via

H:(g,M®M~1~gin

ted g-module

of

ev (-I) < : S(~[~) ~ H~ (g,k) as in I .11. It

the h o m o m o r p h i s m

is a finite map onto

(2)

c

of restricted ~-modules with

is the .....

a morphism

hilator of

(I)

Hopf algebras

integer

n. Then:

3_.2 Consider an

a

in the category

for all (2)

is the smallest

in-

This is however a gene-

Xg(M),

than the others. tiplication),

restricted g-module.

lose too much information when w o r k i n g not

but with the Any variety

Vg(M) Vg(M)

and any homogeneous

occur as

_~g(M) for a suitable

image of

Vg(M)-{O}

which seem to be more accessible is homogeneous

(for the scalar mul-

and closed subvariety

M. If

M

in the projective

of

is indecomposable, space

P(g)

Vg(k)

can

then the

is connected,

cf.[9],

2.2. The only homogeneous

subvarieties

of

~0 = k

as in 1.3 are

and {O}. So

(2) implies

(3)

is not i nj.ective as a restricted ~O-mOdule,

If

M

3~3 The construction

of the

m o m o r p h i s m of p-Lie algebras

(I)

~,(M)

where we regard

c

~g(M) and if

is natural. M

If

as a restricted ~'-module

via

then % ( M ) =

~ : ~' ~ ~

is a restricted

yg(M) M

~O

for any restricted ~O-mOdule:

~.

~O"

is a ho-

~-module,

then

103 For any

x 6 ~, x ~ 0

Lie s u b a l g e b r a of

~

y i e l d one i n c l u s i o n Proposition: (2)

~(k)

with

(" m")

Formula

by

kx

is a p-

(I) and 3.2(3)

(2),(3) below.

= {0} U {x 6 ~ ( k )

M

Ix • O, M

is not injective as a restric-

k x -module}.

(2) is p r o v e d in [13], formula

[8], for

p = 2

(" c")

by

(3) in [9]. This second result (for u n i p o t e n t p-Lie algebras

[13]). In order to get the only inclusion still

one wants to apply

(1) to some e m b e d d i n g

some finite d i m e n s i o n a l v e c t o r space tion

So

One has

had been known b e f o r e in special cases

missing

the subspace

(as above).

= {x 6 ~ I x [p] = O}

~H(M) ted

= 0

gO

in the formulas

and for any r e s t r i c t e d ~ - m o d u l e (3)

x[P]

i s o m o r p h i c to

V

resp.

~ ~ g£(V)

for

to the given r e p r e s e n t a -

~ ~ g%(M). Thus one has to prove only for any finite d i m e n s i o n a l

v e c t o r space

V

that ....~ (k) c £ ~ V ~9_{~v ~ {x 6 g~(V) Ixp = O} and that

~gi(V) (V) c {x 6 g ~ ( V ~ x

= O, x

O, V

not injective

for

kx} U {O}.

This p r o b l e m can be r e d u c e d to a similar one for the Lie a l g e b r a a Borel s u b g r o u p of The proofs

for

b

GL(V)

using the spectral sequences

b

of

from 2.5(I),(2).

are still rather complicated.

3.4 For all r e s t r i c t e d ~ - m o d u l e s

MI,M 2

one gets i m m e d i a t e l y

from

3.3(3):

(I)

~ ( M I ~ M 2) = ~ ( S I) U ~(M2),

(2)

~(M I ~

M 2) = ~ g ( M I) D ~ ( M 2 ) ,

and for each p-Lie s u b a l g e b r a

(3)

h

of

~:

gh(M1) = ~ ( M 1) N h" Any r e s t r i c t e d ~ O - m O d u l e

(as in 1.3) is i n j e c t i v e if and only if

all its i n d e c o m p o s a b l e d i r e c t summands have d i m e n s i o n equal to 3.3(3)

implies

for any r e s t r i c t e d g - m o d u l e

I_~f (p,dim(M))

3.5

Let us assume from now On that we are in the s i t u a t i o n of 2.1 and

any

I £ X(T)

simple roots in

= ~(k).

some a d d i t i o n a l notations.

let

root s y s t e m of

~(M)

M:

(4)

let us introduce

= I, then

p. So

For any T - m o d u l e

V 1 be the w e i g h t space of w e i g h t

G

w i t h respect to R

such

tive roots w i t h r e s p e c t to

T

by

R. Let

that the w e i g h t s of S.

T

V

and

I . Denote the S

in

be the set of u

are the nega-

104

For closed roots ~a

any

in

I c S

of

G

Z I. T h e n

with

I

M^

the

= ~

let

generated ~I

= Lie(GI)

s 6 RI = R D ~

A £ X(T)/Z

(I)

subset

subgroup

I. F o r

GI

be

the

(connected

by

T

and

the r o o t

is the d i r e c t

any

G-module

and

s u m of

M

reductive)

subgroups

and

~

any

for all

a n d of all class

sum

M~

I£A is a G I - s u b m o d u l e same

argument

as

(2)

If t h e r e

is

of for

M

and

3.4(4)

M

is the

direct

sum of all

M^ . T h e

yields:

A 6 X(T)/Z

I

with

(p,dim(MA))

= I, t h e n

{x 6 ~ i l x [p] = O} c ~ ( M ) .

3.6 L e t

(e(1) II £ X(T))

Z [X(T)]

and

let

be

group

ring

there

is a r i n g h o m o m o r p h i s m

ri(e(1)) For ch(M)

the

Z[X(T)/Z

I] . So

rI

for all

any

dimensional

finite

ch(M)

be

basis the

of the

group

ring

canonical

basis

of the

I,~

6 X(T)

e(1) e (~) = e(l+~)

= e(l + ~ I )

is d e f i n e d

canonical

(e(A) IA 6 X ( T ) / Z I )

: ~[X(T)

for

all

] ~[X(T)/~I]

and

with

I 6 X(T). G-module

M

its

formal

character

as =

Z 16X(T)

d i m ( M I) e (I) 6 ~ [ X ( T ) ] .

Then (I)

r I ch(M) Let


v

=

0 6 X(T)~

> = I

B 6 R. F i x

all

a £ S. Set

dule with mula

holds,

(2)

ch(HO(1)) Let

w = WlW 2

=

operating

(1-e(-s)) -I

be

the

subgroup

and

let

with

X I chHO(1)

i.e.

= i n d ~ ( k I) w h e r e

= B

8 6 I. T h e n

with

dominant,

H ~>0

WI

(1-e(-a)) a>O 1-e(-s)

s u m of all p o s i t i v e ~v we w r i t e for the

where

I 6 X(T)

HO(1)

the

kI

with is

dual



k

Then

roots.

l(t).

Weyl's

~ w6W

det(w)e(w(l+p)

root

Z O

regarded

as

So of

for

as a B - m o -

character

for-

cf.[14],II,5.10:

B E I

for all

a 6 S

some

t u 6TU

d i m ( M A) e (A).

be h a l f

for all

any

with

Z A6X(T)/ZI

WI each

wI 6 WI = XIX2 a > 0 =

be

and

by all

set of all

w 6 W

where and

generated the

simple

w 6 W

can be w r i t t e n

w 2 £ W I. F u r t h e r m o r e XI

a ~ Z

(resp. I

X2)

(resp.

is the

with

Z det(w) ~ det(w')e(w' w6W I w'EW I

-p). reflections

with

uniquely we

> 0

in the

form

can write

product

a 6 ~I).

(w(l+p))-P)X21

s8

w-1(~)

of all Then

105

NOW

det(w')e(w'(w(l+p))-P)X21

Z

is the formal

character

of the

w'6W I H OI ( w ( l + p ) - p )

Gi-module HO(1). rI

induced

from

Let us use the a b b r e v i a t i o n

to the c h a r a c t e r

B D GI

in an a n a l o g o u s

w,l = w ( l + O ) - p .

of H ~ ( w ( l + p ) - p )

one gets

way to

If one a p p l i e s

e(w.l + ~ I ) d i m ( H ~ ( w m l ) ) .

Hence: (3) ri(x1 chHO(1)) = Z _ d e t ( w ) d i m ( H ~ ( w , l ) ) w£W ± SO,

if there

' w41

~ Z I for all w' ~ , w ' #

is

w 6 W I with

is not d i v i s i b l e by

by

p, so in that case H O(w.l) I

of

dim H~(w,l) where

the p r o d u c t

for any (4)

If

3±7

holds

n +I

t~(1)

=

and w i t h = I

w.l

then

-

rl(chHO(l))

{x 6 ~ i l x [p] = 0} c ~ ( H O ( 1 ) )

R1

that and

a = ~i - ~j

SO there

are at m o s t

Choose

I c S

P ~ £I

implies

is e q u i v a l e n t Let

W1

to

basis p

such that

are i d e n t i f i e d

of all

W

Lj

from

so

Let

£I = r

and

Lh

with

I +p =

with

if there

O ~ j < p. A root

is some

h

contributes

~. -~. ,~ . . . . . lhl lh2 ~ 2 - ~ih3' of

(for any

with to

a com- ~. lh~"

£I = r ~ p.

A£I_1

x A i 2 - 1 x ' ' ' x A £ -I" O s R I. So (p,dimHi(~l)) = I

w 6 WI).

by all r e f l e c t i o n s

the s y m m e t r i c

young subgroups

i,j 6 L h-

R1

'~lh,tn-1

R%, h e n c e

for all a 6

generated

with

x ... X A m r _ 1 .

partition,

R I is of t y ~

w ( R I) D R I = ~

be the g r o u p

p ~ h.

~ ( 1 ) = ( m I ~ m 2 ~ ... ~ m r > O)

components

p >

a £ R I. One can i d e n t i f y W1

union

with

for

n+1 I (ai+a)£ i i=I Lj = {ill S i ~ n + 1 , a i ~ j m o d p}. T h e n

if and only

Ath_l

a £ R I, e.g.

we can w r i t e

< ih2 <... < i h t h }, then

of type

P}

Hence:

= I.

Am1_1XAm2_1

> O) be the dual

. set

R

w,p.

An. Let us use the n o t a t i o n s

£I S p" Indeed,

is in

L h = {ihl

for all

is a p a r t i t i o n

is the d i s j o i n t

for all

(p,dim H~(w.l))

is of type

ai 6 Z

formula

R I, a > O. Set R ~ = { s 6 R l < ~ + P , a V > 6 ~

is of type

I. T h e n t h e r e

a £ Q

a 6

Rw. p = w(Rp)

p > R

dimension

<w(l+P)'~V>v

all

(~I ~ %2 ~ "'" a l s

{I,...,n+I}

ponent

if

such that

s = m I. We c l a i m with

H a

= 9, then

now that

[3], p l a n c h e of

and

= I

is g i v e n by W e y l ' s

Obviously

R I ~ w(Rl)

Assume

=

is o v e r

~ 6 X(T).

The c o n v e r s e

NOW

(p,dim H~(w.l))

with (p,dim H O(w~l)) I

w

(1) and 3.5(2). The d i m e n s i o n

If

e(wol +~I) .

group

sa

with

Sn+ I. T h e n

corresponding

to dual

WI

106

partitions,

so there is exactly one double coset

w W l w -I D W I = {I}, c f . e . g . [ 1 5 ] , 1 . 3 5 . coset with on the right for all

w(Rl)

WIWW 1

with

This is then also the only double

D R I = @. By m u l t i p l y i n g w i t h an element from

(resp. from

a £ RI, a > O

WI

on the left)

W1

one can assume that w ( a ) > O

(resp. w £ WI). One checks easily that the se-

cond m u l t i p l i c a t i o n does not destroy the first p r o p e r t y as any w I 6 W I p e r m u t e s the p o s i t i v e roots not in We have thus found some claim that any w',l < w,l some and

and

w 6 WI

wI6 W I, w' % w

with

w,l - w',l ~ Z I .

~ 6 R I, a > O

with

RI

w'(a)

and as with

(p,dimH~(w.l))

(p,dim H O(w',l)) I

Indeed,

as

as

w' (s) ~ Z I. T h e r e is ,I

and

w I 6W I

with

W l W ' S a , l - w'sa,16 Z I .

w l w ' s a , l - w',l ~ I .

So

with

that

x [p] = 0

let

Q(~) be the

b e c a u s e of HO(1)

x 6 ~I

W'Sa(Rl)

D RI =

w'sa,l -w',l

~ ~

I

> w',l

and

ri(ch HO(1))

belongs

to

cannot be di-

~_(HO(l)).(Observe

p ~ ZI.) On the other hand i V~(HO(1))

is a G-module.

For any p a r t i t i o n

n

of

is n +I

A d ( G ) - o r b i t of all n i l p o t e n t elements h a v i n g J o r d a n

blocks w i t h sizes given by the parts of

~. Now

~I

x 6 Q(tn(l)). We get therefore:

(I) 2(t~(1)) c yCH°(~)). It seems likely

satisfies

there has to be

Now the c l a i m follows by induction.

p. So any n i l p o t e n t

A d ( G ) - s t a b l e as

= 1. We

= I

w l w ' s a 6 W I. T h e n W l W ' S a , l Z WlW'Sa,l

C o m b i n i n g this w i t h 3.6(3) we see that visible by

w'6 W i w

< O. T h e n also

w'sa,l = w',l - w'(a) > w',l

w's

w ( R I) D R I = ~.

(cf.[13],4.16)

that one has equality.

contains s6me

107

References [I]

H.H.Andersen,J.C.Jantzen:

Cohomology of induced representations

algebraic groups, Math.Ann.269(1984), [2]

D.Benson:

of

487 - 525

Modular Representation Theory: New Trends and Methods,

Lecture Notes in Mathematics

1081, Berlin/Heidelberg/New

York/

Tokyo 1984(Springer) [3]

N.Bourbaki:

Groupes et alg~bres de Lie, chap.4,5 et 6, Paris

1968

(Hermann) [4]

M.Demazure,P.Gabriel:

Groupes Alg&briques I, Paris/Amsterdam

1970

(Masson/North-Holland) [5]

E.Friedlander,B.Parshall: nite groups,

[6]

Cohomology of algebraic and related fi-

Invent.math.74(1983),

E.Friedlander,B.Parshall:

Cohomology of Lie algebras and algebraic

groups, Amer.J.Math.108(1986), [7]

E.Friedlander,B.Parshall:

E.Friedlander,B.Parshall: J.Algebra

[9]

235 - 253

Cohomology of infinitesimal and discrete

groups, Math.Ann.273(1986), [8]

85 - 117

353 - 374

Geometry Of p-unipotent Lie algebras,

(to appear)

E.Friedlander,B.Parshall:

Support varieties for restricted Lie al-

gebras, to appear [10] W.Hesselink:

Cohomology and the resolution of the nilpotent varie-

ty, Math.Ann.223(1976), [11] G.Hochschild: 76(1954),

249 - 252

Cohomology of restricted Lie algebras, Amer.J.Math.

555 - 580

[12] N.Jacobson:

Lie Algebras, New York/London/Sydney

1962

(Intersciene/

Wiley) [13] J.C.Jantzen:

Kohomologie von p-Lie-Algebren und nilpotente Elemen-

te, Abh.Math. Sem.Univ.Hamburg 76(1986) [14] J.C.Jantzen: [15] A.Kerber:

Representations

Mathematics [16] B.Kostant:

Representations

(demn~chst)

of algebraic groups

(to appear)

of Permutation Groups, Lecture Notes in

240, Berlin/Heidelberg/New York 1971(Springer) Lie group representations

Math.85(1963),

327 - 404

on polynomial rings, Amer.J.

108

[17] T.A.Springer:

The unipotent variety of a semi-simple algebraic

group, pp.373 - 391 in: Algebraic Geometry London 1969 (Oxford Univ.Press)

(Proc. Bombay 1968),

On

geometric

invariant

theory

Victor

G.

for

infinite-dimensional

Kac I and

Dale

H.

Peterson 2

Dedicated on

his

groups.

to

Tony

Springer

60 th b i r t h d a y .

Introduction. Let

G be

a complex

finite-dimensional a:

Cx

~ G,

lim a(t)-v t,0 some

a

is

Kempf ~:

set

by

conjugation.

of

the

his

all

says

~(Cv)

FN

Cv

all

denotes

"optimal"

the

However,

one

obtains

immediately

the

•

in

a proper Now

ipartially

parabolic

let

G(A)

be

:

existence

the

subgroup the

set

another

complex

of

(namely,

Kac-Moody

supported

by the

NSF g r a n t

supported

by

Sloan

the

the

any

Foundation.

of a

for

The

Cv

G acts

subgroups

the which

main

field

of

just E ~N

v is

purpose

definition. from

is

the

contained

in @ ( C v ) ) . group

DMS 8 5 0 8 9 5 3

foundation. 2partially

this.

corollary,

stabilizer

and

reductive

subgroups by

of N

of G, on w h i c h

consists

of a o v e r

of G

a G-equivariant

of m a x i m a l

~(cv)

for

= N 0.

subgroups

determined

to s h o w

a-unstable

projectivization

1-parameter

is u n i q u e l y

map

the

subgroup

was

of

the

N

subgroup

~ ~'7-.-.~}. The

constructed

parabolic

E PN,

parabolic

He

are

{v e VI0

fact,

on a

if

which

N =

this.

work

existence

a-unstable

in

operating

a 1-parameter

points

that,

on

group

Given

null-cone

proper

Given

of all

a-unstable;

the

where

of

associated

centralizers

in

V.

called

N O of

elaborated

~ ~,

the

set

algebraic

space

v e ? is

theorem

[i0]

FN

denotes

The

contained

Hilbert-Mumford

map

vector

a point

= 0.

reductive

associated

and

the

to

a

Guggenheim

of

110

generalized

Cartan

be a c o n n e c t e d C,

and

Given

let A be

matrix

its Caftan

of G.

Recall

simply-connected

a commutative

R-poi n t s

A.

Then

(almost)

matrix

algebra

the f o l l o w i n g simple

denote

= G_IE and

(b) G(A)

called

in t).

finite

The direct

type

groups,

analogues

are called

We

introduce

a natural

group

of direct

sums

affine

of finite

type

of weights

then

for any G ( A ) - m o d u l e

the H i l b e r t - M u m f o r d

a-unstable

theorem

class

remains

denoted

Kac-Moody

trivially

of 1 - p a r a m e t e r

non-trivial,

group

by X,

however.

a reductive

If A is not

the c a t e g o r y hence

a of G(A)

This was

X the set

V = N 0 and

The question

subgroups

(a) are

of the

over

G(A).

half-space,

holds.

of Laurent

(b) and their

modules

V from

in an open

extension

groups.

of f i n i t e - d i m e n s i o n a l

of V is c o n t a i n e d

out an optimal

Kac-Moody

of

of example

of example

over

matrix.

is the algebra

generalization,

to the case of an a r b i t r a r y

Caftan

is a central

of groups

and the groups

twisted

category

products

Let

group

by ~R the group

by c x of the group G_~[t,t-i J (where C[t,t -I] polynomials

algebraic

and A its e x t e n d e d

R over C,

(a) G(A)

two examples.

of p i c k i n g

such

that

the s t a r t i n g

v is

point

of our work. The main the absence

of a simple

(the K i l l i n g does,

difficulty

Section

1 is devoted

dista n c e

function.

does

not

We introduce

by p r o p e r t i e s entirely

setup

(i)

and

sole)

always

groups

the "size" exist

a "distance

(ii)

to the study

A (probably

of K a c - M o o d y

which would measure

by Kempf

need not be positive). (defined

and

of a

if it

function"

of P r o p o s i t i o n of p r o p e r t i e s

disadvantage

is

I.I).

of the

of this

function

is

it is transcendental. Using

the distance

G(A)-module contained This

function

form e m p l o y e d

instead

that

in the general

V from

the c a t e g o r y

in a finite

solves

function,

a problem

type

we show

in Section

X the s t a b i l i z e r

parabolic

subgroup

posed by Slodowy

[14].

2 that

for any

of any point

(Proposition The same

2.2

in PV (c)).

techniques

is

11t

allow

us

to

subgroups Bruhat

is

several

G(A)

(Theorem

of

and

contained Our

obtain

Tits in

Theorem

1,

Tits

for

arbitrary

for

G(A)

into

3.1

maximal

complex

Second,

we s t u d y

particular,

we d e r i v e

of

our

3.6

(resp.

with

finite

results

conjugacy

is

every

In

by

a subgroup subgroup.

subgroup proved be

of

G(A)

by Bruhat

an

open

relies

on

classes

subgroup

the

compact)

problem

of

in

[13]

finite

conjugated

into

developed

in

this

may h e l p

K(A))

(resp.

of

the

the

GL ( C ) . n

to

of

algebraic

(resp.

that

the to

in

proof

the

Proposition and

Involutions of

affine Bausch

of biregular

make p r o g r e s s

and,

Kac-Moody

geometry

hopeful

2)

Theorem

automorphisms

[11]

We a r e

In

for

passing

kind)

of

K(A)).

Finally,

first

group

eonjugacy

3.3).

Note

order

is

(Theorem

results

heavily.

every

the

G(A)

whereas

in

a Kac-Moody

Kac-Moody algebras.

problem

be

of

[13].

by Levstein

of

G(A)

above in

developed

subgroup

we o b t a i n

[13],

of

of

(Proposition

obtained use

for

subgroup

(resp.

of

subgroups

we s h o w t h a t

subgroups

do n o t

classified

paper

First,

subgroups

were

known o p e n

reductive

problem.

to

bounded

defined

Borel

theorems

tort)

finite

o f Cn c a n

this

was

bounded

particular,

automorphisms

were

a well

is the

seems

compact

compact

automorphisms order

K(A).

versions

situation

Moody a l g e b r a s

that

3.5).

these

of

reductive

Ad-tocally

conjugate-linear

It

(resp.

G(A)

and

Some o f

deals

(resp.

of

Ad-triangular

infinitesimal

This

conjugacy

form

tort

"global"

of

as

a bounded

(and

theory

unitary

infinitesimal

algebras

[4]

of

a "standard"

(Propositions

in

the

a variety

its

conjugate

that

parabolic.

systems

we u s e

prove

subgroup

system,

cosets

claims

so-called

subgroup,

Tits

double

the

systems).

3,

reductive

of

type

Tits

Tits

and

arbitrary

union

of

A bounded

particular,

affine

2 to

1).

an

a finite

Section

Section group

in

in

and

for

a finite

contained

In

[4]

characterizations

to

Kac[1]). show

automorphisms that the

techniques solution

to

3

112

~I.

A distance

i.I. in

The

[5,

proofs

of

Chapters

Let

I be

generalized integers unique A,

function

up

where

a set

matrix,

further Let

triple space

(~)ieI

<~j,~>

= aij.

r i e Aut

group

Given

subsection

may

be

found

~,

W c Aut

a subset

a.. ii

(~

let

a.. Ij

a.. ji

c ~

are

Q =

are

= 0.

,n,~v), ~ of

Pat

A = (aij)i,je I be

= 2,

over

i e I, b e

~R

J of

and

non-positive Then

called

dimension linearly

z i~i, ieI

Q+

there

the

a of

and

independent Z ieI

exists

realization

2n-rankA,

=

a

sets

Z + a i. H e r e

and

.... }.

Z+={0,1,2

ri-h The

in t h i s

s.. = 0 i m p l i e s ij

is a v e c t o r

on,

cone.

stated

i.e.,

isomorphism

= { a i } i e I c ~R' satisfying

facts

Tits

of n elements,

i ~ j and

to ~R

all

the

3,4].

Caftan

for

on

the

fundamental

= h - a[

generated I, w e

by

denote

all

, h e ~. the

b y Wj

reflections:

r i is c a l l e d

the

subgroup

the Weyl

~roup.

of W g e n e r a t e d

by

(ri}ie J • For and is

J c

only

I let

if Aj

a subset

of

Aj

= ( a i j ) i , j e J.

Then

is a m a t r i x

of

finite

f~n.ite

of

I.

t~Ee

the

type.

group

In t h i s

Wj

is f i n i t e

case

we

say

if that

Let C = {h e g ~ [ < a i , h > be

the

z+(h) One

fundamental

chamber.

= Z Z+<~i,h> ieI

has

(i.I) The

[5,

Exercise

h

w.h

For v

and

Q+(h)

=

> 0,

i e I)

h e C put Z i+(h)a iel

.

3.12]:

e Q:(h)

if h e C,

w e W.

set X =

is a c o n v e x

cone

in g R

called

W-equivalent

to

a unique

point

the

U weW

w.C

Tits

of C.

cone. The

Every

stabilizer

point

of

W h of

X is h e C

is

J

113

Wj,

where

J

is

finite. The

=

{i

Here

cone

1.2. is

(contained

M In

of ~ this

Proposition F:

Int

in

v,H)

a realization

is

by The

in ~ )

the

stabilizer

IntY

S of ~ R

contained

The

further

reflections.

A subset

subset

= 0}.

of ~ * i n d u c e d

fundamental Tits

and ( ~ ,* ~

triple

automorphisms

e II<~i,h>

is

stands

the

for

denoted

called

by

of

of

of

IntX

a set

tA,

Y.

the

corresponding

C v and

if

it

IntX v with

point

matrix

fundamental

admissible

intersection

any

interior

the

the

corresponding

are

the

of

r.1 b e i n g

of

M

chamber

and

X v.

is W - i n v a r i a n t - Q+

for

some

and finite

. subsection

I.I.

Xv

There

, (0,~)

(i)

F is W - i n v a r i a n t

(ii)

for

every

we

shall

exists

prove

the

following

a real-analytic

function

satisfying: and

admissible

strictly S c ~

convex; , the

series

z A~S p~SnC v

converges.

A function a distance Let

{xj}

F satisfying

function. be

Such

(i)

and

(ii)

of

Proposition

i.I

a function

can

be

constructed

as

a basis

of ~R

such

that

F(A)

Z Z j w~W

e
all

xj

are

in

IntX.

is

called

follows.

For

A E ~

let (1.2) To

prove

Lemma e

n

that

i.i.

,n ~ Z

that

the

function Proof.

F is

Let ,span

series f The

is

=

a distance

V be V.

function

we

a finite-dimensional Let

f(u) strictly

convexity

>.

U be

= Z nEZ

the

subset

need

vector of

on

assertions

U and are

lemmas.

space

over

V ~ consisting

e converges.

convex

three

real

obvious,

Then

U

analytic since

is on

e

of

x

R

and

all

convex, IntU.

Is

a

let

u such the

114

positive-valued

and

strictly

convex

complexification

Vc * = V* + iV*

linearity.

the s e r i e s

U + iV*.

Then

Moreover,

convergence U + iV*. (IntU)

Lemma

is u n i f o r m

Hence

f(u)

the

extend

the p a i r i n g

converges

convexity

on c o m p a c t

absolutely

of e x,

polyhedra

is a c o m p l e x

analytic

V* in its

embed

this

on

absolute

contained function

<,> b y

in

on

+ iV*.

1.2.

Z h'eW.h Proof.

If A e Int

Since

Int

X v is the

, w e W,

in q u e s t i o n

X v a n d h e X,

then

the s e r i e s

converges.

e

w . ( I n t C v)

that

using

and

f(u)

We

function.

and

since

is W - i n v a r i a n t

^ e IntC v.

convex

Since

the

and

hull

of the u n i o n

region

convex

X is the u n i o n

of the sets

of c o n v e r g e n c e

(see

Lemma

i.I),

of the w-C,

of the

series

we may

assume

we may

assume

that

h

e C. We have: = e

Z e h'eW.h by

(i.I).

Lemma

for

1.3.

any

Proof.

The

last

series

If h e IntX,

e

S ¢ ~R"

The

in

series

e

Lemma 1.2,

we m a y

admissible

set,

, where

assume

the

it

last

that

proving

the s e r i e s

question

and hence

z

e -

h"eO~(h)

converges,

then

admissible

Z A,peS

e<

h'eW.h

Z e weW AeS peSnC v N

Z

may

be

rearranged

series

by

max pe(IntXV)Nc

h e

IntC.

1.2.

Z e weW AeS peSNC v

is d o m i n a t e d

N =

Lemma

into

the

W v

series

I"

As

in the p r o o f

of

P

By t h e

is d o m i n a t e d

converges,

definition

by N F o ( h )

of Z

an

e-,

115 where h

F 0 is a f i n i t e

exponential

This

sum.

series

converges

for []

e IntO.

Proposition

I.i n o w

follows

immediately

from

Lemmas

I.I,

1.3.

[]

1.3. the

1.2 and

For

a

convex

IntX v

,

exists shall

by

subset hull

of

the

strict

a unique denote

minimum (b)

vector

space

where

We

let R+

(a)

of

PK e Cv"

K be Then

a

IntX v ~

S c ~R be a d m i s s i b l e .

nonempty

finite

subsets

To p r o v e

(a),

its

denote

by

subset

K of

function

absolute prove

F,

there

minimum

our

first

[Y]

on K.

We

key

~ 0}.

nonempty

PK

convex

distance

Now we can

= {r e R l r

of F on the set

Let

the

~ we shall

compact

F achieves

by ~K"

Let

over

a nonempty

convexity

point

that

a

Given

point

i~2.

IntX v such

Y.

this

proposition.

Proposition

Y of

is

the

compact unique

point

(K +

Z R+ai). i~I

Then

for

T of S such

that

any ~ > 0, ~[T]

convex of

subset absolute

the set

e C v and

of

of all

F(~[T])

> ~

is

finite. Proof. A = gK F(PK+

+ u + @' t=)

we w r i t e

where

~ F(gK)

gK

for

+ ~ E K

and @ e z R + a i. i

Da F(~K) F is s t r i c t l y

of e a c h

interval

convex

[g,ri.M],

and

since

PK

form:

Since

> 0.

assumes

> 0

e C v, we

(i.4) Combining

in the

the

same

value

at the

endpoints

we h a v e

Da F(p) 1 Therefore,

(K + Z R + u i) i

0 ~ t < I, we h a v e

(1.3) Since

A E IntX v O

if

> 0.

have

Dp F(~K) ~ 0 (I.3)

and

(1.4),

we

get

Da+ p F(M K)

~ 0.

Since

F is s t r i c t l y

1t6

convex, O,

proving

only it

this

forces

To p r o v e

(b),

a

number

of

finite

show

that

the

finite

finite

to

set. only

But

a

Remarks.

we n e e d

this

finite

is

to

show

(a)

T with

of

up

to

T of

T n

since

equality

iff

a

W-equivalence,

S with

Cv ~ ¢ a n d

by

(ii)

possibilities

If A is a m a t r i x is a d i s t a n c e

W-invariant

(b)

A be

Let

degenerate

of

for

{A e ~R

a matrix

is a d i s t a n c e

Bounded

2.1.

The [5,

function

subgroups

Chapters

Let

A be

of a f f i n e

of

+ p

there

=

B e

are

F(P[T])

> ~.

F(p[T])

> ~ form

Proposition

Hence

1.1,

T n Cv a n d

A

for

of

all

•

Z

any

a

form

on A~.

let

(.,.) form

a

there

for

any

the

function

denotes

a positive

be a non-

on A R w h i c h

on the

the

Rai},

then

(.,.)

Then

is p o s i t i v e

set

function

F(A)

= (-(A,A)) -k

k > ~(n+3).

Kac-Moody

facts

type,

Here

bilinear

stated

group.

in

this

subsection

may be

found

1,3,9,10]. a

generalized

Cartan

realization.

We put • = C ®R ~R

(A e ~ * l < A , h >

~ R for

Lie

algebra

fi'

i • I, w i t h = O;

type,

let ^ • IntX v.

< 0 and A -

proofs

bilinear

W-invariant

on Q and

I(A'A)

of f i n i t e

function.

symmetric

symmetric

semidefinite

[~,~]

that

subsets

clear

number

definite

in

with

Q

= (A,A)+I

12.

> F(PK) ,

A • T.

other

F(A)

+ @)

(a).

suffices

are

F(p K + a

over

all

[ei,fj]

and

h e ~R}.

C generated defining

matrix

by

the

and

identify

let

~

The K a c - M o o d y vector

space

(~R'

algebra ~ and

;

[h,e i] = < a i , h > e i,

[h,fi]

l-a.. (ad ei) 10 e~ =

= - < a i , h > f i for

l-a.. 0 , (ad fi ) ij f.j = 0

h e ~; for

be

its

with

relations:

v = 5ij~i

~,~v)

i ~ j.

~(A)

is the

symbols

e. and 1

117

We the

have

suhalgebra

triangular We ~a

canonical

of ~(A)

have

= Cei

either

the

m ~I~ in ~ +

root

or

then

The A re

A_

set

= {w.alw

for

= ~"

~ 0}.

root

Denote

by A+

and

= -A+

and A+

~ W,

A

ei(resp,

a ~

=

so

that

root is

is an space

then

the

(resp. ieI.

• ~ ~a' ~e~

element ~,

n

We

) be

have

the

where

of

a ~ 4,

called

sets

is c o n t a i n e d

positive

of positive

or

and

negative

negative

c Q+.

is W - i n v a r i a n t .

~ G ~}.

~+

fi ),

#(A)

h E ~}, A root

Each

; the

Let

• ~ S ~+.

all

in ~ _

of r o o t s

the

decomposit~0n

' ~0

~a

~ c ~(A).

by

= n

space

= <~,h>x

~ 0,

respectively. roots;

generated

' ~-a. = C f i 1

1 : ={=

embedding

decomposition:~(A)

= {x ~ # l [ h , x ]

~. A

the

A real

If a E A re , t h e n

root

dim ~a

is an

= i.

element

Put

of

A~ e =

A re n A+,

A ~(A)-module properties (i)

V =

(ii) The

are

module

V.

The

objects

U(~+)v

c

Xv

closed

under

set

Note unless

dim

important

P(V)

if

nilpotent in

the

on

the

following

two

following

are

that

all

for

hE~};

the

of

weights

the

category

elements

for

x of ~(A)-

V which every

tensor

any

are

v ~ V)

~(A)-homomorphiams.

sums,

all

i • I.

called

Also,

~(A)-modules

are

direct

also

# 0}

V for

for

of ~ a

for

V as w e l l .

is f i n i t e - d i m e n s i o n a l

takin~

on

is W - i n v a r i a n t .

inte~rahle

Note

P(V)

nilpotent

= {A ~ ~ * I V A

; the morphisms

quotients. the

work

of X are

(i.e. Int

set

locally

shall

inteKrable

V A = {v E V I h . v = < A , h > v

locally

of P ( V ) :

~ A re a r e We

, where

f. a r e i

elements

called

satisfied:

• , VA Am~

e. a n d 1

V is

e+-locally and

Note

products,

modules.

such that

submodules

finitely-~enerated

module

The finite

that X

P(V)

is and

V of x,

is a d m i s s i b l e .

that ~(A)

the < ~

examples

adjoint (or,

~(A)-module equivalently,

of modules

from

is

inte~rahle,

A is of

category

finite

X are

hut

is not

type).

(some

of)

The the

in x most

118

integrable Let

highest

P+

= {A e ~ R

exists

a unique

admits

a non-zero

all

h e ~.

only

up

This

if A e Int

(2.1) that

for

condition

2.2.

Let For

an

We put

Given

A e P+

, there

#(A)-module

it is

c

L(A)

which

h.v A = < A , h > v A for

in the c a t e g o r y

X if a n d

[W.A]. the

condition

to

the

may be

#(A).

free

in

Ker ~,

to i n t e g r a b l e a G(A)-module,

associated

is

equivalent

is of f i n i t e

of

the of

the

type.

group

the

to

G(A)

associated

properties

of

to

G(A)

[9].

product

d~(x):

= G$/n

IntX v

The proofs

~(x)

= exp

= 0}

construction

found

the

A e

of

the

(V,d~)

additive

the

which

Thusj

we d e n o t e

to the K a c - M o o d y

by

algebra

associated

to t h e

integrable

G(A)-module

associated

to the

adjoint

x e @a'

a e re,

a

a G*-module

intersection

g(A)-modules.

~a'

, x E ~a,a

= Z (dn(x))n/n! n~0 where

groups

we d e f i n e

G(A)-module

by

below.

that ~ + - v A = 0;

is i n t e g r a b l e ;

~(A)-module

naturally

,i E I}. irreducible

vA such

integrable

G(A)

defined

X v since

algebra

G$ b e

associated

group

E Z+

: J = (i e I I < h , a ~ >

below

L(A)

isomorphism

A e Cv ,

Kac-Moody

stated

to

module

We n o w t u r n

the

modules

I

vector

P(L(A))

Note

weight

(V,~)

over

each module

~(A)

(V,~)

@(A)-module

@(A)-module

all

(V,~)

We call

and

by

e A re .

is t a k e n

(V,w).

e ~re

is

G(A)

the

the

(V,dw).

(@(A),ad)

The is d e n o t e d

(@(A),Ad). Given

under

the

an e l e m e n t canonical

homomorphism

we

denote

, G(A)

G*

by

its exp

image x.

in G(A)

We h a v e

by

definition: u(exp Let

Ua

= exp

~a

be

x) the

corresponding

to

U+a . ,

Denote

--

1

i

e

I.

the

= exp additive

real by

root

x 6 ~CX' ct E 4 r e .

dn(x),

1-parameter a.

U+ ( r e s p .

subgroup

T h e n G(A) U_)

the

is

of

G(A)

generated

subgroup

of

G(A)

by

the

generated

119

by

all

Ua

For

(rasp. each

re w i t h a e 4+

U_a)

i e I, we have

a unique

homomorphism

Fi:

SL2(C)

p G(A)

satisfying: ~i(~ Let

ti ) = e x p t e i, ~i(tI

of H. in G.. 1 1

generated

by

Let

the H i (rasp.

The F. are m o n o m o r p h i s m s 1 have

wH

an

isomorphism

identify and w U + w

then

= w-h

.

{ri}iei)

properties

of Tits

the B r u h a t

W

systems

U weWj

type,

parabolic A more

the

special

exists

homomorphisms this

connected

may

be

normal

product

F(ri) sense

= HU

the g r o u p (The

found

in

of G(A)

subgroup

of N.

of the H.. i

is the

coset

We

NiH\H.

to e x p r e s s i o n s

.

the

such

as

w E W and ~ e wH,

We have:

G(A)

is that

definition [3].)

B N N = H. the

quadruple

and b a s i c

In p a r t i c u l a r

we

BwB

(disjoint

system

is that

called

union).

given

J c I, the

a standard

parabolic

parabolic

subgroups

are

called

subgroup

coincides

with

its

group

Pj and

its

conjugates

set

Pj

=

subgroup.

parabolic

normalizer. are

called

If J finite

subgroups. property

G(A) There

B

let N i be

subgroup

If h e ~,

system.

of G(A)

A parabolic

is of f i n i t e type

of a Tits

of s t a n d a r d

subgroups.

about

= we~

is a s u b g r o u p

Conjugates

sequel.

and

decomposition:

property

BwB

that

gives

B = HU+,

is a Tits

the

direct

such

this

in the

facts

N) be

H is the

?;

~ Ex}),

H is an a b e l i a n

~ N/H

We put

G(A) Another

(rasp.

Ni);

using

of the b a s i c

H

and

occurring

(G(A),B,N,

have

~:

W and N/H -i

Ad(n)h One

fix

= e x p t f i (t e C).

G i : Fi(SL2(c)),H i = Pi({diag(t,t-l)]t

normalizer

We

~)

topology

= w e ~ B_wB

a finest

F. are 1

group

then (cf.

is the

(disjoint

topology

continuous

on G(A);

topological

of G(A)

and

G(A)

on G(A) G(A)

Birkhoff union). such

that

the

is a t o p o l o g i c a l

is a H a u s d o r f f

[8]).

decomposition:

group.

connected

We

simply

120

2.3

In t h i s

Then

V has

v # 0. v =

subsection

the

We

V is a f i x e d

structure

decompose

of

a distance

Proposition (b)

function

2,1.

(c)

if a n d

There all

(d)

Proof.

G(A)-module.

weight

space

category Fix

X.

v E V,

decomposition:

~(g') (a)

is as

= ~(g)

(c),

we

Using

~ F(~(g))

let

g'

g,

decomposition, Let

write g"

may

for

Now,

all

• G(A) g'g

= b'g.

1.3).

g e G(A),

e C v,

then

then

~(ng)

F(~(bg))

= w-~(g).

~ F(~(g)),

with

such

that

~(g)

e C v and

F(~(g'))

~ F(~(g))

(c)

and

where

if g'

E G(A),

J = (i • I[

then

g'

is as

< ~(g),ai>

in

(c)

= 0}.

case.

F is W - i n v a r i a n t .

(b)

follows

from

assume

-i By

that

1.2(b),

V = U(#(A)),v,so there

G(A)

g'

E

be

as

in

= bnb', (b),

exists

Using

(c). where

~(g")

(a),

Using b,b'

g • G(A) we

the

can

such

take

is that

~(g)

Bruhat

E B and

= ~(g).

P(V)

that

n • N,

Similarly,

say

~ ( b - l g ') =

Hence:

(2.2)

~(ng")

w-~(g)

(d).

Section

put

= ~(g).

Proposition

(c).

~(g'),

(see

and

~(g)

in t h i s

since

proves

~(g').

in

e Pj,

This

n ~ wH.

g.v]

g ~ G(A),

1.2(a).

admissible.

C v.

Given

If n e w H

g , G(A)

is c l e a r

To p r o v e

F(~(g'))

~ P(V) lv ~ ~ 0).

X v.

if ~ ( b g )

if g , g - i

Proposition

of

the

the

e G(A).

only

Moreover,

(A

= ~[supp

only

If g ~ G(A)

if a n d

=

b s B and

exists

g'

v

F on

(a)

If g ~ G ( A ) ,

equality

But

to

from

Z v , and put AEP(V) A

~(g)

for

associated

v relative

supp

Fix

the

~(A)-module

= w.~(g")

hence The

w E Wj "if"

= ~ ( b - l g ') = ~ ( g ' ) .

= ~(ng") amd

part

~(g)

by

(a).

Using

= ~(g').

This

follows

immediately

(2.2)

we

proves from

(a)

get:

the and

w-~(g)

"only (b).

if"

= part D

121

Let ~ d e n o t e G(A)

with

the a c t i o n

Proposition such

2.2.

g e G(A),

and

type

There

(c)

The

Proof.

by

There E Cv

F(p

'

parabolic

exists

v 0 = g.v

, where

2.1(d)

it f o l l o w s

(b)

follow

immediately

and

(c)

Corollary

of the p r o o f

if v e V is s u c h

Therefore,

by

Theorem

v 0 on

the o r b i t

) > F(p [supp

G(A).v

) for all

g.v0]

in the

where

map

in a f i n i t e

that

for K a c - M o o d y i.

that

The

g'

from

type

parabolic

supp

= ~(g),

= (g,g)g-i

v is c o n t a i n e d

parabolic

main

If n o w

and b y

E Pj,

proving

< ~ and

dim

in an o p e n

parabolic

theorem,

first

(c).

(a).

O

If dim @(A)

in a p r o p e r

our

~(g'g)

2.1

(a).

[i0]).

in a p r o p e r

can p r o v e

in P r o p o s i t i o n

hence

the H i l b e r t - M u m f o r d

is c o n t a i n e d

Now we

e £g-v,

(cf.

G ( A ) c v is c o n t a i n e d

theory

Pj,

g is as

Proposition

2.4.

of

of G(A). Let

O(A)cv

subgroups

line Cv 0 is c o n t a i n e d

is c o n t a i n e d

g'g.v

then

v0]

a G(A)-equivariant

of Cv

parabolic

= 0}

g ' . v 0 G Cv 0 , then

and

a point

of the

a~>

type

conjugation.

[supp

subgroup

v0]'

finite

exists

the s t a b i l i z e r

stabilizer

subgroup

of all

of G(A)

v0]

J = {i e If< @ [ s u p p (b)

set

(a)

that P [ s u p p

finite

the

subgroup

half-space, of G(A).

if 0 • ~

, then

subgroup.

result

V < ~,

[]

on g e o m e t r i c

invariant

groups.

following

conditions

on a s u b ( s e m i ) g r o u p

P of G(A)

are

equivalent: (i)

P is c o n t a i n e d

in a f i n i t e

(ii)

P is c o n t a i n e d

in the

BwB,

(iii)

w

type

union

parabolic

of a f i n i t e

subgroup;

number

of d o u b l e

cosets

~ W;

for

contained

every

G(A)-module

in a P - i n v a r i a n t

V from

the c a t e g o r y

finite-dimensional

X,

every

subspace;

v ~ V is

122

(iv)

P leaves

invariant

some

G(A)-module

(v)

P leaves

from

the

Proof.

V from

The

systems).

The

Proposition

implication on

V.

(ii)

of

x; subspace

The

implication

m-dimensional

Finally,

of some

G(A)-module

(iii) of

of Amv;

implication

(by p r o p e r t i e s

is also

subspace

subspace

the

is c l e a r

==, (iii)

2.2.

Definition. equivalent

(i) ==, (ii)

1-dimensional

--- (v).

category

subspace

V

~.

U is a P - i n v a r i a n t P-invariant

finite-dimensional

a 1-dimensional

implication

locally-finitely

(iv)

the

invariant

category

a non-zero

clear

==, (iv) V,

then

this

since

B acts

is o b v i o u s . Amu

proves

(v) --~ (i)

of Tits

If

is a

the

follows

implication from

[]

A sub(semi)group properties

(i)

-

P of G(A) (v)

satisfying

of T h e o r e m

one

of the

i is c a l l e d

a bounded

subgroups

by p r o p e r t y

sub(semi)group. Remark.

Bruhat

and proved

the

and

(i)

an open

problem.

also

is e q u i v a l e n t

We n o w called

i ~ I;

one

Given ~' (A)

[4]

equivalence

Whether

2.5.

Tits

need

of to

~(A)

J c I, we d e n o t e

generated

and

(ii)

(ii)

for

an a r b i t r a r y

b y ~'

on ~ ' ( A ) ,

algebra.

= ~'(A)

+ a+

bounded

(i)

a digression

a Kac-Moody has:

define

for

the

any

Let ~'

by ~j (resp.

(resp. g')

= ~(A) ~(A)j)

and

Tits

derived

It is g e n e r a t e d

+ A.

affine

by n ~

the

the el,

(i)

Tits

system.

system

remains

algebra

of ~(A),

the e. and 1

f., i

( = Z Ca[). iEI subalgebra

of

fi w i t h

i ~ J. A #'(A)-module locally

nilpotent an

V is c a l l e d on

Note

that

that

the ~ ( A ) - m o d u l e s

V for all

integrable

integrable i E I;

~(A)-module

L(A)

remain

then

is an

if the ~'

e. and 1

is d i a g o n a l i z a b l e

integrable

irreducible

f. are 1

when

on V.

~'(A)-module,

restricted

to

and

123

~'(A). same

The

group Let

AutA

construction G(A)

AutA

(cf.

of S e c t i o n

the

group

invariant

the

sets

action ~+

defined

G(A)

by a . e x p

x = exp Q-x,

denote q

H = Hom(o,Ex). way,

the

is the

obvious

can

this

group

complex

group

form

this

,: ~&

Aura

x Aura

and

been

by

to Q and

acts

i.e.,

on ~ by

leaves action

of

lifts

to

which

a ~ 4 re.

on ~'(A)

to an a c t i o n

acts

~ ~ AutA

= fa( i)'

and

on ~(A)

on G(A).

of C:

on ~(A),

satisfying the

in a n a t u r a l

explained,

so

<,(a~),

algebra

h E ~o

x AutA

acts

Z Rat, ieI

to F ( ~

a Lie

Now,

(i)

lifts

: (Aut E x Aut

is a f i n i t e

isomorphism

AutE

acts

AutC

a (non-canonical) an

the

A,

matrix

corresponding

a'fi

conjugation.

&Rv'=

complementary

~'(A)

action

Let

in an AutE

AutC

= (l,q},

where

~'(A)

and G(A)

in an

way

on H M G(A),

so

that

group, ) and

~o M ~'(A)

Clearly,

choose

[h,x]

extends

as

a subspace

and G(A)

realization with

~R'

gives

to an a c t i o n

are

a linear

map

of ( ~ ' ) *

~Ro

where

0

of all ~ ( A ) -

Let

= C ®~ ARo and

Put ~o

This

We

, is A u t A - e q u i v a r i a n t .

x e ~a n of A, we

which

on #(A).

and G ( A ) - m o d u l e s

compatible;

have

extends

us an a c t i o n

of G(A)

conjugate-linearly): of ~(A)

follows.

of the

@(A).

on ~'(A).

Define

= <~,h>x,

of A~ @ ~

with

let X be the c a t e g o r y

actions

by

identification

which

on ~(A)

: E @RA~.

= ai~.

By the u n i q u e n e s s

on ~(A),

(canonically)

AutA-invariant.

of ~o M ~'(A)

(~ acts

~'

we m a y

(H ~ G(A)).

acts

action

that

aj>

A) ~

G(A)

to a ( n o n - c a n o n i c a l )

, (~')*

define

group

Then

the

automorphisms

AutE

just

Z Ra[, ieI

Since

gives

the g r o u p

As has

~R' =

We have

x ~ #a c #, (A),

This

of the

linearity

of c o n t i n u o u s

G(A)

extend

to ~'(A)

way.

The we

and

by

by a.e I• = e o(i)'

on ~'(A)

obvious

extends

and ~+re.

Aura

Let

of a u t o m o r p h i s m s

= a..,i Ij ' j ~ I}.

= {a E A u t I l a a ( i ) a ( j ) This

applied

[9]).

denote

o . a i = aa(i).

2.2

V

of

to

124

(ii)

V is in X and

(iii)

H acts

sums,

Using Ai,

action

locally-finitely

The m o r p h i s m s direct

the

of X are tensor

the

of G(A)

is

induced

by

that

of ~'(A);

on V.

the o b v i o u s

products,

ones.

submodules

identification

Then and

A = ~o @ ~,,

x is c l o s e d

quotient

define

under

modules.

fundamental

weights

i e I, by: Ai{

Then for

o . A i = Ao(i). all a e Aut

(Here

we

let

L(p),

where Note

can

form

p =

that

Hence,

A,

g(v)

o : O,

then

: 6.1j.

if A = Z kiA i e P+

L(A)

is from

= g(A-A)v Z Ai, iGI

1 J

category

x

that

ko(i)

= ki

if A E IntX v.

if g e H and v e L(A)A. )

is

the a c t i o n

the

is s u c h

For

example

in ~.

of Auto

x Aura

on • n o r m a l i z e s

W,

so

that

we

a group = AutA M W.

The

group

W leaves

X v invariant;

~(~) we

get

a W-invariant

Let B = (Aut we have

the

=

distance

C)HB,

following

B_

putting z o~AutA

(cf.

(1.2)):

F o a,

function.

= (Aut C)HB_,

variants

Pj

= (Aut

of the B r u h a t

C)HPj,

etc.

Then

and B i r k h o f f

decompositions: G(A)

=

I I BwB

(disjoint

union),

weW

~(A)

=

(disjoint union>.

{ { ~_w~

wE The Theorem

following I.

The

variant

following

of T h e o r e m conditions

I is u s e f u l

for

applications.

on a s u b ( s e m i ) g r o u p

P of G(A)

are

equivalent: (i) (ii)

(iii)

P normalizes

a finite

P is c o n t a i n e d

for

in the

any G ( A ) - m o d u l e

type

parabolic

union

V from

subgroup

of a f i n i t e

the c a t e g o r y

of G(A);

number

X,

any

of d o u b l e

v e V is

cosets

125

contained (iv)

P

in a P - i n v a r i a n t leaves

invariant

some

G(A)-module

(v)

P leaves

from

the

Proof

V from

invariant

category

is

finite-dimensional a non-zero

the

subspace;

finite-dimensional

category

subspace

of

5;

a 1-dimensional

subspace

of

some

G(A)-module

X.

essentially

the

same

as

that

of

Theorem

i.

0

~

~

A subgroup called

union

(a)

of

Using

Theorem

of

(b) holds

the With

in

(c)

satisfying

one

of

(i)-(v)

of

Theorem

systems,

it

is

1 is

union

of

We c o n j e c t u r e

subgroup

of

G(A)

the

BwB.

that

bounded (cf.

the

applying

of to:

Tits

U g•P

gBg - 1

is

contained

clear

that

in

a finite

if

U seS

same formulation to

is

all

called

h set s.P

if

end

only

to

conjugacy

if

c.

bounded

is

contained

of it

Theorem

of

if

it

automorphisms

bounded

a subsemigroup

proof,

automorphisms

S of

is

and

for

every

automorphisms normalizes

of

G(A)

bounded of

G(A)

a finite

is

in

a

called

subset

P.

is

type

parabolic

[4]).

Applications

In

equivalent

essentially

bounded

uniformly

properties

BwB.

a version

uniformly

3.1.

G(A)

the

1 is

A subset

finite

13.

P of

bounded.

Remarks. (ii)

V

order

to p r o v e

our

theorems.

next

theorem,

we

need

the

following

two

lemmas. Lemma Y be

3.1. an

Let

P be

integrable

finite-dimensional

subspace

V'

P-v Let

type

@'(A)-module.

(3.1) Proof.

a finite

P = gpjg-1

where

parabolic Then

of

for

Y such

subgroup every

of G(A)

v • V,

there

that

c U(n+)V'. g e G(A)

and Pj

is

a standard

and

let

exists

a

126

finite-type Write

v'

parabolic.

= bll-v.

Then

(3.2)

Write

g = blnb2,

(3.1)

is e q u i v a l e n t

bl,b 2 G B and

to

n ~ N.

:

n P j n - l . v , c U(e+)V'

Now, i t

is easy

to c h e c k

that Pj • v o c

for

where

any v

e V.

We have

the

U(pj)v

vector

°

space

decomposition

O

@J

Recall

e I = ~+ n A d ( n ) - l , + ,

~2

that

finitely

~'

acts

finite-dimensional locally-finitely

locally and

are

on V.

3.2.

elements images

Let

such

w k in W are

Suppose

(3.3)

o supp(Ad k

Let ~k be

that

all

V.

obtain

the

bk,

~I c ~+.

Putting

ak)#a

of m i n i m a l

be an

height

V'

infinite

and

n k ~ N are

Then

the

linear

is a f i n i t e

and

are

hence

act

=

D

b~ ~ B,

for

4 2 and ~3

vectors,

(3.2).

then

O pj.

root

i G I, k=l,2, ....

contrary;

= ,

Clearly,

by real

Ad(n)

distinct.

(Ad ak)fi,

Proof.

on

a k = b k n k b ~ , k = 1,2,...,

of G(A)

(Ad ak)e i and

spanned

we

where

= ~+ N A d ( n ) - l ~ _ , ~ 3

Finally,

nU(~')U(~2)U(~3)n-l-v', Lemma

~i ® a2 @ ~ 3 '

= ~'@

is

such

span

sequence that

of

their

of all

infinite-dimensional•

all a G ~:

set.

in s u p p ( A d

ak)~a.

It

is clear

that

we

have: (3.4) From

height(Wk.a ) ~ (3.3),

(3.5)

and

(3.4)

for - a

we d e d u c e

[ h e i g h t ( W k . a ) I ~ c(a),

where

c(a)

determines

Given

a

(3.4)

height(Pk).

standard

opposite

is a c o n s t a n t w E W,

(3.5)

J c I, the

opposite parabolic

of ~ ( A ) - m o d u l e s ,

depending contradicts

set

Pj

:

u w~Wj

on a but the

B wB

Since

its

subgroups.

We a l s o

may

are ~ _ - f i n i t e

conjugates introduce

w.~ D

of the w k.

is a s u b g r o u p

subgroup;

objects

on k.

distinctness

parabolic

whose

not

of G(A)

are the

integrable

called

called category

modules

x

V such

127

that

P(V)

c -Int

parabolics be

Theorems

conditions Remark.

G(A))

2.

(i)

The

1

1 or

S can

following

and

exist

We

is c a l l e d

antibounded.

(resp.

Pj o w P j , w - l ( r e s p ,

first

and

action

give

Let

where

by

of S on ~'(A)

the p r o o f

gSg -1

3.1

.

The Now we

G(A). (ii)

1

and

there

g2Sg2 1 c Pj,,

where

their the

c

Pj The

implication

implication explain

(iii)

(iii) how

to

(i) ==, (ii)

is p r o v e d

normalizers

obvious

-1

(resp.

S of G(A)

(resp.

analogue

of

J and

proves

(ii)

==~ ( i i i )

modify

we

The

and

a subgroup

then

J'

in G(A)

are

finite we

how

to adapt

of G(A). such type

that subsets -i g2gl

can w r i t e

=

implication is

clear

from so

Theorems by

implication

Lemma 3.2.

explain

subgroup g2

the

follows

for G(A)

finite.

,

arguments

use

that

Putting

= nb~lg2)

(i)

I and w e W s u c h

locally

gl and

This

as above

in G(A).

is

n e N. (

---

of

decomposition,

, b ~ B and

N nPj,n

J'

antibounded

exist

to the B i r k h o f f

b_ e B

To p r o v e ~.

below

of G(A);

for G(A)

S be a b o u n d e d i and

J and

an e l e m e n t

g = bg I

==, ( i i ) .

on a s u b g r o u p

into

According

(i)

will

antibounded).

be c o n j u g a t e d

by T h e o r e m s

get

these

the e q u i v a l e n t

be b o u n d e d

subsets

glSgl I c Pj

we

would

type

it to G(A).

b_n-lb,

, etc;

satisfying

finite

adjoint

the

I.

x by x

replace

antibounded;

(iii)

of

if we

A subgroup

conditions

Pj and w P j , w -I)

Then,

1 hold

B by B_,

terminology

normalizes

Proof.

1

1 and

equivalent:

there

that

and

of b o u n d e d

S is b o u n d e d

(ii)

Theorems

parabolics,

adequate

in p l a c e

are

I

of T h e o r e m

A more

Theorem

Then

by o p p o s i t e

called

above)

X v.

by

Lemmas

that

3.2

they

T and ~replacing (iii)

Lemmas and

will

The

and

3.2

apply

to

implication

Pj and

==, (i)

3.1

Pj,

follows

by from

128

Remarks.

(a)

Using

subgroup

S of

G(A)

finitely

on

every

(b)

We c o n j e c t u r e

then

the

Let

G(A))

is

into

called

a direct

Similarly,

I be

= ~+

if

our

U

Let

Then

the

4j

(-w.4j,).

o

group

The

proof

~ublemma

P = Pj

of

of

all

Fix

(a)

If u

• U_,

(b)

If J c (i)

(ii)

for

Pj n w w ' U Proof. may

To

write

-i u+uln Inn (3.6)

prove

J'

be

3.3

finite

exists

e W,

i

~+J

= ( z j~J of

I

of

G(A)

I,

and

~ P,

type, such

#'(A)/c

(resp.

decomposes

subgroup

is

a reductive

e i,

fi'

lemma.

zaj)

n

finite by

G(A)j

define i

of

a

e J.

Given

J c

I,

~+. type

H and

and

the

let

w e W.

Ua w i t h

a

sublemmas.

put

then

G(A)

the

all

on t h r e e

S of

Similarly,

more

generated

J' c

there

e J

and

one

u_n

in

~'(A).

Then

by A'

is

w ~ Wj

locally

representations.

subgroup.

is b a s e d

n ~ N and

w'

type.

subsets

n wPj,w -1

-I P = WlPj,w I .

u_,

Then:

n E P.

then: that

exists

Pjn

U+ c w P w -I,

w E Wj

such

that

w ' - l w -I c U+.

(a),

choose

-i n I u_n I = u+u~, I e Pj,.

Ad S ,

of

need

and

w I e W and

all

to

generated

I is of

there

respect

red uctive

0 ~)

2 acts

divisible

A subgroup

U_a " w i t h 1

we

any

bounded.

finite

Ua. , 1

result

Lemma

3.3.1.

is

subalgebra

Z~j) and

g e G(A)

a reductive

next

J

Theorem

~'(A).

with

jeJ Lemma 3 . 3 .

of

show that

module.

g is

of

to

finite-dimensional

~'(A)

(( Z

easy

irreducible

a subset

of

is

conditions

element

by

a standard

~(A)j

To p r o v e

an

center

by H and

called

subalgebra

4j

the

it

9'(A)-

if

reductive

generated

subgroup,

put

be

sum of

J c

the

generated

we d e f i n e

Let G(A)

that

c c ~'

below,

satisfying integrable

subgroup

3.2.

Lemma 3 . 3

Since

n I ~ WlR.

where u+

u+

E U+,

e U+ c Pj,, u~nllnnl

Since u~ we

~ Pj,.

[9]

e U_. get

wllU_wl So:

c U+U_,

we

129

Using

the

system

fact

from

that

that

Pj,

inherits

of G(A)

[9],

the

structure

we have

Pj,

of a r e f i n e d

= ~

Tits

(U_ N P j , ) n ' U + .

We

n'eNNPj, also

have

[9],

G(A)

= II

U_n'U+.

We d e d u c e

from

these

and

(3.6)

that

n'eN -i n I n n I E Pj,,

so that

n e P and hence

To p r o v e

(b)(i),

choose

for

j e J,

generated Pjn

~(rjwwl) by

> ~(WWl) J , a e 4+

the U

w • Wj

ww'U_w'

(b)(ii),

-I -I w

Then

that

Pj N U_ N U'

Pj

U_

n

U'

= {I}.

proving w • Wj

3.3.2.

~(WWl).

subgroups

U

Proof.

proceed

We

trivial. [9],

Let

we h a v e

by

e(w)

(b)(i) ~(ww')

A U+).

since,

So, by

a n d put

we m u s t

the

induction choose

U2

, where

h e H = H o m ( Q , C x)

write:

u = UlU2,

u21h-nu2

then

, which

> 0;

U*

=

show

choice

of w,

if U is a s u b g r o u p

of U

: = U

U is c l o s e d

and

~ wU w +

is g e n e r a t e d

-i

--

by the

it c o n t a i n s on ~(w).

The

i e I such

cases

that

~(w)

e(riw)

= 0 or

< ~(w).

1 are Using

a homeomorphism

Uw = U 1M Define

b y ~,

, a ~ 4 +re

is

S

If w e W a n d

is n o r m a l i z e d

Then

Pj A U+

W

which

(a).

that

to m a x i m i z e

is c l e a r

proves

Since

c w w ' U + w ' - I w -I.

Sublemma

This

to m i n i m i z e

= (U' N U_)(U' This

u_ • P.

e WWl(4+).

, we deduce

choose

[9]

so as

and so =j

U+ c w w I U + w l l w -I c w P w -I To p r o v e

also

where

by

U 1 = Ur. and U 2 = r.l U r . w 1 1

<~j,h

u k e U k.

> = exp(l-6ij). Then

Fix u E U and

for n = 1,2 ....

h n = u - l h - n u h n ~ U ~ U 2 and

lim h - n u 2

-i r.l

we h a v e

h n = i.

Since

U n U2

n~

is c l o s e d follows

by

that

induction. Sublemma

assumption,

U = (U n Ul) M

3.3.3.

parabolic

inductive

(U A U2),

we d e d u c e and

so

the

that

u 2 e U N U 2.

sublemma

follows

It by

B

W-conjugates

Proof.

the

If W 1 a n d W 2 are p a r a b o l i c

of s u b g r o u p s

subgroup

Choose

of the

f o r m Wj,

subgroups

J c

I),

then

of W W 10

(i.e. W 2 is a

of W.

hl,h 2 e X with

stabilizers

W 1 a n d W 2 in W.

Choose

130

t > 0 such possible Wh,

that

since

proving

of

Proof

w - ( h I + th2) card

the

Lemma

~

= h I + th 2 i m p l i e s

> card

W,

and

put

w • W 1A

h = h I + th 2.

W2, Then

sublemma.

3.3.

which

is

W1 n W2 =

[]

Let

Sublemma

3.3.1

be

Sublemma

3.3.1

with

+ and

may

assume

- interchanged. Put

P'

= w P S , w -I

Pj n U_ c P nH

e Wj

n,u+

For

and

• P'

Sublemma Pjn

But

Pj o U _

by

the

P'

U

's

3.3.1

(a).

= (Pjn

and

Ua's

contained

is p r o v e d ,

since

Proposition into

some

Proof. P:

=Pjn

Every

of

, we

Pj have

as

by

containing [9,

so

Pj

that

that

we

F(P)

get c P.

a homomorphism Therefore,

where

u_

• P',

and

that

• Pj n U_, so

n U+).

that

and P'

Sublemma

reflections

by

n U+

using is

ra,

a • 4 re,

generated

Combining

these

statements,

in Pj O P'

subgroup

iff ~ • aj n

S of G ( A )

by

and

so

H and

the

lemma

(-w.~j,).

can

be

Q

conjugated

G(A)j.

2 that

S is a s u b g r o u p

Denoting

by

U J the

a e ~ +re , t h a t

Ua,

generated

3.3.3,

of G(A)

3.3.

all

nu+

we

subgroup

Theorem

Proposition

(3.7)

see

subgroup

in L e m m a

have

using

reductive

reductive

assume

w P j , w -I

subgroup G(A)j

may

the

we

some

is c o n t a i n e d

standard

We

in

in Pj n P' U

3.1.

by

we

n P' n N ) ( P '

3.3.2,

generated

g = u_nu+,

U a s it c o n t a i n s ,

Finally,

Pj O P' O N is c o n t a i n e d the

the

(b)(i),

So:

U_)(Pj

by

it c o n t a i n s . is

write

u_ e P',

(b)(ii)

(Pj A P' N N ) / H

P',

3.3.1-

Since

is g e n e r a t e d

3.3.1

Sublemmas

Sublemma

g • Pj n

u+ e U+.

by

By

are

of smallest

not

normal

contained

in

4.6]:

= G(A)j ~:

P

~

U J,

, G(A)j.

P = P' ~ U',

where

Using P'

Lemma

3.3,

= P N G(A)j

we

and

U ° = P A UJ . Let

Z c H be

the

center

of G ( A ) ,

so

that

G(A)/Z

acts

faithfully

see

131

on # ' ( A ) / 6

[9].

i e I, and

let

Let

H and

on w h i c h

some

algebraic subgroup G/Z

the

V = (V'+c)/c.

finite-dimensional, ~'(A)/c

V' be

so that

subgroup o f P/Z.

be a m a x i m a l

By L e m m a s

acts

(Lemma

U"/Z

this

and

(3.7),

But that

the

first

G/Z

factor

Let ~ be completely

reducible

so of P/Z.

But

any

reductive

reductive

reductive

P/Z

P/Z-conjugate.

are

subgroup,

Proposition

the

3.3. called

of

P is g e n e r a t e d

P/Z

by

as a c o n n e c t e d

is a c o n n e c t e d

radical

of P'/Z,

and

of P ' / Z

algebraic and

let

so that

the s e c o n d

is u n i p o t e n t ,

in GL(V).

is a r e d u c t i v e

any

one

so

of P/Z.

of SZ/Z

subgroup

Hence,

following

3.2.

Every

subgroup)

is a m a x i m a l of G(A)

P'/Z

subspace

of P/Z

Since

subgroup

V is a

of GL(V)

is c o n t a i n e d

two m a x i m a l

S is P - c o n j u g a t e

reductive

and

in a subgroups

to a s u b g r o u p

of

of G c

Q

We have

connected

regard

subgroup

~

and

P,c G ( A ) j .

Since

V is

(U"/Z ~ U'/Z).

closure

~-module,

maximal

3.1-,

that

= G/Z ~

the Z a r i s k i

A d P . e i, A d P . f i,

= G/Z M U"/Z.

is r e d u c t i v e

is a m a x i m a l

we m a y

subgroup

we deduce P/Z

and

the u n i p o t e n t

P'/Z Using

3.1

Similarly,

be

reductive

of the

faithfully.

3.3),

of G L ( V ) . Let

span

V is a f i n i t e - d i m e n s i o n a l

A d P = P/Z

of the U

linear

Given

complex

of G(A)

complex

which

corollary

of G(A). b y two

a representation

~-triangular

if e v e r y

finite-dimensional

subspace

For e x a m p l e ,

w e W,

given

torus

(i.e.

Every

elements

= of G(A)

on

into

~(R)

the s u b g r o u p

B

w

H.

The

subgroup

Ad-diagonalizable is G ( A ) - c o n j u g a t e

V,

a subgroup

v E V is c o n t a i n e d on w h i c h

3.1:

Ad-diagonalizable

is G ( A ) - c o n j u g a t e

torus

is g e n e r a t e d

of P r o p o s i t i o n

H

subgroup into

H.

R of G(A)

is

in an R - i n v a r i a n t

is t r i a n g u l a r

in s o m e b a s i s .

: = B N wB w -I of G(A)

is

132

triangular where

in any

integrable

module.

U w = U+ N w U w -I acts

locally

This

is b e c a u s e

unipotently

on

B w = H ~ U w,

any

integrable

~'(A)-module. Another Proposition

3.3.

conjugated Proof.

corollary

into

Since

parabolic

of T h e o r e m s

Every one

of

Ad the

type

,(A)-triangular,

its

Ad

(A)j-triangular,

hence

into

the B o r e l

conjugated argument

as

conjugated

Remark. proved

3.4.

B.

some

Now we

shall

Theorem

Let A b e

3.

If ~

b y G(A)

type

If ~

an

be

2 now

a

Since

R is

by using

(3.7)

is

It f o l l o w s

gives

into

R c Pj.

b y an e l e m e n t

of P r o p o s i t i o n s

subsets

into of

finite

into

that

3.2

I and

is a r e d u c t i v e

of G ( A ) j

that

R can be

B

The

same

3.3 h a v e

been

.

R can be

and

J and

J'

G generated

are of f i n i t e

Caftan

of ~ ' ( A ) ,

(Ad n)@~,

of ~' (A),

J is a f i n i t e finite

Hence, by

@j n

of T h e o r e m

2 and

matrix.

Then:

then ~ can

, where

J,J'

be are

n ~ N.

where

ad~ c Ad G(A). group

subalgebra

subalgebra

is an a d - l o c a l l y

version

generalized

a subalgebra

If ~

where

conjugated

R can be c o n j u g a t e d

a symmetrizable

Proof.

the

that

defined

conjugated

infinitesimal

some ~(A)j,

that

assume

n B of G(A)j.

versions

into

exp

be

can

B

by G(A)

[13],

it can be

in G ( A ) j

of T h e o r e m

is an a d - l o c a l l y

conjugated

(b)

can

R of G(A)

w

B w.

prove

3.1.

finite

We m a y

subgroup

[13].

Proposition

(a)

2),

Similarly,

Infinitesimal in

(Theorem

G(A)j

in the p r o o f into

B

image ~

subgroup

into

subgroups

Pj.

Ad

2 is

,(A)-triangular

R is b o u n d e d

of finite

1 and

exp type

type

subalgehra

by applying ad~

then ~ can be c o n j u g a t e d

of ~'(A),

Theorem

is c o n t a i n e d

and n E N.

subset

But

of

I.

then,

2, w e m a y

in P j n then

G,

by assume

n P j , n -I, and

hence

133

normalizes If,

in

pj

addition,

Proposition ~,

and

3.5

One

# is

3.1

normalizes

finite-dimensional

so

Proposition

3.4.

some

Proof.

Let

inclusion

that

G is

Kac-Moody

examples

x be

a nilpotent

homomorphism

then

(a).

so by G,

and

hence

= #(A)j.

D

of s e m i s i m p l e

however

algebra in the

which

affine

~'(A).

Then

~

into

a subalgebra

are not

case:

finite-dimensional

can be c o n j u g a t e d J of

proving

and

But

of a K a c - M o o d y

algebra

subset

reductive

~ c pj 0 pj

is b e t t e r

type

(Ad n ) p j , ,

n

G c G(A)j.

Let ~ be a s e m i s i m p l e

and h e n c e

finite

that

subalgebras

The s i t u a t i o n

~ c pj

then

construct

reductive.

subalgebra

that

reductive,

and ~j,

easily

of an a f f i n e

so

we may a s s u m e

pj

can

(Ad n ) p ~ ,

subalgebra

is a r e d u c t i v e of ~(A)j,

for

I. element

~: #

of the

....P #'(A)

Lie

induces

algebra

~.

The

a homomorphism A

W:

~

, ~' (A)/c

But

a simple

finite

dimensional

Laurent L®c

series

~ over

(this

over

E.

that

~

m a y be

~(x)

of L ®E ~

algebra

and

induces

a homomorphism

dimensional

of a r g u m e n t

on L ®C6 , so

is a s u b a l g e b r a

Lie

Thus,

of f i n i t e -

type

~' (A)/c

(over

found

is n i l p o t e n t

in

L)

L is the

field

Hence

on @ ' ( A ) / c

®E ~

Lie

~(x)

and F(x)

~ is

of formal

F:

semi-simple

[12]).

' where

'

algebras

is n i l p o t e n t

is n i l p o t e n t

on ~'(A). But

~

generated

is g e n e r a t e d by e l e m e n t s

it f o l l o w s Remark.

stated

there

is not

3.6.

In this

form

K(A)

its

nilpotent

ad-nilpotent

that ~ is a d - l o c a l l y

In the c a s e

conjugacy

by

quite

that

~

correct

group the

on ~(A).

finite,

3.4

hence

F(#)

is

By

[6,

Lemma

on p.

170]

proving

the

proposition.

S

type

X~ of X~ I),

is a s u b a l g e b r a

Proposition

subsection

of the

Recall

by

when

elements,

of

is c l a i m e d

in

[12];

the

the

proof

however.

we

derive

conjugacy

theorems

for

the u n i t a r y

G(A).

Lie a l g e b r a

#(A)

carries

a unique

conjugate-

134

linear -I,

involution

w(ei)

of G(A),

= -fi also

a ~ A re . and

in

, w(fi)

fixed

is c a l l e d

found

some

[9].

T and

Kw,

the

w e W.

by w,

point

of the

Put

One

such

set

i e J;

Furthermore,

prove

have

the

results

K(A)

by u s i n g

of G(A)

following

properties.

connected

abelian

K(A)

: K(A)

G(A) topology

for

= exp w(x)

is d e n o t e d

proofs

T = H O K(A),

of w h i c h

E(A)j

= TKi;

is the

x e #a,

by

K(A)

may

be

= G(A)j

K(A)j

disjoint

O K(A),

is g e n e r a t e d

union

of the

n Pj.

decomposition,

(3.9)

The

x)

we have:

lwasawa

about

wl~ ~ =

involution

involution

of K(A),

and

K(A)j

also

that

corresponding

K i = Fi(SU2) ; K(A){i}

K(A) i w i t h

such

of G(A).

K i = G i N K(A), Then

the

w(exp

of this

form

involution,

has

that

properties

(3.8) We

the c o m p a c t

= -e i.

the u n i t a r y

K w = BwB O K(A). by

called

denoted

The

Recall

w,

related

the

results

a topology

subgroup

subgroup);

often

allows

about

one

to

G(A):

= K(A)B.

induces

The

which

on K(A)

T is a torus

K

are

locally

which

(i.e.

has

the

a compact

closed;

a closed

w

subset

of K(A)

number

of

is c o m p a c t

the Kw;

and

3.5.

(a)

if and

the

only

subgroups

if it

K(A)j

intersects

with

only

J of f i n i t e

a finite type

are

compact.

Proposition conjugated

into

(b)

torus

Every

torus

of

{gnln

~ 1}

(c)

K(A). is

of

Let

the

K(A)

compact

from

e be

of

can

in the

compact

K(A)j

can

Any e l e m e n t

The s t a b i l i z e r

G(A)-module (d)

one

Every

be

K(A)

Let R 0 = {v ~ fllF(~[supp

of

in v])

J

K(A)

is

such

conjugated

category

a G(A)-orbit

where

conjugated

g of

be

subgroup

any X is

into

K of

K(A)

a finite into

T,

that

the

can

type which

be

subset is

closure

of

I.

a maximal of

T.

finite-dimensional

subspace

of

a

compact.

a G(A)-module is m a x i m a l

V from

and ~ [ s u p p

the v]

category ~ Cv}"

X. Then

~:

135

= P[supp

v]

is i n d e p e n d e n t

of the

choice

of v e n0,

=0

is a B W B - o r b i t

and R = K.~ 0 . Proof.

(d)

follows

from

Proposition

If v e Y and e = G(A).v, k.v

~ e 0.

Hence,

G(A)

lies

Ck.v

in K(A)

using (a)

the

also

G(A)

in Pj,

exists

lies

a G(A)-module

contained

But,

in K(A)j

follows,

using

connected

Lie

Remarks.

(a)

K(A)j, of

easily.

with

G(A),

J

for

for

of

the

Ca)

S denote

any module

the

type,

the

Thus,

set

of

the

the

all

category

subgroup

of

G(A)

is

(c)

Every

compact

subgroup

of

K(A)

(resp.

unique

subgroup (resp.

finite-type (resp.

G(A))

subsets

of

The p r o o f

and

following

the

Lemma. some

(a)

Every

of K(A) G(A))

subset

of K(A)

I.

J of

if and

K

w

of T h e o r e m

.

i.

subgroup

Hence that

stabilizer

is

of E k . v

about

of

there Ev

kKk -I c K(A)j. facts

by

is

Finally compact

K(A). X,

of

Then,

there

the

subgroups

as

in

exists

a

the

case

be

I.

type.

G(A))

G(A)).

conjugated

K(A)j

facts

well-known)

subgroup

is

contained

Every onto

is b a s e d

in

maximal K(A)j

is a m a x i m a l

if J is m a x i m a l

of these

finite

reductive.

(resp.

can

only

(presumably

J c I of f i n i t e

follows

~ S.

compact

of K(A)

of

(c)

conjugates

group

Every

subgroup

stablizer

v ~ V such

corresponding

of

V from

map PV

compact

in

==~ (v)

of the

that

of C k - v

is a b o u n d e d

z and

(b)

maximal

the

such

D

finite

K(A)-equivariant

k ~ K(A).

from

(iv)

argument,

k ~ K(A)

(3.8)).

number

category

above

some

(a),

a finite

the

the

groups.

Let

by

from by

implication

decomposition.

stabilizer hence

(see

K from

Iwasaws

exists

the

type,

= K(A)j

the

the

there

2.2(a),

Indeed,

it is c o v e r e d

K-invariant.

(b)

proving

and

(d),

J is of f i n i t e

in Pj n K(A)

argument

since

by

by P r o p o s i t i o n where

follows

then,

2.1

among

for

compact all

a

compact a subgroup

finite-type

on P r o p o s i t i o n

3.5

lemma~

of W is W - c o n j u g a t e

into

Wj for

136

(b)

If

J

is

maximal

maximal

finite

(c)

J

If

subgroup

and J'

are

J ~ J',

then

Proof.

To p r o v e

(a),

h e Int

X.

the

subgroup

Wj.

Then

suppose

containing h'>

(b),

h"

= h'

Wh,

= Wj by

not

W0 b e

WO,

finite-type

a finite

I,

then

Wj i s

a

of

I and

if

in W of

proving

Wj,

By the m a x i m a l i t y

- h. the m a x i m a l i t y

of J,

let

is a f i n i t e

finite-type

parabolic

subsets

of W s t r i c t l y

h ° ~ Int

h'

• C.

Among

This

Int

contradicts

stabilizer

the

X and Wh,

of

containing

X with

--l<~i,h>,

h" E C N

of J.

all

and

minimal

Then

W, a n d

(a).

subgroup

stabilizer

of

Z w.h w~W 0

W 0 is a f i n i t e

one w i t h

subsets

subgroup

among

choose

of

W-conjugate.

stabilizer

h ~ C with

< 0,

subsets

let J be m a x i m a l

that

W O.

among all

are

let

finite-type

W.

maximal

of W c o n t a i n i n g

Choose


of

Wj a n d Wj,

To p r o v e I, and

among all

i e I with

and

put

m Wj,

so

that

r i E Wh. ~ p r o v i n g

(b). To p r o v e finite

type

Wj,

Int

on

cannot

The

also

true.

(e)

Every

X are

~ ~lg(Q)

of

by the

of

particular,

In

by

{t

action

for

of

reductive

of

C,

so

sets that

of

actions

~ c I

Itl

= 1}).

group

of G(A),

of w, that

subgroups

G(A)

is

of

G(A),

AurA,

K(A)

is

the

of

K(A),

actions

the

we b e l i e v e

fixed-point

of

I of

o f Wj a n d Wj a n d Wj, O

by

the

subsets

(c).

(c)

the

the

among a l l

subsets

automorphism

that

generated

Then

automorphism

generated

We c o n j e c t u r e

E.

Remark

continuous

s I :

be maximal

disjoint

proving

generated

:

J'

J g J,.

continuous

automorphisms

and

nonempty

analogue

Every

(g)

J

and satisfy

automorphisms

{g

let

be W-conjugate,

(d)

(f)

(c),

al! and

in

the

in

AutA,

of

group

every

group

of

is

of

H, ~ a n d w. group

of

q and

automorphisms the

G(A)

of all

automorphism

G(A)

is

automorphisms of

G(A)

t37

leaves

{gBg-llg

• G(A)} U {gB g - l l g

(h)

The

(i)

If g G AutA

bounded (j)

closure

then

of a s u b g r o u p

example

the

subspace

3.5(c)

if #(A) ~'/6

and

fails

is of a f f i n e

of ~' (A)/c

a connected

group

let

K be

the

the box

then

group

the K a c - M o o d y [7].

Let

group

K

T be

torus

of K and, by u s i n g into

be

T.

integrable

G(A)-modules.

the c e n t r a l i z e r

in K(A)

of S 1 into

K with

finite

by an with

the

torus

Proposition Let ~ :

of B.

compact

of

the s u b g r o u p

3.5(b),

SU 2

of c o n s t a n t

then

of K;

by S 1 of the

it is a m a x i m a l

we see

, K,

Fourier

Caftan m a t r i x

extension

of K;

Lie

inclusion

extended

is a c e n t r a l

a maximal

by g is

simple

(induced

K(A)

generated

connected

K is i d e n t i f i e d

group

K is c o n j u g a t e

then

iff K is b o u n d e d .

compact.

of maps

of K • Let A = (aij)ei,j=0

loops

type

simply

topology

c Mat N ( C [ t , t - l ] ) .

subgroup

arbitrary

is not

K be

with

if the

for

Let

series,

is c o m p a c t

a G(A)-conjugate

Example. and

K of K(A)

(H ~ G(A))

g normalizes

Proposition

For

M

e G(A)} i n v a r i a n t .

that

any

i = i,...,~,

torus

of

and ~0:

1 SU 2 ~

K be h o m o m o r p h i s m s

highest

and

root

0.

let K i = F i ( S U 2 ) ,

subgroup

connected

A by d e l e t i n g compact

subgroups

nonconjugate t=l

defines

cyclic by

that

subgroup

by all

row

maximal injective

Denote

K. w i t h 1

of K w h o s e

and

column•

by KJ,j

i ~ j (so Caftan Then

conjugated

by R e m a r k

compact

to the s i m p l e

roots

and

to the

: K by:

of K can be

Moreover,

of o r d e r

SU 2

subgroup

the j - t h

K J.

F0:

i = 0,...,£.

of K g e n e r a t e d

a compact

any

Define

associated

(c),

subgroups

homomorphisms

~j:

= 0,...,~,

that

matrix

K 0 = K).

one

K j is

is o b t a i n e d

by P r o p o s i t i o n into

the

from

3.5(a),

of the

compact

the K ~ are m u t u a l l y of K. Kj

Note

that

, K and

evaluation

that

= I ( K j)

at is

a v.. N o t e that the q u o t i e n t of the root l a t t i c e of K J of ~ j ( K J) is c y c l i c of o r d e r aj. It is e a s y to s h o w that K j

138

can

be

if

a. 3

conjugated =

1.

(The

[5, C h a p t e r

3.7.

for

the

2(c)]

is of

and

Recall

are

Proposition

{i

called

the n o r m a l i z e r

Let

A be

order

into w N

(c)

finite

where

o

order

order

2 we m a y pj

and

l'st

assume, its

w ~ W and J,J'

a symmetrizable

of

2'nd

hence

of ~'

the

l'st

and

(resp.

kind).

if a . ~ +

Note

is

in G(A)

[g].

is ~

(N

generalized

of the

Theorem

Ad G(A)

only

= Ad(AutA

conjugate-linear

kind

2'nd

Cartan

matrix.

of ~' (A)

kind

can be

of ~ ' ( A )

can be

automorphism

of the

l'st

kind

automorphism

of the

2'nd

kind

into O N O •

conjugate-linear

Let a be a f i n i t e of the

if a n d

Let

automorphisms

from

(resp.

[13,

A.

.

be c o n j u g a t e d

finite

kind

from

all

in Ad G(A)

automorphism

can he c o n j u g a t e d

normalizes

only

2 a conjugacy

symmetrizable

Those

of H a n d

Put No

automorphism

order

conjugated

Theorem

and

conjugate-linear

contains

kind

of ~'

~ ( N ~ H~.

finite

automorphism

l'st

N is the n o r m a l i z e r

x AutA)

can

of the

~_).

(b)

Proof.

if

in

1 and

It f o l l o w s

of ~'(A).

(resp.

into A u t A M H.

of 9 ' ( A )

found

#' (A) w i t h

G(A)

with ~+

finite

Every

be

Theorems

and

of ~' (A).

2'nd)

3.6.

Every

from

algebra

, w} M A d

conjugated

(d)

automorphism

a v. m a y J

automorphisms

(resp.

that

= Ad~AutC

of ~ ' ( A )

continuous

we d e r i v e

l'st

that

It f o l l o w s

Every

a. J

automorphisms

the

commensurable

Every

the

order

the g r o u p

f r o m w Ad G ( A ) )

(a)

of

involution

conjugate-linear

N:

a

of a K a c - M o o d y

compact

that

that o

= K by

subsection,

finite

automorphisms be

values

K0

4].)

In this

theorem

into

into w N order

kind

o

.

automorphism

of ~ ' ( A ) .

replacing

~ by

finite-dimensional c I are

finite

type

or c o n j u g a t e -

T h e n o ~ Ad G(A) its

conjugate,

subalgebra subsets.

linear and b y

that

@ = pj m By the

(Ad w ) p j ?

139

finite-dimensional Cartan

subalgebra

subalgebra

of @,

normalizes

~'.

is

we

of

may

the

the

assume

that

o

2'nd

(B)

some

finite

It

follows Writing we

we

be

o-invariant.

3.7.

= fu .i' By

the

o

ow-invariant, Va and

[7] ow

also

owI~ , =

i,

real

eigenvalues.

be

it

is

form

we

form (. [.)

Put

~

that =

= gB

o by

7]

(the

R so

that

(d).

¢

to

some

involution

that

ow

is

H becomes

the

2'nd

a

to

diagonalizable is

an

A: =i.

--

~ wN

o

.

Then

for

+ ~ - a ) is positive

invarinat

respect

((ow)2) I/4", t h i s

of

u m Aut

= - ( x l w - y ) is

a standard with

that

o . f .i = +e.I if u . i

(~a + ~ a ) + ° ' ( ~ a

is

deduce

a conjugate-linear

involution

assume

H(×,y)

of

Q.

i; o . e .i = ±fi' may

its

proof

of

for

for

subgroup

by

element

g-1

Birkhoff

we

an

of

that

well)

self-adjoint

follows

completes

as

conjugated

Va : =

(where

Theorem

4.5],

eigenvalues

Ad-triangular

a conJugate-linear

3.6(d)

Hermitian

[2,

the

automorphism

Replacing an

if o

o

Theorem

This

a.B

the

o

complete

assume

that

E q N

automorphisms

and

if u . i

subspace

the

on

can

° ' f l. = e u - i

the

Applying

following

Proposition

~ A+,

~'(A)),

o be

stable

c

positive

may

to

b+Hb: 1

leaves

(b)

3],

according

linear

proves

Then

of

H.

we

that

To

conjugate-linear (a)

assume

[2,

~+.

a

a Caftan

and

By

and

normalizes of

(c).

with

~'

By

~

may

~ ~(A)j.

Theorem

a conjugate

Let

of ~ ' ( A ) .

involution

o

conjugate

This

Proposition

that

by

[13,

B f] o . B

containing

replaced

or

we

proves

of A'

o

part

automorphism

/;j/~J

g = b+nb_

have:

obtain

for

an

This

order.

from

reductive

normalizes

of

a

on

o

and

works

definite

that

4.5]),

conjugate,

element

automorphism

which

a

an

Theorem

the

is

on

an

= B f] o - B

each

acts

o be

conjugate,

Proof.

if o

o

let kind

is

its

(a).

decomposition,

°'ei

o

fixes

follows

g E G(A).

kind

~ N

by

of

= B.

can

a

~'

[2,

automorphism. that

Now

R:

o

note

proof

e.g.

Since

replacing Hence

It

(see

@.

(a),

on ~ + / ~ J .

a

of

a conjugate-linear

proof

2

theory

this

bilinear form.

on ~ ' ( A ) automorphism

form

Since with of

140

$'(A) back far

and to

we

one

E.

checks

Caftan).

have

not

used

of

the

Theorem

we

obtain

parabolic normalizes This

completes

that

l'st

subalgebra ~'

~-io~

Thus,

involution 2,

that

we m a y o

kind

that

commutes

is

of

assume the

of ~ ' ( A )

J is

of

and ~+, b y the a r g u m e n t the

proof.

w

(this

that

o

commutes

2'nd which

a K(A)-conjugate

flj w h e r e

with

kind.)

Then

normalizes of ow

finite proving

argument with aw

is

K(A).

normalizes

type,

and

goes w.

(So

an Using

a

hence

Proposition

3.6(a). O

141

References.

I. Bausch affines,C.R. 2. Borel algebras,

J.,

Automorphismes Acad. Sci. Paris

des alg~bres (1986).

de Kac-Moody

A . , M o s t o w G . D . , On s e m i s i m p l e automorphisms Ann. Math. (2) 61(1955), 389-405.

3. Bourbaki N., Groupes Hermann, Paris, 1968.

et

alg~bres

de

4. Bruhat F., Tits J., Groupes r~ductifs P u b l . M a t h . IHES, 4 1 ( 1 9 7 2 ) , 5-251.

Lie, sur

Ch.

of

Lie

4-5,

un corps

local,

5. Kac V . G . Infinite-dimensiona] Lie algebras, Progress in Math. 44, Birkhauser, Boston, 1983, Second edition: Cambridge University Press, 1985. 6. Kac V . G . , Constructing groups infinite-dimensioanl Lie algebras, conference on I n f i n i t e - d i m i n s i o n a l MSRI P u b l . #4, 1985, 167-216.

associated Proceedings Lie groups,

to of the Berkeley

1984,

7.Kac V.G., Peterson D.H., Unitary s t r u c t u r e in r e p r e s e n t a t i o n s of i n f i n i t e - d i m e n s i o n a l groups and a c o n v e x i t y theorem, Invent. math. 76(1984), 1-14. 8. Kac V.G., Paterson D.H. Regular functions on certain i n f i n i t e - d i m e n s i o n a l groups, Arithmetic and G e o m e t r y (ed. M. Artin and J. Tare), Progress in Math. 36, Birhauser, Boston, 141-166, 1983. 9. Kac V.G., P a t e r s o n D.H., D e f i n i n g relations of certain i n f i n i t e - d i m e n s i o n a l groups, P r o c e e d i n g s of the Caftan conference, Lyon 1984~ Asterisque, Numero hors aerie, 1985, 165-208. 10. 108

Kempf G., Instability (1978), 299-316.

in invariant

theory,

Ann.

Math.

II. Levstein F, A c l a s s i f i c a t i o n of ~ n v o l u t i v e a u t o m o r p h i s m s of affine K a c - M o o d y Lie algebras, Thesis MIT, 1983. 12.

Morita

the affine

J.,

Conjugacy

Lie a l g e b r a

13. P e t e r s o n D.H., c o n j u g a c y theorems, 1778-1782.

. K e(i)

classes

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preprint,

Xe in

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Kac V.G., Infinite flag v a r i e t i e s and Proc. Natl. Acad. Sci. USA 80(1983)~

14. Slodowy P., An adjoint quotient for certain groups attac h e d to K a c - M o o d y algebras, P r o c e e d i n g s of the c o n f e r e n c e on I n f i n i t e - d i m e n s i o n a l groups, B e r k e l e y 1984, MSRI Publ. #4, 1985, 307-333.

142

V i c t o r G. K a c D e p a r t m e n t of M a t h e m a t i c s M.I.T. Cambridge MA 02139 U.S.A. D a l e H. P e t e r s o n D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of B r i t i s h C o l u m b i a Vancouver Canada

ETALE LOCAL STRUCTURE

OF MATRIX INVARIANTS

AND CONCOMITANTS

Lieven Le Bruyn (*) University of Antwerp,Belgium Claudio Procesi University of Rome,Italy

(*) : work supported by an NFWO-grant

0: INTRODUCTION

One of the basic problems in linear algebra is to study the equivalence classes of m-tuples of n by n matrices under simultaneous conjugation. This problem is readily seen to be equivalent to that of studying n-dimensional representations of the free algebra ~ < X1, ..., Xm > upto equivalence. Geometrically, one has to sudy the orbit structure of mn2-dimensional afllne space X,n,,, = M n ( ¢ ) ( ~ ... ~

Mr,(¢)

under action by componentswise conjugation of the general linear group

GL,,{¢}. The space of

orbits of X.,,n is not Hausdorff due to the existence of non closed orbits. The classical approach

144

is to approximate this space using also be denoted

GL.(~)

Xm,,~/GL,~(~). The

invariants as parameters of a variety

Vm,. which

will

resulting map

: X.~,. --* Vm,.

is surjective and the fiber ~r-l(p) of each point p e V,~,,~ contains a unique closed orbit. Thus V.,,. parametrises naturally the closed orbits of the action which, by Artin's fundamental paper [Ar] , are in one-to-one correspondence with the isomorphism classes of semi-simple n-dimensional

representations of the free algebra ~ < X1 .... , X,n >. From [Pr2] we get that the coordinate ring of V.,,. is the center of the so called trace ring of m generic n by n matrices [IF,n,,.. By this we mean the following : consider the coordinate ring

~[X,n,.] =

~[zo.(1) : 1 < i , ] _< n; 1 _< I <__rn]

then we can consider in M,~(~[Xm,.D the so called generic matrices

x~ = (=~;(0),,,' ~ M . ( ¢ [ X , . , . ] ) The ~-algebra generated by these elements is called the ring of m generic n by n matrices, (]~,n,. • Then 3rrn,. is the subalgebra of M . ( ~ [ X m , . ] ) generated by ( ~ m , . and Tr((g~m,.). It is known that V~,r~ is a unirational variety and that the Formanek center of [Ir,n,n (see [Pr3]) determines an open smooth subvariety of Vm,.. In this paper we aim to initiate the geometric study of the varieties

Vm,,. both

globally and

locally based on some powerful results of D. Luna [Lu]. We will now briefly describe the main results : We say that a point ~ 6 Vm,. is of representation type • -- (el,k1; ...;er, kr) if the corresponding isomorphism class of semi-simple representations is built from r distinct simple components of dimensions ki occuring with multiplicities ei.If r is such a representation type we call Vn~,.(r) the subset of Vm,. consisting of all points of representation type ~r. We will prove that the sets Vn,,r,(~) form a finite stratification into locally closed smooth subvarieties, Furthermore, we prove

145

that

Vm,,~(r)lies in the closure of Vm,,~(r')ifand only if r is a degeneration ( or refinement) of ~'

(theorem II.1.1). Next, we determine explicitly the Stale local structure in a point ~ E

Vm,,~ of representation

type (1, ]¢I;-**;i, ]gr)(theorem II.2.1). A generalization of this result to arbitrary representation type can be phrased in the setting of representations of quivers and their invariants in the following way. Let A = (A0, At) be a finite quiver where A 0 is the set of vertices and A1 the set of directed arrows between these vertices (we allow loops and multiple edges). A representation of A is a couple V = (V=, Va) where each ]7= for x E A0 is a finite dimensional vector space and the

Va

for a E AI are linear maps between the corresponding vectorspaces, d = (dim(V=))z is called the dimension vector of the representation. Conversely, for any dimension vector d we can look at the space of all representations of A with this dimension vector. This is an affine space admitting a natural action of the reductive group

GL(~, d) = H aLa. (¢) and we denote the corresponding quotient-variety by

V(A, d). Now, let ~ 6 Vm,. be of represen-

tation type (el,kl; ...;er,kr) then we can form a quiver A~ consisting of r vertices {xl .....xr} and (m - 1)k~ + 1 loops in vertex z~ and dimension vector

(m- 1)k~kj directed edges from vertex x~ to zj. Let d~ be the

(el.....et). Then, one can show as in II.2 that a neighborhood of the origin in

V(A~, de) is analytically isomorphic to a neighborhood of ~ in Vm, n. Furthermore, we claim that the coordinate ring of¢hese quotient varieties V(A, d) are generated by traces of oriented cycles in the quiver. In this paper we prove this claim for the special case A~, d~ if ~ is of representation type (1, kl; ...;1, kr) using the results on tori-invariants. In section II.3 we combine these results to prove that the singular locus of V,~,~ is determined by the Formanek center of 3rm,~. In the final chapter we give an explicit description of the Stale local structure of the trace ring of m generic n by n matrices 'Irm,n in a point ~ E Vm,~ of representation type (1, kl; ...;1, kr). W e prove that this 'noncommutative slice' is a Cohen-Macaulay module and that its Poincar6 series satisfies a specific functional equation. In the final section, these results are applied in order to

146

solve the regularity problem for trace rings of generic matrices :

gldirn('Irm,.)

< co if and only if

rn or n is equal to one or (rn,n) = (2, 2), (2, 3) or (3, 2).

Acknowledgement

We thank D. Luna (Grenoble) for advising us to use his ~tale slice method to tackle regularity questions for trace rings of generic matrices and their centra. The results of this paper were obtained while both authors visited the University of California at San Diego. We like to thank the department of mathematics and especially L. Small for the

hospitality. I : PRELIMINARIES

For the reader's convenience, we have collected in this first chapter the basic results which will be used throughout the paper. Further, we fix notation and terminology which we will use freely in the next two chapters.

1.1 : T H E E T A L E S L I C E T H E O R E M

Throughout this paper, the basefield will be ~. Let G be a linear algebraic group. A representation of G is a finite dimensional complex vectorspace V (the representation space) together with a homomorphism of algebraic groups

p : e --~ C L ( V )

We denote this representation by p or (V, G). We will always assume that G is reductive, that is, every representation of G is completely reducible. An action of G on a complex algebraic variety X is said to be rational if the canonical map

GxX-*X

147

is a morphism of varieties. If the action is rational, G acts on the coordinate ring ¢[X] of X and one can define an algebraic variety X/G by ¢[X/G] = ¢[X] c , the invariant ring under this action which happens to be affine. The natural inclusion •[X] G c ¢[X] gives rise to a morphism of varieties ~'x : X -~ X/G

which is onto and separates disjoint G-stable closed subsets of X, [Mu, Ch.l,2]. Let ~ E X/G, then it follows from these facts that the fiber r ~ 1(~) contains exactly one closed orbit (that of minimal dimension} which we will denote by T(~). Therefore, the points of the quotient variety X/G parametrize the closed G-orbits in X. If x is a point of X , we will denote by G(x} the orbit of x under G and by Gx the isotropy group of x, that is, the stabilizer of the point x. The theorem of Matsuchima [Ma] asserts that whenever the orbit G(x) is closed, the isotropy group G= is itself a reductive group. Before stating the ~tale slice theorem, we need to recall some facts about fibers. Let H be a reductive group acting rationally on an affine variety Y and let G be a reductive group containing H as a subgroup. The group H acts on G by translations on the right which makes G into the total space of a principal fibration over H {locally trivial in the ~tale topology}. That is, there exists an affine variety Z, an ~tale and onto morphism Z ----,G/H and an H-isomorphism

H x Z "~ G xG/z.z Z

Recall that a map between smooth complex algebraic varieties is ~tale if its differential is everywhere an isomorphism. Now, H acts on G x Iz by h.(g, y) = (gh-1, by) and we denote the quotient under this action by G x ~/Y. The action of G on itself by translations on the left passes through this quotient in such a way that the projection

G ×~ Y ----,G/H commutes with the action of G. The action of G on G ×/~ Y is totally determined by the action of H on Y and (G ×~ Y)/G is identical to Y/H.

148

Again, let X be a rational affme G-variety and let X ' be a G-invariant subset of X. We say that X' is a G-saturated subset of X if ~rxl(~rx(X')) = X'. If X ' is open (resp. closed) we also have that the natural map X ' / G ---, X/G is an open (resp. closed) embedding. We can now state a version of Luna's slice theorem [Lu,p.97] or [Sc,Th.5.3,p.55]

T h e o r e m I.l.1

: Let X be a representation space of G. Let z E X be such that the orbit G(x)

is closed. Choose a Gz-splitting of X ~- Tx(X) as Tx(G(x)) ~9Nx and let ~b denote the canonical equivarlant map

~:GxGffiNz--* X [y.,,l -" 9.(, +

.)

Then there exists an ai~ne open G-saturated subset Y of X and an affine open Gz-saturated neighborhood Bz of 0 in N , such that the maps

4 : G x G f B~-~ y

~ : (G x c. B.)/c -~ Y/G are both ~tale, where ~ denotes the map induced by ¢. Also, ¢ and the natural map G x G* Bx --,

Bz/Gz give rise to a fiber product diagram G

x c:

Y

B,

---*

---,

B./Gx ~- (G

x G'~

B.)/G

Y/G

with a G-isomorphism G x G" B , -- ¥ xWG B , / G .

Whereas ~ is a morphism between smooth varieties, ~ usually is not. We recall that a morphism between two affine complex algebraic varieties is Stale in a point iff it is a local isomorphism

149

of analytic spaces in that point,ie, a homeomorphism which induces an isomorphism of the structure sheaves (in the analytic topology,obtained after embedding both varieties in a suitable affine space). Therefore, it is sometimes profitable to look at the following realization of X/G. Let Yl, ..., Yd be generators of (~[X] G and let y = (y~ .... , yd) : x - .

Cd

The image of y ,which we will call Z, is closed in c d and naturally isomorphic to X / G , see [So,Prop.5.2.1]. If X, G, y, Z and d are as above, we call y and the quintuple (X, G, y, Z, d) orbit maps. Now, let H be a reductive algebraic subgroup of G. Then Z(//) will denote the set of points in Z whose corresponding closed orbit has an isotropy group which is conjugate (in G) to H. If Z(H) # ~, then we say that (H) is an isotropy class of (X, G). If H and H ' are subgroups of G, we win write (H) _< (H') iff H is conjugate to a subgroup of H'. As a consequence of the ~tale slice theorem, one obtains the next result of G. Schwartz,[Sc,lemma 5.5] :

T h e o r e m 1.1.2 : Let (X, G, y, Z, d) be an orbit map. (1) : {Z(H) : (H) an isotropy class of (X, G)} is a finite stratification of Z into locally closed irreducible smooth algebraic subvarieties. ( 2 ) : If (H) is an isotropy class of ( X , G ) , then cl(Z(H)) =

U

(H)<_(H')

Z(tt,).

Here, cl(-) denotes the Zariski closure in Z, but since Z(H) is constructible this closure coincides with the closure in the analytic topology. If (H) is an isotropy class of (X, G) and if y : X ---, Z is an orbit map, we denote by X(H) the inverse image y-l(Z(~)). X(E) is equipped with the structure of a locally closed subvariety of X, but is not necessarely reduced.

150

1.2 : I N V A R I A N T S

01~ T O R I

Let ~) be an r by s matrix with integer coefficientssuch that rk(~) = r <_ s. Let ct be an r-dimensional (column) vector over the integers and define the following sets

E+ =

{p E ~. +.~ :

=

o}

E , , ~ = {# E I N s : + . ~ = ~} It is clear that E# is a submonoid of INs and that E+,~ is an E ~ - m o d u l e in the sense that E+ + E+,a C E®,a. CE+ will denote the monoid algebra of E+ over ~ , that is the subalgebra of ~[xx ..... zs] generated by the monomials x a for all fl E E+ where we denote x~ x ...x~" = x a for any/~ E / N s.

Similarly, we define ¢E+,a

to be the q'e-subvectorspaceof ~[X

1 ...., xs]

generated by the mono-

mials x a where ~ E E¢,a. It is clear that eEv,~, is a eEl-module in the obvious way. Both the ring e E ¢ and the module eE+,a have an interpretationin terms of invariant theory. Suppose that ¢ = b~,--., ~,l where each "Yi is an r-dimensional vector, then we define

r, = {aiag(u', .....u~'): u ~ (¢*)'} where for u = (ul, .... ur)" and 7~ = {~i, ..... ~ , ) " we denote u ~' = u "/il 1 ,.+U~ir, Because rk(d~) = r, Tr is a subgroup of G L , ( ~ ) an r-dimensional torus.

isomorphic to (¢*)r and hence by definition

Tr acts in a natural way on the polynomial ring ~ [ z l .... , xs] , viz. if

r~ = d i a g ( u ~ , .... u 7.) E Tr, then

"ru.f(xl ...., xs) : f(u~Ix I .....u~'xs)

One verifieseasily that ~ E + is equal to the ring of invariants ~[xl .....xs]T'. Since T, is linearly reductive, it follows from this that ~ E + is an affine Cohen-Macaulay algebra, [HR].

151

W e also wish to interpret the module

¢E¢,a

in terms of invariant theory. Suppose that the

equation ¢.fl -- a has at least one integral solution fl • ~

, then the m a p

X~ : Tr--~ ~* defined by Xa(~',~) -- ua is a one- dimensional representation (or character) of Tr. Then, one can verify that CE¢,a is the module of semi-invariants or relative invariants of Tr with respect to the character Xa, i.e.

CE~.a= { f ( x l ..... x,) e ¢[xl ..... x , ] : Vru e T r : r u . f ( x l ..... xs) = Xa(ru)f(xl, .... x,)} From this one can deduce that CE~,a is a finitely generated CE¢-module. However, it is not true in general that CE,l,,a is a Cohen-Macaulay module. Stanley [St,Th.3.2] has proved the following sufficient condition :

Proposition

1.2.I

: Suppose there exists a rational solution fl =

• ./~ = a satisfying - 1 < ~ < 0 for all i, then

~]E~,a is a

(fll,...,fl,)r •

(~° to

Cohen-Macaulay module.

In [St] a n d [Ho] it was shown that ring- and moduletheoretic properties of C E v a n d ¢ E ¢ , ~ are closely linked to topological properties of the (s - r)-dimensional convex polyhedral cone C¢ in JR' consisting of the set of all solutions fl E JR' to ¢./~ -- 0. Let Pc be the non-degenerate cross-section C¢ A {(fll, ..., /9,) • ]1%' : ~ f l ~ = 1} of C¢ , then Pc is an (8 - r - 1)-dimensional convex polytope. If fi e IR~., we define its support supp(fl) = {i : fl~ > 0} and if/9 • ]1%" we define its negative support supp_ (fl) = {i :/9~ < 0}. If 7 is a face of P c , then all elements of the relative interior of ~r have the same support, supp(3:). It follows that the faces of Pc are in one-to-one correspondence with the supports of elements/9 • E v and that two faces ~r ~ satisfy ~r c ~ iff supp(7) c supp(~). If v is a vertex of Pv , then those elements fi • E ¢ satisfying supp(fl} = supp(v) are IN-multiples of a unique element/gv • E¢ , the so called completely f u n d a m e n t a l solution corresponding to the

152

v e r t e x v. Recall t h a t ~ e E ¢ is said to be completely f u n d a m e n t a l if whenever mfl = -~ + ~ where rn _> 1 and "1,6 6 E ¢ , then "7 = ij9 for some 0 < i < rn. W i t h C F ( E o ) one denotes the set of completely f u n d a m e n t a l solutions. T h e Krull dimension of ¢ E ¢ is equal to the dimension of the Q- vectorspace spanned by C F ( E ¢ ) , i.e. is equal to s - r. Let < E ¢ > denote the group generated by E ¢ in 1~ and let < E¢,a >=

< E . > +E¢,a, the coset of < E ¢ > i n l f containing Ec,a. Now, let P~ be the dual polytope of P¢ and define

r. = U{z': z face of P¢ s.t. supp_(fl)C supp(~r)} From [St2,p.47]we recallthat if(~E¢,a is a Cohen-Macaulay module of dimension d, its d-th local cohomology module with respect to the irrelevantideal of ¢ E ¢ is given by

The CE¢-module structure on it is given by

x ~.x ~ = x ~+~ if r ~ + ~ =

= 0 if r~+ a # ¢ for all -y E E ¢ and x ~ E H d ( ~ E ¢ , ~ ) . Suppose there exists ~ -- ('~1, ...,'y~) 6 Q* with - 1 < -y~ _< 0 such that ~ . ~ = c~ , then by Prop.L2.1, ~ E ~ , a is a Cohen-Macaulay module.

In this case, H d ( ~ E ~ , ~ ) can be described as

follows : choose m 6 IN such that m'~ is integral and let ~ E < E¢,a > , then ~ . m ( f l - "y) : 0 and supp_ (~) = supp_ (~ - ~1) = supp_rn(p - ~I). Since F~ depends only upon ,upp_ (~) we have F~ = r,~(a_~).

If we assume,further,that there exists a solution 6 = (6,, ..., 8~) 6 E o such that

6~ > 0 for all i , then since rn(~$ - ~/) 6 < E o > we have [St2,p.47] : r.~(a_~) = ¢ iff rn(~ - 7) < 0. Therefore, we have

nd(¢E¢,,~) = ¢ { x a : fl 6 < E¢,~ >: fl < 0}

153

1.3 : F I N I T E D I M E N S I O N A L

REPRESENTATIONS

An r~-dimensional representation of the free algebra in m wriables is an algebra morphism ~:¢<

x1 .... , x ~ > - . M . ( ~ )

Note that this is equivalent to giving m elements ~{XI) ..... ~(Xm) in Mr,(~) , i.e. n-dimensional representations of ~ < X1, ..., Xm > can be parametrised by an affine variety Xm, n which is just mr~2-dimensional afflne space. Two n-dimensional representations are said to be equivalent if they differ upto a 4" automorphism of M~(~). So, the projective linear group PGLn(~) acts on Xm,,~ and the orbits under this action are the equivalence classes of representations. If ~ is an n-dimensional representation, then ~('~) becomes a ~ < X1, ..., Xm >-module via ~. If ~(r~) is completely reducible as such, then ~ is said to be semi-simple. In general one has a composition series

o=V, cVt_~ c

... c V~ c V 0 = ¢

(")

of ~ < X1 ..... Xm >-modules. Then, I~ = (B(Vi/Vi+I) is completely reducible and r~-dimensionaL

For a suitable choice of basis of ~(n)~b can be expressed in the matrix-form

,,,here the ~s: ¢ < X~..... X ~ >--. M ~ ( ¢ ) for k; -- d i m ¢ ( ~ ' - d ~ ) are the i~ed~cible components. With ~ we can associate the semi-simple representation

isomorphic to W as module. Artin [Ar] proved that ~" lies in the closure of the orbit GL,~(~)(~),Moreover he proved that the closed orbits in X,~.,~ under action of representations. Therefore, the quotient variety

GLn(~)

correspond precisely to the semi-simple

154

parametrizes the equivalence classes of semi-simple n- dimensional representations of ~ <

X1 ..... X.~ >. A concrete description of the coordinate ring ¢[Vm,.] can be given as follows. Consider the polynomial ring

:'-.,. = ¢ [ = ~ A O : 1 _< i,i < ,~;1 < z < m] and let O~.~,n be the ring of m generic n by n matrices, that is the subalgebra of M,,(Pm,,,) generated by the m elements

X,

= (xii(1))i,jE

Mr,(.Pm,n)

The ring of matrixinvariants ~,n,,~ is the subalgebra of Pm, n generated by the traces of elements of (]~. . . . See for example [Pr] for a proof that ~ m : , is an affine algebra and ¢[Vm, n] = ~ . . . . Not much is known about the geometry of Vm,n.

In [Pr] it was proved that the points

corresponding to the equivalence classes of irreducible n-dimensional representations form an open smooth subvariety V~,r, irr of dimension (m - 1)n 2 + 1. If one is not only interested in the semi-simple representations but also in their irreducible components one has to study a certain 'noncommutative algebraic variety' U,,,,~. The trace ring of rn generic n by n matrices is the subalgebra of Mn(Pm, n) generated by ~m,n and (~m,n and will be denoted by 'Irrn,n. The points of Urn, n

are

the maximal twosided ideals of the noncommutative but affine p.i.-

algebra ']r,n,n. We can equip Um,n with the usual Zariski topology [Pr3], that is a typical open set consists of those maximal ideals not containing a given twosided ideal ofqirm, n. Since ,l~rn, n C rJ~rrt, n is a central extension, there is a canonical continuous map

i : U,~ m --* Vm, n

In [AS] the fibers of i were described in the following way. Let ~ ~ Vm,~, then ~ corresponds to the equivalence class of a semi-simple representation

155

Let Xz, ..., Xr be the distinct irreducible components, where X~ is a k~-dimensional representation occuring with multiplicity e~ in ¢. That is, ~ ei = t and ~ el.k~ --- n. We can always assume that /=i

]¢1 >-- 1c2 >-- ...

/=I

>-- kr and then we say that ~ or ~b is of representation-type (el, kl; e2, J~2;---; er,

kr)"

The fiber i-1(~) consists of r points (¢, ~i) each corresponding to one of the distinct irreducible components. The morphism i is then given by sending a point ~ = (¢, ~i) to ~.

II : T H E

VARIETIES

V.~,.

In this chapter we aim to initiate the geometrical study of the varieties V,n,~.

In the first

section we will show that the different representation types give a finite stratification of V,n,r, into locally closed smooth subvarieties. In the second section we describe the/~tale local structure of V,n,,~ in points corresponding to semi-simple representations with distinct irreducible components. In the last section we show that, except when (rn, n) = (2,2), Vrn, r, is always singular and the singular locus is precisely the difference V m . , - Vr~.r,.~rr

ILl : STRATIFICATION

OF. V,n,,,

Recall that a point ~ E Vm,, is said to be of representation-type r = (ez, kz; ...; er, kr) if the corresponding semi-simple n-dimensional representation has r distict irreducibld components Xi of dimension ki and multiplicity e~. I I. I Another representation-type r' = (el, kl, ...; %, k~) is said to be a refinement Qf • if there is a

permutation a on {1, ..., r'} such that there exist natural numbers

jo = l < j l < j2 < ... < j r = r '

such that for every 1 < i < r we have

eik i =

¢1

ei l 'e=(y) for all j i - l

kt

< j < ji

156 This defines a partial ordering on the set of all representation- types for n-dlmensional representations : RT~,. For example, RT4 has the following Hasse-diagram : (4, 1)

(3,1; 1,i)~ l (2, 1; 1, 1; 1, I)

(2, 1; 2, 1)

(1, 1; 1, 1; 1, 1; 1, 1)

(2, 1; 1,2) J

(I,1;1, 1;i,2) (1, 1; 1, 3) ~ /

(2,2)

~

(1, 2; 1, 2)

(1,4)" For a representation-type r E T•, we denote by V,n.~(r) the set of all points ~ E Vm, r, of type r. The main result of this section can now be stated as :

Theorem

I I . l . l : W i t h notations as above we have :

(1): {Vm,.(r) : r E RT,~} is a finite stratification of V,n,. into locally closed irreducible smooth algebraic subvarieties. (2) : Vm,.(r 0 lies in the closure of V.~,.(r) if and only if r ' is a refinement of r.

Proof : Let r = (el, kl; ...; er, kr) and ~ E Vtn,n(r), then the fiber of ~ under the morphism ~r,n,~ :

Xm, r, "-* Vm,,~ contains one closed orbit T(~). In this orbit, one can find a point x = (xl ..... z,~) where each n by n m a t r i x x~ is of the form rnl ~ ) le~ 0

0 rn2 {~ le~

) 0 •.~ x i

0

mr ~

1~ r

where each rni E M~, (~). We will now compute the isotropy group in this point : GLn((~)x. An element a E GL,~((~) leaves z fixed if and only if it commutes with each of the xi. Therefore, GLr~((~)x is the multiplica-

157

tire group of units of the centralizer of

Mk, (¢) @ 1,. which is the algebra generated by the xds by assumption. It is easy to verify that this group is equal to G L , ( ¢ ) , _ GL~,(¢) × ... x

GLe,(¢)

where the embedding in GLr,(¢} is given by

aL,,(¢).lk,

× ... × a L ~ . ( ¢ ) . l k .

Of course, a different choice of the element x in T(~) gives a group conjugated to GL,,{¢)= in V L , ( ¢ ) . Further, if =' e T { ¢ ) is chosen such that

GL,,(¢)=, = GL~I(¢).I~I x ... x GL~i(¢).lk/

and if GLn(¢)=,

is conjugated

to

GLn(¢)=

a permutation cr on {1 .....r} such that

in

GL,,(¢), then

it is clear that r = r' and there exists

(ei,ki) = tte,a(#),~a(i) j l ' '~ , Le.

~ and ~' belong to the same set

vm,.ff). The statement now follows immediatly from Theorem 1.1.2.(1). (2) : In the first part w e have shown that the isotropy corresponding to

V,,,,,~(r), where

r = (el, kl; ...; er, kr), is the conjugacy class of GLe, (¢).1k~ × ... x GLe,.lk, de=~GL,((~)r in GL,,(¢).

From theorem 1.1.2.{2} we know that Vm,~,(r'} lies in the closure of Vm,,(r) if and

only if the group GLn((~)~ is conjugated to a subgroup of GL~,((~)r,.It is easy to verify that this happens precisely when r I is a refinement of r.

For example, the closed subvariety of Vm,, determined by the Formanek center of the trace ring of rn generic n by n matrices ( in out terminology V,,,r, - Vm,r,(1,n)} is in general reducible.

158

Each of its [ ~1 irreducible components contains an open set induced by Vm,,,(1, i; I, n - i). Finally, we note that the dimension of the subvariety V,n,,,(r) where r -- (el, k~; ...; er, kr) is equal to (m - 1)(k~ + ... + kr2) + r.

II.2 : L O C A L S T R U C T U R E

O F V,~.,

According to the &ale slice theorem,the local structure of the variety Vm,,, = Xm.r,/GL,,((~) near a point ~ is isomorphic to that of the quotient of the slice representation near the origin, i.e. with Nz/GL,,((~)~ where x • T(~) and N~ is the normal space in Xm, n to the orbit GL,.,((~)(x). Suppose ~ is a point of type (el, kl; ...; er, kr), then we can take for x = (xl .... , zm) • Xm,~ such that each of the z~ has the form

xi =

0

" .

0 mr ~ le,

where mi E Mk, ((~). In the foregoing section we have calculated the isotropy group in such a point

GLr,((~)x = GLe, (¢).lk, X ... X GLe.((~).lk.

The t a n g e n t space T~(GL,.,(q3)(x)) in Xra,,-, to the orbit GLr,(¢)(z) is equal to the image of the linear map M n ( ¢ ) --* M,,(¢) G

"'" ~

M,(¢)

y -" [y, xl] 0 . . . (~[y, =m] see for example [Mo]. The kernel of this m a p is clearly the centralizer of the subalgebra of M,,(~) generated by

So, we obtain an exact sequence of GL,~(~)~-modules

o ~

o , --, M . ( ¢ )

-. T~(GL.(¢)(z)) -,

o

159 where C~=

0

"-

0

M~, (¢) ~ 1~, But then, since

GLr,(•)x

is a reductive group (so every

GLn((~)x- module

isomorphism by its irreducible components), the normal space N~ to

is determined upto

Tx(GL,.,(¢)(x)) is isomorphic

to the GLn(¢)~-module

Nx = M,.,.(¢) G "'" ~ ) Mn(¢) ~ C x where we have rn - 1 copies of Ms(C) and the action of the isotropy group

GLr,((~), is,of course,

given by componentswise conjugation. The ~tale slice is then the variety corresponding to the ring of invariant polynomial mappings from Nx to ¢ under this action cf

GLr,((~)x.

We will now describe this ring in the special case that all the irreducible components Xi of the to ~ associated semi-simple representation are distinct, that is, ~ is of type (1, kl; ...; 1, kr) where ~ k i = n.

In this case, the isotropy group of x is the r-dimensional torus Tr which is embedded in

GL,~(¢)

aS

T , . = ¢ * .I k, x ...× ¢ * •i k, Clearly, Tr acts trivially on the following subspace N1 of Nx

0".

0

~=i

@

Mk, (¢)

...

0 ¢. ik,

so, the 4tale slice is N , I T , = ,~d x N21T,. w h e r e d = ( m - - 1 ) ( k l2 + . . . + k r 2 ) + r and

N2 = 6 ~ i where V~j is an ( m - 1)kiky-dimensional vectorspace on which an element t = (c~l .... , a t ) E Tr acts by sending an element v E V~i to

a~a'~lv.

Let

s = ¢[N2] = ¢[v,,.(~) : 1 < d # 3 < r; 1 _< ~ < ( - ~ - 1)k,k~]

160

t h e n t h e a c t i o n of Tr on S is d i a g o n a l a n d is therefore d e t e r m i n e d b y a n r b y s m a t r i x w i t h integer coefficients w h e r e 8 =

gdirn(S)

= 2 ( m - 1) E , ' . # j

k~kj.

T h e c o l u m n c o r r e s p o n d i n g to t h e variable v i i ( a ) consists of zeroes e x c e p t a t t h e i - t h row + 1 a n d a t t h e j - t h row - 1 . O n e easily verifies t h a t t h e last row is a linear c o m b i n a t i o n of t h e o t h e r s so we c a n restrict a t t e n t i o n to the r - 1 b y 8 m a t r i x ¢ w h i c h is o b t a i n e d b y erasing this last row. O n e verifies t h a t rk(¢) = r-

1.

T h e ring of invariants, (~[N2/T~], is o b t a i n e d from t h e set of i n t e g e r solutions fi E IN s t o ¢ . f l = 0. As we have seen before, it suffices to consider t h e f u n d a m e n t a l solutions. In t h i s case, t h e y are also completely f u n d a m e n t a l . T h e corresponding m o n o m i a l s in S = ¢[N2] are o b t a i n e d b y the following p r o c e d u r e : Let 2 < k < r a n d let (il .... , ik) be a cycle of k distinct elements from {1 ..... r} s.t. its m i n i m a l e l e m e n t is i l . T h e n , w e get t h e i n v a r i a n t s

V i i i 2 x V i i i , x ... x Vik-l~k X V i ~ l

w h i c h are g e n e r a t e d by t h e e l e m e n t s

~,,,, ('~, )...~',k-,,~ (,~k-1),~,,, (,~k ) w h e r e t h e a i r u n over all admissible values. Finally, we n o t e t h a t

dim(~T~/T,)

= 8 - (~ - i ) = 2(m - i) ~

k,k~. - ~ + 1

w h i c h is c o m p a t i b l e w i t h the fact t h a t

( m - 1),~ ~ + 1 = d ~ m ( Y , , , , ,

= d + d~m(g2/T,)

In t h e n e x t section we will give a more precise d e s c r i p t i o n in t h e special case t h a t r = 2. Let us s u m m a r i z e t h i n g s in

161

Theorem

11.2.1 :

If ~ is a point in V,~,,~ of type (1, kl; ...; 1, kr), then a neighbourhood of ~ is isomorphic to a neighborhood of the origin in .~1d x N2/Tr where d = (m - 1)(kl2 + ... +/¢~) + r, N2 = (~ir#i Vii

where VO is ( , ~ - 1)k, ki-dimensional and T, acts on it by (~1 .....

~,).~

= ~,~1~.

Further, the coordinate ring ~[N2/Tr] is the subring of ~[N2] = ~ [ t ~ i ( a ) : 1 < i # j < r, 1 _< a _< ( m -

1)/~k/] generated by all monomials of the form v~i~ (O~l).,.Vi~_li~(O~k--1)t~ikil

(O~k) where

(i~, i2 .... ,/k) is a cycle of length 2 < k < r of distinct elements from {1 ..... r}. Its Krull dimension is 2(rn - 1) ~ k, k i - r + 1.

11.3 : S I N G U L A R

LOCUS

O F V,~,~

The main result of this section states that the closed subvariety of Vm, n determined by the Formanek center of the trace ring of rn generic n by n matrices (or,equivalently, the set of reducible semi-simple representations) is precisely the singular locus of Vm, n. If rn or n is equal to 1, V,n.~, is clearly nonsingular, so we m a y assume that rn and n _> 2.

Proposition H.3.1 : The variety V,n,,, is singular except when (rn, n) = (2, 2).

Proof : Assume that V,~,,~ is nonsingular. Since ~m,n is a positively graded afiine algebra, it has to be a polynomial ring in (rn - 1)n 2 + 1 variables over ~. So, the Brauer group Br(Vm, n) is just Bd¢)

-- 1.

R. Hoobler proved in [Hb] the Auslander-Goldman conjecture stating that the Brauer group of a smooth afllne variety is determined by the codimension one irreducible subvaxieties. Therefore, we have

B~(V~,.)--

N B~((~,.)~) pEX(1)

where X (1) is the set of all height one prime ideals of ~m,n and the intersection is taken in the

162

Brauer group of the field of fractions ]~,,,~. Now, we know that the localization (2rm,,,)p of 2r,~,n at any height one prime ideal p of ~m,,, is Azumaya except for (m, n) --- (2, 2) and p =

(XlX2 - X2X1) 2.

For, the dimension of the closed subvariety determined by p is equal to (rn - 1)n 2. Suppose that the corresponding localization of ~l]?m,, is not Azumaya, then the points lying on this closed subvariety correspond to reducible semi-simple representations. It follows from our stratification result (Th.II.l.1) that such a variety of maximal dimension has an open subset consisting of semi-simple representations with two irreducible components of dimensions r and n - r . Therefore, the dimension of such a variety is at most {m-1)[r2+(n-r)2]+2. Clearly, the equation ( m - 1)[r~ + (. - r) ~] + 2 = (.~ - i)~ ~

has only an integer solution if (rn,n) = (2, 2) and r -- 1. It is well known from 19-th century algebra that J~2,2 = ~J[Tr(X1),Tr(X2),D(X1),D(X2),Tr(X1X2)].

Therefore, the class of the

generic division algebra A,~,,, in Br(Km,,~) belongs to Npex(1) Sr((J~)p) provided (rn, n) # (2, 2), and so we obtain a contradiction.

We will now investigate when V,n,,, is smooth in a point ~ corresponding to an equivalence class of a semi-simple representation having two distinct irreducible components.

Proposition II.3.2 :

The 6tale slice of Vm,,~ in a point ~ of type (1, r; 1, n - r) is ~ d x W where

d = ( m - 1)[r 2 + ( n - r) 2] + 2 and ¢[W] = ¢[t,j : 1 < i, 3' < (m - 1)r(n - r)]/I2 w h e r e / 2 is the ideal generated by all 2 by 2 minors of the generic matrix {tO~)i,y.

Proof : By the calculations of the foregoing section we know that the slice in ~ is equal to

.~d × N2/T2

163

where N 2 = V12 (~ V21 and both components are (m - l)r(n - r)-dimensional and an element (c~,fl)E T2 acts on a generator z~ E S(V12) (reap. y~' E S(V~I)} by sending it to a~-Izl (reap.

c~-lflyj). Therefore, the invariant ring ~[N2/T~] is generated by the monomials z~yj for all admissible values for i and j. The relations among these invarlants are easily seen to be generated by the 2 by 2 minors of the matrix (x~yy)i,j. Sending the indeterminate ti# to z~yj we get the required statement.

If ~ is a point of Vm, n of type (I,r; I,n - r), then Vm, n is singular in

Proposition II.3.3 :

except when (m, n) = (2, 2).

Proof : By ~tale descent, it sufficesto show that the ~tale slice is singular in the origin. This follows from the fact that 12 is a nontrivialideal (if (m, n) ~ (2, 2)) not generated by degree one elements (in the obvious gradation on ~[t~'].

Using this fact and the stratificationresult of ILl, we can now prove the main result of this section :

Theorem

H.3.4 : The singular locus of the variety Vm.n coincides with the complement V,n,n -

irr V~. n except when (m, n) = (2, 2).

Proof : Let F~lrm, n be the Formanek center of the trace ring of m generic n by n matrices, i.e. the ideal defining the open set V~r,~. Let ~

be the variety defined by ~[V~---~,~= ~ m , n / F q r . . . .

then by theorem II.l.1 we know that each of the irreducible components of Vm, n has an open set determined by seml-slmple representations having two distinct irreducible components. Suppose that V" m ltea n

,

the open set of all regular points in Vm ~ n , is strictly larger than V~, nirr

,

164

then V reg induces a proper open subvariety in at least one of the irreducible components of Vr, ,. This entails that V reff contains points corresponding to semi-simple representations having two distinct irreducible components, but this is impossible by proposition II.3.3.

Ill : TRACE RINGS OF GENERIC MATRICES.

In this chapter we will investigate the ~tale local structure of the trace ring of m generic n by n matrices, ~Irrn,n. If ~ E V~.,~ irr , it is well known that this ~tale local structure is just n by n matrices over a commutative (regular} domain. We will describe explicitly the structure when ~ is a point corresponding to a semi-simple representation with distinct irreducible components. It will turn out that the 'noncommutative slice' in such a point is Cohen-Macaulay and its Poincar~ series satisfies a certain functional equation. In the final section, these results are applied to solve the regularity problem for trace rings, i.e. gldirn(V~rn, n) < co iff rn or n is 1 or (rn, n) = (2, 2), (2, 3) or (3, 2).

III.1 : L O C A L S T R U C T U R E

O F ~Tm,.

Recall from [Pr2] that the trace ring of rn generic n by r~ matrices is the ring of equivariant maps :

i.e. polynomial maps such that for every a e GLn(~) the following diagram is commutative :

Xm, n

~¢

Mn(¢)

X m , r~

- -¢ -,

M.(¢)

where the action of GL,~(~) on M . ( ~ ) is given by conjugation. Now, let ~ be any point in Vrn, n and x • T(~), then we know that the diagram below is defined and commutative in a neighborhood

165 of x

GL~,(¢) XGL~(¢) ~ N~ --* N:,/GL,~(¢)~ -~ (GLn(¢) x cL~(¢)" Nx)/GL,,(¢)

where the morphism affine

Nz. -'* Xm, r, is defined by sending a point n to x + n. There exists an open

GLr,(¢)~ stable neighborhood N ° of the origin of N~ and an open affine GLr,(¢) stable

neighborhood X ° n of x in X,n.,~ so that

GL,.((~) x cL~(¢)" N~° -= X~,no xu~,,~(N °x/GLn((~)x) From this the following can be easily proved : let B be the coordinate ring of is an algebra over

NO/GLr,((~)x which

(~[Vm,n] then ~]]?,~,.( ~ ¢ [ v . , . ] B (the noncommutative ring ~Irm,n localized in the

given ~tale neighborhood of ~) is isomorphic to the ring of equivariant maps from

GLr,(¢} × cL.(¢)x

N ° to M,,(¢). Furthermore, we can assume that N ° is the set of elements of Nx where an invariant polynomial / (under

GL,~(¢)=) on N= is not zero.

Then, if R is the ring of equivariant maps from

GL.(~) x c L ' ( ¢ ) , N~ to M . ( ¢ ) we have that

¢[vm,.I The ring R can be called the noncommutative sfice (in the point ~). We will now restrict attention to the case that ~ is of representation type (1, kl; ...; 1, kr), that is when G L n ( ¢ ) x = Tr. Then we have to describe the ring of equivariant maps

f : M,,(¢) ( ~ ... G M,~(¢) ( ~ C . ---, M,,(¢) where Tr acts on every component by conjugation. This study is essentially the study of all polynomial maps g : M . ( ¢ ) ( ~ ... ( ~ M,~(¢) ( ~ C . ( ~ M,~(¢)" --~ ¢ which are invariant under Tv and homogeneous of degree one in the indeterminates corresponding to the component Mr,(¢)*. As a Tr-module, M,,(¢)* decomposes into a direct sum of one-dimensional

166

vectorspaces ~ Ce~y. If

kl +... + ko < i <_k~ +... + ko+l kl + ... + k~ < j <_kl + ... +k=+~ t h e n v = ( U l ..... ur) acts on ~e O. by sending e~y to u-~luteij. This allows us to determine the part of the invariant ring which is homogeneous of degree one in the variable corresponding to e~j : z~y. A typical element is of the form h.z~y where h is a s e m i i n w r i a n t on N= with respect to the character

x.

: T, --,

¢*

determined by sending r = (uz, ..., u,) to u,.u~ 1. W i t h notations as in II.2 this module of semiinvariants is ¢[E.,~,][yl, ..., Yd] where d = (m -- 1)(k~ + ... + k~) + r and c~ is the (r - 1)-dimensional column vector with a --1 on the i-th row, + 1 at the j - t h row and v.eroes elsewhere. Therefore, (~[E¢,a] is the subvectorspace of ¢[N2] consisting of all polynomials h such that h.vt,(a) E ¢[N2/Tr] for any 1 < a < k,k,(rn - 1). These observations prove

Theorem

III.l.l

:

T h e n o n c o m m u t a t i v e slice of the trace ring of m generic n by n matrices,

~]rra.,~, in a point ~ E Vm, n corresponding to a semi-simple representation of type (1, kl; ...; 1, ]%) is isomorphic to a polynomial ring in (m - 1)(k~ + ... + k~) + r indeterminates over the ~ - algebra

[ MIc, (¢[N2/Tr]) F,.,,,,,(~) = [ W2, \

W,,

W,2

...

Mk,(¢lN2/n])

...

W,.2

...

Wlr W2,Mk,(¢[Nu/T,])

"~

)

where Wi 5 is a k~ by ks. block of ¢[N2/Tr; Xiy] which is the module of semi-invariants on N2 with respect to the character XO' : Tr -4 ¢* or, equivalently, the subvectorspace of ¢[N2] consisting of polynomials h such that h.vsi(a ) e ¢[N2/Tr] for all 1 < a < (m - 1)kiky.

167

III.2 : T H E F U N C T I O N A L

EQUATION

In [Le] it was shown that the trace ring of rn generic 2 by 2 matrices is always a CohenMacaulay module over its center , i.e. a (graded} free module of finite rank over a polynomial subring of the center. Unfortunately~ the ~tale-sllce machinary cannot be used to prove Cohen-Macaulayness for arbitrary p.i.-degree. The reason is that the study of the noncommutative slice in a point of type (r~, 1) is as hard as the study of the trace ring in the origin. Nevertheless, we can give some weight to the conjecture that trace rings are Cohen-Macaulay by proving that the Cohen-Macaulay locus is large.

Theorem

III.2.1 : The trace ring of m generic n by n matrices is a Cohen-Macaulay module in

a point ~ of

Vm,, of type

(1, kl; ...; 1, kr).

Proof : B y ~tale descent, it is sufficient to prove Cohen-Macaulayness of the noncommutative slice in 4By the Hochster-Roberts theorem and theorem III.l.1 this amounts to showing that ~[N~/Tr; Xij] is a Cohen-Macaulay module. In the foregoing section we have seen that ~[N2/Tr~ X~j] = ~[E¢,a] where a(i) = -1,a(j) -- 1 and a(k) = 0 elsewhere.By proposition 1.2.1 we know that ¢[E.,~] is Cohen- Macaulay if there exists a rational solution fl = (81, ....~,)~ to ~.fl -- a satisfying -1 < ~ _< 0. It is easy to see that such a solution always exists (with entries ½ or 0) if n > 2 or m>2. The remaining case, ~r2,2 is easily seen to be a free module of rank four generated by {1, xl, x2, xlx~} over its center which is a polynomial ring.

W e will now study the Poincarfi series of the noncommutative slice, or equivalently that of r .... (4).Usually, it is rather hard to determine the power series P (Fm,,~(~); t). There is, however,

168

one i m p o r t a n t exception :

Proposition

III.2.2 :

Let ~ E V,~,,, be a point of type (1, k; 1, l), then the n o n c o m m u t a t i v e slice

in ~ has the following Poincar~ series

)(

(i ld)d.{2k/. ~--~ (m- 1)kl+s1 8 a>O

"

q_(/¢2÷i~)~--~ ( m - l )

÷s-I

(rn - l)]cl 8 -- --{1 8- I

)

t2'-i

2t~,,}

0~0

Proof : We will first determine the Poincar~ series of the ring of all polynomial maps

e.o(¢/e which are invariant under T2.

o)

¢01,

~

M,~(¢)" --* (p

Here we give the indeterminates corresponding to the first rn

factors degree t and those to the last factor degree x. This invariant ring is a polynomial ring in (rn - 1)(k 2 + l 2) + 2 indeterminates of degree t and k 2 + l 2 indeterminates of degree x over the ring of invarlants of

(V12~ W21)~ (If21( ~ W12) where d i ~ ( V ~ ) = d~m(V2,) = (,',,- l)kl and ai,.(w~2) = di,~(W~) = kl ~nd on

(., ~) e

W~ ~cts

the firstcomponent by multiplicationwith 0 ~ -I and with a-i~ on the second. The ring of

invariantsisthen the subringof ¢[zi, y~:

I<

generated by all products xiy~, x i v y ,

i < kl(m-

y~uy, y~vy.

1);uy, uy:

1_

3"--~ kl]

The ideal of relations between t h e m is generated

by the 2 by 2 minors of the matrix

I

xiyi xkl(m-l)Yi Yi U k l

•--

Zlyk/(m-i)

• ..

Xkl(m--i)Ykl(m-l)

•..

ykl(rn--i) ttl

• ..

ykl(m-i)Ukt

XlVi

...

XiVkl

Zkl(m--1)~l UI~I

...

mk/(m-- i) t;kl

...

UIt)kl

UklUi

...

UklVkt

169

So, all relations are homogeneous in the (t,x)-gradation. It is then fairly easy to see that the Poincar~ seriesof this invariant ring is (use Plethysm formula) co

s. (¢"-) ® s. (¢~'~) where ~klm = U {~ V with dim(U)

=

(m - 1)kl, dim(V) = kl and the coordinates of U (resp. V)

have degree t (resp. x). So,

s'(u G v) = ~ s'(~) ® s'-'(v) /=0

± ((.,,_,,,.+,-1) (,.+.-,-, ] d=,,-i ~=0

i

\

s- i

/

Therefore, the Poincar~ series in the (t, x)-gradation of the invariant ring is equal to

• :0

i

/=0

S -- i

ti~s--i

and therefore, this expression multiplied with (1 -- t) -((rn-D(k~+I')+~) ( I -- x) -(k2+t~)

is the Poincar~ series of the total ring of invariants in the (t, x)-gradation. The Poincar~ series of the noncommutative slice is then the partial derivative of this expression with respect to x and evaluated at x -- 0, which gives us the claimed expression.

We have seen in the foregoing section that ¢[N2/Tr; X,:5] = CEv,a, for certain a, is a CohenMacaulay module which is clearly of dimension 2(m - 1) ~ k~kj - r + 1 = h. Assume that 81 .... ,0h is a homogeneous system of parameters for ¢[N2/Tr; Xiy] and let S = ¢[01 ..... 9hi, then the canonical module of ¢[N2/Tr; X~Y] is defined to be

fl(¢[N2/Tr; x,y] = HOMs(¢[N2/Tr; X~y],S) which is Z-graded in the obvious way. Form ISt,th 4.4] we retain that 12(¢[N2/Tr; X~y]) is the subvectorspace of ¢[N2] spanned by all monomials YI v~y(k) b'y(~) s.t. (fl~y(k)) e ~

(for notation see 1.2) and r_~,¢(k) = ¢.

170

But, if (/9ii(]¢)) e E¢,a then ( - f l , y(k)) E E . , - a

and since ¢ [ E ¢ , - a ] = ~,[N2/T,; Xi~] we have

again a solution to ¢./~ = - a satisfying - 1 < fl~ < 0 for all i , whence we can apply the argument at the end of 1.2 to ensure that F(_/~,Ak) ) = 0 iff -fliS(k) < 0 for all i, j, k. So, we have shown that

n((~[N2/T,; X,y]) = ( ~ ) { ¢ H

v'J(k)~'Ak) : fl,j(k) > o}

Luckily, there is a unique strictly positive solution to ¢./9 = 0 namely fli = 1 for all 1 < i < 2(m - 1) E k~k~.. Therefore,

£t(¢[N2/T,; X,i]) = H v'i(k)'¢[N2/T'; X,j] Translating this equality to Poincar6 series gives us

P (t2(¢[N2/Tr; XO']); t) = t'.2(¢[N2/Tr; XO']; t) where s = 2 ( m - :) E,~k,..

Applying :St2,p.58t we get the functional equation I = (-1)u.t'.P(¢[N2/T,; XO']; t) ~ ' ( ¢ [ N 2 / r ~ ; x~,]; :-)

where h = 2 ( - , - :) E k,k; - ~ + 1.

This concludes the proof of the following

Theorem

III.2.3 :

Let ~ E Vm, n be of representation type (1, kl; ...; 1, kr), then the Poincar4

series of the noncommutative slice R at ~ satisfies the following functional equation 1

P(R; :) = (-:)d.t("~-:)'+'.P(R;0 where d = Kdim('~m,n) = (rn - 1)n 2 + i.

171

III.3 : REGULARITY

OF ~r.,,~

W h e n m or n is equal to one, 2r,~,,, is a commutative polynomial ring and hence has finite global dimension. In [SS] L.Small and 3.W. Stafford proved that gldirn(2r~,2) = 5. In [LV] it was shown that for n _< 4 , gldirn(2rm,•)

< co if and only if (re, n) = (2,2);(2,3) or (3,2). In this

section we aim to show that these are the only n o n c o m m u t a t i v e trace rings of generic matrices having finite global dimension. The strategy of the proof is the following : choose a suitable point ~ in Vm,,~. Ifgldirn(~m,,~) < c o , then the global dimension of the noncommutative slice in ~ has to be finite,too. Suppose that is chosen in such a way that all indecomposable graded projective modules of the noncommutative slice have the same Poincar~ series, then finite global dimension can be tested by the fact that the Poincar~ series of such a projective module needs to have the rational form I - ~ for some f(t) • l~t] and k • IN, by the s t a n d a r d argument.

T h e o r e m I H . 8 . 1 : The trace ring of rn generic n by n matrices, ~m,,~, has finite global dimension if and only if (1) : rn or n is equal to one

(2) : (rn, n) = (2,2); (2,3) or (3,2)

Proof : Consider a point ~ E Vm, n corresponding to a sum of n distinct one-dlmensional irreducible representations. By theorem II.2.1, the central slice is a polynomial ring in rn.n variables over the invariant ring ¢[N2/Tn] described in II.2.1. By theorem III.l.1 ~ the noncommutative slice is also a polynomial ring in m.n variables over the ring r , , , n ( ~ ) described in III.l.1 . From this description it is clear that every indecomposable

172

graded projective left module over the noncommutative slice is of the form n

P, = •

¢[S~/T,; xj,}

¢lN=/r,i

@ ¢lN~/r,I

j'=i

Further, by symmetry it is clear that the Poincard series of all ¢[N21Tr; Xii], i ~ ], are equal. So, all indecomposable graded projectives have the same Poincard series and we have to verify when

~(p~;t) 1

- (1 - t) m" .p(¢[N2/T,~]; t) + (n - 1)p(¢[N2/T,; )/12]; t)] has a rational expression of the form (f(t)) -1 where f(t) e ~ t ] . It follows from the description of ¢[N21Tn] and from the fact that ¢[N21T,; X12] is a finitely generated module over ¢[N21T,,] that their Poincar~ series have the rational form

p(t)

~(¢[N2/T,];t) = (1 - t ) " ' . . . ( 1 - t'~) ~-

q(t)

~(¢[~/T,; for some p(t), q(t)

e~t]

x,~]; t) = (1 - t)-,...(1 - t - ) ~

and a~ e IN. Therefore, f(t) has to be a product of irreducible factors of

1-t iwherel_
f{t} ----(1 - t ) a H ( irreducible factors ~ 1 - t)

In any case, fCt) -I satisfies the functional equation 1 (1): 7¢(~) =

(--1)dtdef(1)f(t)

On the other hand, we have seen in the proof of theorem III.2.3 that the Poincar~ series of ~[Nx/T,~] and qJ[N2/Tr,; X12] @ ¢[N~,/Tr,] satisfy the functional equation

(2).~(-;~) = (-l)dt(m-')-'+-~(-;t)

173

Therefore, if the noncommutative slice has finite global dimension, or if ~ has finite projective dimension, we get from a combination of (1) and (2) that

&g(f) = ( m - 1)n 2 + n whence f(t) = (1 - t)a(1 + alt + a2t 2 + ... + a,~_lt"-l). This entails that the first terms in the power series expansion of f(t) -1 are (3): 1 + ( d - a l ) t + ( d ( d ; 1)

dai + a ~ - a2)t 2 + ...

On the other hand, it follows from the description of q~[N2/Tn] and q;[N2/T,; X12] that

~(¢[N21T,]; t) = 1 + (m - 1) ~ t

2 + ...

~(¢[N2/T,; XI~]; t) = ( m - 1)t + (n - 2 ) ( m - 1)2t2 + ... Therefore,$v(Pi; t) is equal to 1 ( 4 ) : -(1' - ---- ~t)1

+ (n -- 1)(m -- 1)t + (n - 1)(rn - 1)( 2 + (m -- 1)(n - 2))t 2 +

Comparing the coefficient of t in (3) and (4) gives the equation ai = ( n - - 1 ) [ ( n - 1 ) ( r n - 1) -- 11 But we know that ai has to be smaller or equal to the n u m b e r of irreducible factors in the remaining part of f(t).Therefore, al _< n - 1 giving the inequality (n-1)(rn-1)-l_ 1 are (m, n) = (2, 2), (2, 3) or {3, 2) So, in all other cases ~ is of infinite projective dimension over the noncommutative slice R, and so also R/R+. In view of the slice theorem, we have to verify that

~[v,~,,,I

174

has infinite global dimension. Now, assume R[~] has global dimension d . If k > d and M is an R-module we have

k i = ExtR(R/R+, M)[T] = Ext~(R/R+, M) since ](0) ¢ o and

R/R+ is annihilated by R+.So, we obtain a contradiction.

In view of [LV,Prop 1,Prop 2,Prop 7] this finishes the proof.

We will conclude this paper with an example. Consider the trace ring of 3 generic 2 by 2 matrices and let ~ • V3,2 of representation type (1, 1; 1, 1). Then, ¢[N~.] = ¢[vl, v2, v3, u4] and ¢[N2/T2] is the subring generated by the monomials

vivi, i ~ j. Hence, ¢[N~/T2] = ¢tx, y, z, t]/(xy - zt)

and therefore l+t ~ ~(¢[N2/T21; t) = (1 - t2)3

Let P be the height one prime ideal of ~[N2/T21 generated by (x, z), then

r3,~(~) ~ (¢[~}T2] Since

P

¢[N2/T2] )

vl.+[N2/T2; Xt~] - P and the quotient ~[N2/T2]/P is the polynomial ring in z and t we

obtain 1 + t2

t ~ ( ¢ [ N 2 / r ~ ; x ~ ] ; t) :

(I - t2) 3

1 2t 2 (I - t~) ~" = ( i - t2) s

and 1

~,(P~; t) - (1 - t)e ~,(¢[N~/~];

t) + ~,(¢[N~/T2; X~2]; t)

1--1-2t ÷ t 2

1

(I - t)~(1 - t~)~

(i - t)s(1 - t~)

Since ~3,2 has finiteglobal dimension, r3,2(~) is regular too, giving an example of a reflexive Azumaya algebra of global dimension three having an height one prime ideal (r3,2(~)(x,z))** which is not generated by a normalizing element since (x, z) is the generator of CI(~[N2/T2]) ~-Z

175

REFERENCES

:

[Ar] : A r t i n M. ; O n Azumaya algebras and finite dimensional representations of rings; J.Alg

i i (1969),pp 532-563 [AS] : Artin M.,Schelter W.; Integral ring homomorphisms; Adv. Math 39 (1981),pp 289-329 [Hb] : Hoobler R. ; When is

Br(X) =

Br'(X)? ; Springer LNM 917 (1982),pp 231-244

[Ho] : Hochster M. ; Rings of invariants of tori,Cohen-Macaulay rings generated by monomials and polytopes; Ann. Math 96 (1972),pp 318-337 [HR] : llochster M.~Roberts J.; Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay; Adv. Math 13 (1974),pp 115-175 [Le] : Le Bruyn L. ; Trace rings of generic 2 by 2 matrices; Memoirs AMS (to appear) [Lu] : Luna D.; Slices ~tales; Bull Soc Math France M~m 33 (1973),pp 81-I05 [LV] : Le Bruyn L.,Van den Bergh M. ; Regularity of trace rings of generic matrices; J.Alg (to appear) [Mo] : Morrison K.; The scheme of finite dimensional representations of an algebra; Pac. J Math 91 (1980),pp 199-218 [Mu] : Mumford D.; Geometric invariant theory; Springer (1965) [Ma] : Matsuchima y.; Espaces homog~nes de Stein des groupes de Lie complexes; Nagoya Math J 16 (1960),pp 205-218 [Pr] : Procesi C.; Finite dimensional representations of algebras; Israel J Math 19 (1974),pp

169-182 [Pr2] : Procesi C.; Invariant theory of n by n matrices; Adv. Math 19 (1976),pp 306-381 [Pr3] : Procesi C.; Rings with polynomial identities; Marcel Dekker (1973) [Sc] : Schwartz G. ; Lifting smooth homotopies of orbit spaces; Publ IHES [SS] : Small L.,Stafford J.T. ; Homological properties of generic matrices; Israel J Math (1985) [St] : Stanley R.; Linear diophantine equations and local cohomology; Inv. Math 68 (1982),pp 175-193 [St2] : Stanley R.; Combinatorics and commutative algebra; Birkh~user PM 41 (1983)

Fourier transforms on a semisimple Lie algebra over Fq

George Lusztig M.I.T., Cambridge MA 02139 USA and IX Universit~ degli studi di Roma

To Tonny Springer on his 60 th birthday

I.

Let g be the Lie algebra of a reductive connected algebraic group G over k,

an algebraic closure of the finite prime field F . In this paper we shall assume P that p is large. We assume chosen a non-singular G-invariant symmetric bilinear form < , > : g x g + k. If we are given (a)

an F -rational structure on G (hence on g) with Frobenius map F, such that q < , > is defined over F , q we can define the Fourier transform of a function f : gF ÷ ~f to be the function (b)

f : gF ÷ Q l

'~(~) =

~

~'E~

P <~,~'> f(~'), where ~: F

~ Ql is a fixed q

non-trivial character. (c)

Let N be the variety of nilpotent elements in g. The purpose of this paper is to describe those G F- invariant functions f:gF÷ Q1 such that both f and f vanish on gF _ N F" ^

It turns out that there are very few such functions, other than O. They are very closely related to the cuspidal character sheaves of [L2]. For example, if g = SP2n(k) (resp. S On(k)), there is ( ~ i t°+1)a scalar) at most.one function f # 0 as above; it exists if and only if

n

2

(resp. n = i 2)

for some integer i > O, and in that case it is supported by the nilpotent elements with Jordan blocks of sizes 2,4,6,...,2i

(resp. 1,3,5,...,2i-I)

in the standard

representation of G. The study of the Fourier transform of GF-invariant functions on gF has been initiated in Springer's work iS] in connection with the geometry of nilpotent orbits. He obtained very interesting applications to the theory of Green functions of reductive groups over F and the representation theory of Weyl groups. (Earlier, q Harish-Chandra has discovered the connection of Fourier transforms on real and p-adic Lie algebras with the character theory of Lie groups.) This has been further pursued by Kazhdan [Kz]. The theory of D-modules and perverse sheaves [BBD] has provided some new tools for the study of Fourier transform, see [B], [HK], [KLa]; in this paper we shall

178

make use of this theory as well as of the results in [L 2] on character sheaves. Here are some of the notations used in this paper. We shall denote by M(X) the abelian category of l-adic perverse sheaves on an algebraic variety X over k; we assume that £

is a fixed prime # p. If G acts algebraically on X we have the concept

of G-equivariant perverse sheaves on X see [L2, 1.9]; these form a full subcategory MG(X) of M(X). In particular, MG(g) is defined in terms of the adjoint action (g,~) + Ad(g)~ of G on g. 2.

In [LI] , [L2] we have studied a class of irreducible perverse sheaves on G

called admissible complexes. We wish to define an analogous concept for g instead of G. We first define the process of induction. Let: (a) P be a parabolic subgroup of G with Levi subgroup L and unipotent radical U; let p, I , u

be the corresponding Lie algebras; let p:p+l

be the canonical

projection. Consider the diagram 1~

VI

~

V 2,

.>g

where V I = {(~,h) E g x G I Ad(h-1)~ E P } V 2 = {(~,hP) £ g x G/P I Ad(h-1)~ £ P} ~" (~,hP) = ~, ~'(~,h) = (~,hP), ~(~,h) = p(Ad(h-1)~). Let A be an object of ML(/). There is a well defined perverse sheaf A I on V 2 such that ~ A ~ ~' A I. (Here ~, ~' denote inverse images with a shift, as in [L2, (1.7.4)]. We define .G

zL A = H i' At. This is a complex of sheaves on g ; it is said to be obtained from A by induction. Let K E MG(g) be irreducible. If G is semisimple, we say that K is cuspidal if its support is a closure of a single nilpotent orbit in gand if for any P ~ G as in (a) we have p!(KIp) = 0 as a complex of sheaves on I (notation of (a)). We now drop the assumption that g is semisimple and write 0 (b) g = z @ g ' where z is the Lie algebra of Z G (= connected centre of G) and g' is the Lie algebra of G/Z~; we say that K is cuspidal if it is of form K! ~ K 2 where K 2 E M

0 ~')

is cuspidal in the sense of the previous definition and

GtZC K I E M (~)is (up to shift) a local system of the form h k-

l"inear

form

" is

• defzned

E~

and E, the local system on k by •~ _. v by ~0' a fixed imbedding Fp c-+ Ql"

where h:z ÷ k is a 0

the F -covering P

x ~ - x = y of k and

An object A E M G ~ )

is said to be admissible

if it is irreducible and if there

exists P, L, p, 1 as in (a)and a cuspidal K E ML(/) such that A is a direct summand

179

G of iLK. In particular,

a cuspidal perverse sheaf on g is admissible.

3.

Here are some properties of admissible objects in MG(g).

(a)

G If P, L,p , l are as in 2(a) and A 0 C ML(/) is admissible then iLA 0 is a

direct sum of finitely many admissible objects in MG(g);

if in addition, we have

P ~ G, then any direct summand K of l.G L A0 satisfies supp K ~ z + N (see 2(b)) and hence is not cuspidal. (b)

If A E MG(g) is admissible then AIN extended by zero on g- N (shifted by

codim

N) is a semisimple object of MG(g).

(c)

Induction of admissible perverse sheaves is transitive.

(d)

Let A E MG(g) be irreducible. Write g = z @ g' as in 2(b). Then A is ad-

missible if and only if it is of form h*E~0 A' f M

(e)

m A' where h*E~o is as in no. 2 and

o(g ') is admissible. G/Z G If G is semisimple,

there is at most one cuspidal object in MG(g) on which

the centre of G acts by a prescribed character. (f)

Assume that G is semisimple and that K E MG(g) is irreducible with support

the closure of a single nilpotent orbit C. Then there exist P, L , p , l as in 2(a) and a cuspidal object A 0 E ML(/) such that extended by zero on g - N (shifted (g)

K is a direct summand of i~(A0) IN

by codim N).

Let G, K; C be as in (f); assume that K is cuspidal. Then the restriction of

K to ~ - C is zero. (h)

Let G, K, C be as in (g) and let A C MG(g) be admissible,

non-cuspidal.

Let

L be the irreducible local system on C such that KIC is L (up to shift). Then no homology sheaf of A restricted to C contains L as a direct summand. We now make some comments on the proofs of (a) - (h). Let logarithm map as in [BR]. From the de£initions, log

log: G ÷ g

it follows that for G semisimp!e-

defines a bijection between the set of cuspidal objects in M G ~ )

of "strongly cuspidal" perverse sheaves

[L2, II(7.1.5)]

and the set

on G whose support is the

closure of a single unipotent class of G. By [L2, I 6.9(b), V(23.](b))] dition "strongly cuspidal" above is equivalent

be a

to "euspidal"

the con-

[L2, II(7.1.1)]

and

to "cuspidal character sheaf" [L2, I 2.10, I 3.10]. Hence the classification of cuspidal objects in MG(g)

(for G semisimple)

is the same as the classification of

cuspidal character sheaves on G with support in the unipotent variety of G. Hence (e) follows from [L]], [L2]. Similarly (g) follows from [L2, V 23.1(a)]. Similarly, using the definitions, we see that the restrictions of G to the unipotent variety of G correspond under log of admissible objects in MG(g). Therefore

to the restrictions

to N

(b), (f), (h) follow from analogous pro-

perties of character sheaves on G, see [L], (6.6.1)], Properties

of character sheaves

[LI, 6.5], [L2, III(14.3)].

(a), (c) are proved in an entirely similar way as the corresponding

properties of character sheaves on G, see [L2, I 4.4(b), (d) follows from definitions.

(4.3.2), 4.2]. Property

180

4.

Let V be a finite dimensional k-vector space with a given non-singular

bilinear form

V × V ÷ k. Deligne has defined the Fourier transform FK of a perverse

sheaf K 6 M(V); then FK C M(V). The definition is in terms of a fixed embedding Po: Fp + Ql" We refer to [B] and [KLa] for the precise definition and properties of F. We shall use this construction for g

and < ~ > . It is known that F is

additive and (a)

F FK m j K, where j:g ÷ g

is defined by j ~ = -~. It follows that F takes

irreducible (resp. semisimple) objects in objects in

M(g).

M(g)

to irreducible (resp. semisimple)

When considering the transformation F on a subalgebra I

as in 2(a), we shall take it with respect to the restriction of < , >

of g

to £.

Note also that F takes an object of MG(g) to an object of MG(g). An irreducible object of MG(g)

is said to be orbital

if its support is the closure

of a single G-orbit in g. An irreducible object of M G ( ~

is said to be anti-orbital

if it is of the form FK where K 6 MG(g) is orbital. 5.

Theorem. Let A 6 MG(g) be irreducible.

(a) A is admissible if and only if it is anti-orbital. (b) I f G is semisimple and A is cuspidal, then FA m A. Thus A is both orbital and anti-orbital. The proof will be given in no. 9. 6.

Assume given a n F -structure on G, g as in 1(a). If K is a perverse sheaf on g , q , such that F K m K, we choose an isomorphism ~:F K $ K and we define the characteF ristic function XK,~: g ÷Q1 by XK, ~ (~)=Zi (-1)irr(~' H~i K) where H i K are the cohomology sheaves of K and the subscript ~ denotes the stalk at ~. We shall use several times the following principle. If K, K' are two semisimple perverse sheaves on g, in order to prove that K m K', it is enough to check that one can choose an F -structure as above and ~:F*K $ K, ,

*

:F K' ÷ K' such that X i = X, i :g K, ~ K',~9'

i:

Fi

q

+ Q£

for i = 1,2,3,...; here

(Fi)*K $ K is defined by iterating ~ and ~,l is defined similarly. The Fourier

transform f of a function f:gF + ~ £

is defined by 1(b) where ~:Fq÷ Q£

is

~ T r F q / F p . If f = Xk,~ (as above) then f = XFK,~, for a suitable ~':F * FK ~ FK.

7.

Let K be an orbital object in MG(g) with support ~ where C is the G-orbit of + ~ 6

g (o semisimple, ~ nilpotent,

[o,~]= 0). Let L be the centralizer of o in G,

P a parabolic subgroup of G with Levi subgroup L and let

U,l,p,u

be as in 2(a).

Let Ko be the orbital object in ML(/) whose support is the closure of the L-orbit Co of ~ + ~ and is such that KoI~ o is (up to shift) the same as KI~ o. Assuming that 5(a) holds for L, we shall prove that (a)

FK m i~(FKo). L

181

Note that FKo is anti-orbital hence by our assumption, it is admissible in ML(/). Hence i~(FKo) is a semisimple perverse sheaf on g, see 3(a). Since FK is a semisimple (in fact irreducible) perverse sheaf on g, to prove (a) it is enough to prove the equality of the corresponding characteristic functions (see no.6) for larger and larger F . Choose an F -rational structure on G (hence on g) with Frobenius map q q F such that P , L , o , ~ < , > are defined over F and such that there exists ~: F K S K . ,

Let fo = XKo ' ~o where

~

: F Ko $ Ko is defined by ~ and let f = XK ' .

ffo(Ad(g)~),

if Ad(g)~s s = ~

We have

for some g 6 G F

f(~) ,

otherwise

here ~ss is the semisimple part of ~. It is enough to show that

(b)

F

7 (~) =juFI ]pFI-I

~ fo(p(Ad(g)~)) g ff G F Ad(g)~ 6 p

5 6 g

.

We have f(~) = ILFI-I

~

g E / FGF r/6 rl n i l p .

~ <~, A d ( g - 1 ) ( O + q ) >

fo(O+T]).

Fix a coset uFgo, go 6 G F, and let g run only over this coset. Note that u -~ Ad(u -|) ( o + q ) - ( ~ + n )

is a bijection UF + u F, hence this part of the sum is

F ~ <~' A d ( g ° - 1 ) ( ° + T ] + ~ ) >

fo(O+ q)

~Eu I ~Eu 0,

F

~
N>%~ f o ( O

+ n )

unless Ad(go)~ 6 p .

Hence

f(~)

= ILFI -I

~
O + q>fo(O

+ n)

g 6 GF Ad(g)~ 6 p 6 1F nilp. and (b) (hence (a)) follows. 8.

Let L, P, U, l, p, u

A = h ~o

W

A]

be as in 2(a) and let A 6 ML(/) be cnspidal. Write

as in 3(d) for 1

instead of g. Here h is a linear form on the

centre z I of i. Let ~ 6 z I be defined by h(~) = - < o , ~ > Assume that o is in the centre of

g.

for all ~ 6 z I.

182

Let N o = o + N. Assuming also that 5(b) holds for L/7°, we shall prove that (a)

F(iGA) miGAINo

extended by 0 on g - Nj, shifted by codim N d.

(The special case where L is a maximal torus and

A = Q£ appears in [B].)

We can easily reduce the general case to the case o = 0. Assume now that o = O. Using 3(a),(b) we see that it is enough to prove instead of (a) the equality of the corresponding characteristic functions (see no.6) for larger and larger F . q Choose an F -rational structure on G (hence on g) with Frobenius map F such that q , P, L, < , > are defined over F and such that there exists q0 = F A J~ A. Let f° = XA,q0: 1 F ÷ Q1 " From 5(b) q for I~/ZL° it follows that (b)

~°(~°) =

I~

f°(~°)'

nifi l ~° p ° tCe n IF t o t his erwise

where X is a constant. It is enough to prove that IPF[-I

~ F ~ <¢' ~'> Z GF fo(P(Ad(g)~')) ~'Cg gC Ad(g)~' C p yluFI

IpFI-I

Z g C GF

fo(0(Ad(g)~)) , if ~ C g

~0

F

,

is nilpotent

Ad(g)~ C p , otherwise

; this follows easily from (b). 9.

Proof of Theorem 5. The theorem is obvious when G = {e}. We can assume that

G ¢ {e} and that the theorem is already known for all Levi subgroups of proper parabolic subgroups of G. Moreover, using 3(d) we can assume that G is semisimple. We first show: (a)

I f A = FK where

K E ~G(g) is orbital and non-cuspidal,

then A is admissible,

non-cuspidal. Let d + ~) 6 C, L, P, U, l, p, u be associated with K as in no.7.

If

P # G, by the induction hypothesis and by 7(a) we have A = FK m iG(FK o) where K

C ML(/) is orbital; moreover, by the induction hypothesis, FK O

is admissible hence O

iG(FKo ) is a direct sum of admissible non-cuspidal objects in MG(g) , see 3(a). Since iL(FK o) is isomorphic to A, it follows that A is admissible, non-cuspidal. Therefore we may assume that L = P = G so that o is in the centre of g . Since G is semisimple, it follows that (7 = O so that the support of K is the closure of a single nilpotent class. Attach P, L, p, l, Ao to K as in 3(f). (These P, L, p, l, are not the ones considered above). Since K is not cuspidal, we have P # G. From 3(f) and 8(a) (which is applicable by the induction hypothesis) we see that K is a direct summand G .* .G of F(i G Ao) hence A = FK is a direct summand of FF(i A o) = j (ZLAo) , see 4(a). Then j*A is a direct summand of iGA ° hence it is admissible, non-cuspidal,

see 3(a).

183

But one checks easily that j

permutes the admissible non-euspidal objects in M G ~ ) ,

so that A is admissible non-cuspidal, as asserted. Next we show: (b)

If A E MG~)

Let L, P,

1,p

is admissible, non-cuspidal,

be as in 2(a) and let A' C ML(/)

then A is anti-orbital. be a cuspidal object such that A

is a direct summand of i~A'. Since A is not cuspidal we have P # G. By the induction hypothesis, we have A' ~ FK where K E ML(/) is orbital. Let ~ + ~ E C c

1

be

attached to K as in no. 7 (for/ instead of g ). Let L' be the centralizer of ~ in G. Then L' D L since ~ is central in 1.

Let P' be a parabolic subgroup of G with Levi

subgroup L' such that p' ~ p. By 8(a), F(i~'A')- is a direct sum of orbital complexes on l' with support contained ,e v

in

{6 E /'l~ss = 0}. Hence lL A' ~ ~ FA s {5 E /'I~ss = -0}. We have i~,(FA a ) ~

support contained in {~ E

where_ A s f ML,(/')_ are orbital, supp A s = F A a where A s

E MG(g) is orbital with

g1~ss = -0}. (See 7(a); this is applicable by the in-

duction hypothesis if L' # G and is trivial if L' = G.) Hence, using 3(c), we have •G A , ~ L' iL = i , (iL A') = ~ i ,(FA s) = ~ F As" Since A is a direct s u ~ a n d of i~ A', we must have A m F ~ a for some g, so that A is anti-orbital, as asserted. It remains to show that FK ~ K for any cuspidal K E MG(g). We now fix an F -rational structure on G (hence on g) with Frobenius map F such q , that < ' > is defined over F q and F K ~ K for all cuspidal K C MG(g). (This is possible since there are only finitely many cuspidal K). Let I be the set of pairs (C, L)where C is an F-stable G-orbit in g and L is a G-equivariant irreducible local system on C (up to isomorphism) such that F L is isomorphic to L.

g

F

For each i = (C, [) E I we choose a definite -- * + Q£ by ~ T r ( ~ , L~), if ~ E C F fi (~)

~: F L $ L

and define f.: l

= , if

~ C

gF

_

C F.

It is easy to see (cf. for example [L2, V(24.2.7)]) that fi(i E I) form a basis for F F the vector space V of all G -invariant functions g + Q£. Now let K i E MG(g) , (i = (C,E) E I), be the orbital perverse object such that K]C = E up

supp K, = C i

and

to shift.

Then F •K. $ K. ; we choose a definite isomorphism and we let f'. be the characi l i teristic function of K. with respect to this isomorphism. i

i

Then f*l can be expressed in terms of the fi by means of a triangular matrix with non-zero elements on the diagonal. In particular f'. (i E I) form a basis of V. I

Let I0 b e the set of those i E I for which K i is cuspidal (All euspidal K E MG(g) are among the Ki, i C I). Let ( , ) be the non-singular bilinear form on V defined by (f' f') =

% F f(~)f'(~)" Cg

184

We shall need the following properties. (c)

fh =

chfh

(Ch = constant) for h C I0.

(This follows from 3(g)) (d)

If i C I - I0 and

h E I0 then (fi,fh) = 0

(This follows from [L2, V 24.4(d)] if

i = (C,E) with C nilpotent and is trivial,

otherwise ) (e)

If i E I - I 0 and h C I0 then (f'i' fh ) = 0.

(Indeed, from [L2, V 24.4(d)] it follows that f'iiN F is a linear combination of ftlN F where t E I - I0, and it remains to use (d).) We now choose for each i E I

an isomorphism

F FK i $ FK i and denote by f". i

the corresponding characteristic function. We have ^ (f) f"'l = d.f~l l ' (di = constant), i E I. (g)

If i E I - I0 and h E I0 then .(f"i, fh ) = O.

(Indeed from (a) and 3(h) it follows that f". INF is a linear combination of l f INF where t E I - Io, and it remains to use (d).) t A

^

From (g) it follows that (f~, fh ) = 0

for

i C I=- 10S h C I0. Since f ÷ f

preserves ( , ) up to a scalar, it follows that (f~oj, fh ) = 0 {f~° j},

for

i E I - I0, h E I0.

(f[,

fh ) = 0

hence

When i runs through I - I0 the functions

{f~} coincide up to order and up to scalars. Hence (f!, fh ) = 0 for i i

i E I - I0, h C I0. By computing dimensions, we deduce that

{fhlh E I0} span the

subspace of V orthogonal to all f~, (i E I - I0). Similarly from (e) we see that {fhih E I0} span the subspace of V orthogonal to all f!l (i E I - I0)° It follows that

{fhlh E I0} span the same subspace as {fhlh E I0}. In particular each

fh(h C I 0) is zero on zero on

gF _ N F

Using (f) and (c) it follows that f"h(h E lo)iS

g F - N F" Letting now F

become bigger and bigger, we deduce that q is zero, or in other words, supp FKh c N for h E I0. Since FK h is

FKh i g - N

G-equivariant, irreducible, and N consists of finitely many G-orbits, it follows that

FKh is orbital, (h C I0). Assume that FK h is non-euspidal, h E I0. Applying (a) to K =

F FKh = j

Kh

FKh we see that

is non-cuspidal hence K h is non-cuspidal, a contradiction. Thus

FK is cuspidal whenever K is cuspidal. Note that F preserves the character by which the centre of G acts on an irreducible object we see that FK m K theorem.

of MG(g) ; hence using 3(e),

whenever K is cuspidal. This completes the proof of the

185

10.

In the following results we assume that we are given an F

on G (hence on g) with Frobenius map F such that < , >

rational structure q is defined over Fq; we

assume also that G is semisimple. * Corollary. Let K E M~(g) be cuspidal. Assume that there exists ~ : F K C~ K ~F -^ and^let f = XK,q~: g * Ql" Then f = xf where ¥ is a constant. In particular both f, f are concentrated on nilpotent elements. his follows immediately from 5(b). The corollary reminds us of a recent result of Kawanaka [Kw, (3.3.9)] stating that a certain function on a prehomogeneous vector space associated with the exceptional group G2, F 4 or E 8 I I.

is essentially invariant under Fourier transform.

We now state the following result which is a complement to Cor. 10. (The

setup is that in no.|0). Theorem.

. Let f : g F -~ Qo be a G F -invariant function such that

flg F- N F ~ O, flg F - N F =- O. Then there exist and isomorphisms

q0i : F K i $ K i

functions

such that

f

K i C MG(g) , cuspidal, is a

(I < i ~< m)

Ql -linear combination of the

F XK i, qOi: g

÷ Ql' (I ~< i ~< m).

Proof. Let I be the set of all pairs (C, L) where C is a nilpotent orbit in g and L is a G-equivariant irreducible local system on C (up to isomorphism). For each i E I we denote by A i

the irreducible object in MG(g) with support

such that AiIC is L up to a shift; let A~ E MG(g) be defined by F A~ = A.. i

l

i

Let J be the set of all pairs (L, Ko) up to G-conjugacy where L is the Levi subgroup of a parabolic subgroup of G, 1

is its Lie algebra and Ko E ML(/)

cuspidal. For each j E J, j = (L, Ko) we denote extended by zero on g -

is

K~ = iG (Ko) and K. = K~

N, shifted by eodim N. There is a unique map T : I -~ J

with the following property: for any j, K. is a direct sum of irreducible objects ] of MG(g) , isomorphic to some Ai, i C T-I(j); each A i (i C T-~(j)) is isomorphic to a direct summand of K.. 3 According to 5(b) and 8(a) we have F(K;) = Kj, (j C J). Hence K! i s a dire.ct -I J sum of irreducible objects of Mg(g) , isomorphic to some A;, i E r (j); each A!; (i E r-1(j)) is isomorphic to a direct summand of K~. ] On the set I we have a natural action of the Frobius map F. If i = (C, E) E I F then F A.l ~ A.I and F A!l ~ A'l ; we choose isomorphisms ~Oi: F A i * Ai, , ' qOi: F A lt. +^A;. Then f is a linear combination of the functions

(since } = 0 on gF _ N F) XA i ,q0i

Hence f is a linear combination f =

Let h ÷ h

Z i E IF

ci

XA; ' ~i

(c i E ~ l ) .

be an automorphism of Ql which corresponds to complex conjugation under

some isomorphism Q l ~

~ .

186

We shall need the following fact. (a)

If A', A" 6 MG(g) are irreducible direct summands of K',, K',, respectively . J J (j', j" 6 J) and q)': F A' -~~A', q)": F*A" ÷~ A!' are isomorphisms then

~ 6 g F XA',~'($)XA", Q0''(~) = qm(j') ~6~ N FXA'" q),(~) XA", ~0,,($) and both sides are zero unless j' = j"; here m:J + IN is defined by m(L,Ko) = dim ~ L. This can be proved by expressing XA,, q), (~), XA,,' q),,(~) in terms of "generalized Green functions" as in [L2, II 8.5, (10.4.5)] and then using the orthogonality relations [L2, II 9.11, V (25.6.2)]. The formula (a) is applicable to (A', ~0') = (A.~, ~0i), (A", k0") = (~, q)h), i, h 6 I F . Hence we have (b) $6

gF f($)f(~) =

E iF CiEh E gF XA~, q)i(~)XA~, q)h(~) i,h6 $6

E IF Ci~h qm(~[(i)) E NF XA[ ' £01($~X~ ' % ( ~ ) = i,h 6 ~ 6 " I'(i) = ~(h) =

where f'(~)3

E j 6J

qm(j)

E N F fj (~)-fj(~) $6

= i £~ T-I(j): ci XA~ ' %°i ($)" F(i)

i

=

On the other hand, since f = 0 on gF _ N F, we have (c)

~ gF f(~)f(~) =

~6

Z N F f(~)f(~)

Se

Z

i,h 6 IF

i,h 6

c~7h

IF CiCh

~6z ~F ×Ab

~i(OXA~,%(0

$6

~(i) = T(h) E j 6 J

~6

NF

f: (~)f. (~). J J

Comparing (b) and (c) we obtain E (qm(j)_" I) E NF fj($)fj(~j" j 6 J $ 6 It ~ollows that

= 0

f INF is zero whenever m(j) > 0. Hence J

f(~) = i 6~ IF c i m(T(i)) = 0

XA[, ~0i(~)

($ 6 NF).

187

The condition m(~(i))

= 0 is equivalent

to the condition that A! is cuspidal. l

The

theorem follows. 12.

We now state a variant of the previous

theorem in which functions are

replaced by perverse sheaves. Theorem. Assume that G is semisimple. such that

supp K c N,

Let K E MG(g) be a semisimple

object

supp FK c N. Then K is a direct sum of cuspidal objects in

MG(g). Proof. We may assume that K is irreducible. shall find a contradiction.

Since

we see that FK is admissible, This contradicts

our assumption

Acknowledgement.

supp K c

non-cuspidal.

We assume that K is not cuspidal and we N, we see that K is orbital. From 3(a) we see that

From 9(a)

supp FK ~ N.

that supp FK c N, proving the proposition.

This research has been supported

in part by an N.S.F.

grant.

188

REFERENCES

[BR]

P. Bardsley,

R.W. Richardson:

groups in characteristic

[BBD]

A. Beilinson,

J. Bernstein,

J.L. Brylinski, de Lefschetz,

[HK]

R. Hotta, M. Kashiwara, Lie algebra,

[KLa] [Kw]

Inv. Math.

N. Katz, G. Laumon, N. Kawanaka,

transformation 295 - 317.

pervers,

Ast~risque

de France.

Transformations

transformation

Soc. 51(1985),

P. Deligne,Faisceaux

100 (1982), Soci~t~ Math~matique

[B]

Etale slices for algebraic

p, Proc. Lond. Math.

canoniques,

dualit~ projective,

th~orie

de Fourier et sommes trigonometriques, preprint The invariant holonomic

75(1984)

system on a semisimple

327 - 358.

to appear in Publ. Math.

I.H.E.S.

Generalized Gelfand - Graev representations

of exceptional

simple algebraic groups over a finite field, preprint.

[K~]

D. Kazhdan, Proof of Springer's hypothesis. Israel J. Math. 28(1977) 272 - 286.

[e I ]

G. Lusztig, Math.

[L 2 ]

Intersection

75(1984)~

G. Lusztlg, II Adv. in Math.

57(1985)

T.A. Springer, representations

Inv.

205 - 272. Character

IV Adv. in Math. 59(1986),

Is]

cohomology complexes on a reductive group,

sheaves,

I Adv.

in Math. 56(1985),

226 - 265, III Adv.

in Math.

193 - 237,

57(1985)

266 - 315,

I -63, V to appear in Adv. in Math.

Trigonometric

of Weyl groups,

sums, Green functions

Inv. Math.

36(1976),

of finite groups and

173 - 207.

COMMUTING AND

SPHERICAL

OPERATORS FUNCTIONS

I.G. M a c d o n a l d of M a t h e m a t i c a l Sciences Queen Mary College L o n d o n E1 4NS.

School

i.

DIFFERENTIAL

ZONAL

Introduction The

subject

of m y

talk

is a c l a s s

of p o l y n o m i a l

symmetric

functions

Jl(Xl, .... Xn;~) indexed

usefully)

~,

as a

which may positive

For p a r t i c u l a r (i)

when

when

of

they

SLn~R)/SO(n)

when

functions,

(iv)

when

SL

~ = 0

of the

Finally,

The follows.

involving or

occur

factor)

the c h a r a c t e r s

n

zonal

(more

the

"in n a t u r e " : Schur

of the p o l y n o m i a l

polynomials

These

SU(n)/SO(n)) (or

can

are

familiar

zonal

defined

to the

spherical

functions

by p o l y n o m i a l

SU(n))

again

be i n t e r p r e t e d

symmetric

we have

I'

e = ~,

and

space

as z o n a l

spherical

SLn[~)/Sp(n)

or its

SU(2n)/Sp(n)

jugate

function

sn,

polynomials

(up to a s c a l a r

the

on the

of e l e m e n t a r y

for

these

[4]).

dR)

they

time

product

(v)

are

(or on

~ : ½

form

~

i.e.

e.g.,

of

this

length

as an i n d e t e r m i n a t e

number.

of

are

of

regarded

GLn({)

(see,

representations

compact

they

~ = 2

statisticians

.(iii)

real

sl(xl,...,Xn),

representations

on

be

values

~ = i

functions

(ii)

I = (Ii,...,I n)

by p a r t i t i o n s

a parameter

partition

when

and

suitably

reduces

m l ( x I .... ,Xn) cases Let

Jl(Xl ..... Xn;0)

symmetric

(i)

-

G

be

functions

1

([I03,

normalized

= e l , ( x I .... ,Xn),

corresponding

Chapter

I)

Jl(Xl ..... Xn;~)

in t h i s = Z

(iii)

case to the m o n o m i a l x ~ l . . . x nIn

above may

a connected

be d e s c r i b e d

non-compact

the

to the c o n -

makes

sense

symmetric

uniformly

semisimple

as

Lie g r o u p ,

190

and

K

root

system

a maximal

denote

possibilities

are

= 1,2

of t y p e

= 2/m on

U/K,

Jack

where

U

[12].

We

shall use

[I03.

Let

acts on

An, ~

the

~

or

(type

G)

m

the o n l y

and

possibility EIV

on

arising

with

in C a r t a n ' s

J~(Xl,...,Xn;~)

functions

of

Let

n > 3

~,

is a n o t h e r

= 8

of

with

G/K from

(or a l s o finite-

G.

;~)

were

first

introduced

to his definition eigenfunctions introduced

~[Xl,...,Xn~

in

§3.

of

variables;

by p e r m u t i n g

of

[2] and

symmetric the the

by Henry Here we

of a f a m i l y

by Debiard

and t e r m i n o l o g y

be i n d e p e n d e n t

of

symmetric

(~l,...,~n)

In p a r t i c u l a r ,

functions

symmetric x i,

group

and we write

let

~ =

e(w)

is t h e

let

t

= A -I

=

sign

6 ~n

,

(n - l , n

let

in

xe

Xl,...,x n

.

denote

- 2,...,1,0);

x ~n l ._. . x then

an

the Vandermonde

e ( w ) x w~, n

of the p e r m u t a t i o n

n Z c ( w ) x w~ H w~S n i=l

D i = xi~/~x i .

=

~ weS

be a n i n d e t e r m i n a t e ,

is a l i n e a r d i f f e r e n t i a l

D(L,~)

polynomials

is

H (x i - x~) i<j 3

D(t;~)

where

(If

= ~ [ x I ..... X n ]Sn

determinant

Now

the n o t a t i o n

the r i n g

~ =

A =

back

operators

Xl,...,x n

subring

If

come

the restricted An_ 1 .

D (~) r

as s i m u l t a n e o u s

Sekiguchi

where

form

representations

J l ( x I .... ,x n

them

m

spherical

The polynomials

obtain

F = ~, there

and

operators

differential

for

as z o n a l

shall

root.

the polynomials

is a c o m p a c t

commuting

Sn

where

The d i f f e r e n t i a l

a n d we

such that is of t y p e

restricted

F4,

Then

G,

G/K

n = 3

of t y p e

polynomial

[7],

shall

K

If

[53).)

c a n be v i e w e d

dimensional

2.

of e a c h

4.

of

space

G = SLn(F) , or

E6,

classification

subgroup

symmetric

the m u l t i p l i c i t y

m = d~F G

compact

of t h e

n ~ trD (~) r=0 r

The

and

(i + t ( ~ D i +

coefficient

operator

which

of

w

.

let

(wd)i))

tr(0

we d e n o t e

,

~ r ~ n) by

D

r

in

D(t,@)

191

If

f c An

is h o m o g e n e o u s

D 0(e)f = f,

l-(~)f = u

D2(c~) =

where

C

Remark:

When

invariant

differential U/K).

(or

with

each We

Define

([2],

,

n,

d

and

and

~)

or

½

the

x

(cases algebra

operators

.

i - x , 3 ~x i

(i) -

(iii)

of r a d i a l

on the

in §i)

corresponding

D~~)-

(or

the

components

~ U + V)

of

symmetric

space

is e s s e n t i a l l y

operator.

[12]):

The

operators

r-(e) (0 _< r _< n) u

commute

no

the d e t a i l s .

other.

shall

sketch

a scalar

for the p r i m e scalar

on

S i~j

In p a r t i c u l a r

the L a p l a c e - B e l t r a m i

Proposition

V = •

generate

G/K

- l))f

we have

+ V))f

(depending

~ = I, 2

D r(~)

d,

2 xi

n 2 22 Z x i= 1 i ~x2 1

operators

(d~ + ½n(n

(C - e ( ~ U

is a c o n s t a n t

U = ½

of d e g r e e

a proof,

product


but we have v>'

in the n o t a t i o n

on

A

is t h a t we

space

n~ shall

for

all

as f o l l o w s later

(the r e a s o n

define

another

product):

'e

= I

u(t)v(t)IA(t)I

2/~ dt

Tn where

Tn

is the t o r u s

T n = {t =

and

dt

is the H a a r m e a s u r e ,

Alternatively, c

where

-I

the

constant formal

each

preserving; indexed

'

u ( t ) v ( t -I)

Then (a)

(t I ..... tn ) ~ ~n

is ~

c

: itil

= i,

normalized

the

constant

i _< i _< n}

so that term

'

= I.

in

(i - t i t j ) i / ~

is d e t e r m i n e d

computations

operator

D (~) r m o r e o v e r if

b y the p a r t i t i o n

by Dyson's

conjecture

([3~,

[Ii]).

show that

maps

An

m I = Zxl

and is d e g r e e n,~ is the m o n o m i a l s y m m e t r i c f u n c t i o n

1

Chapter

~R

([10],

into

A

I)

then

192

= Z a Dr ( e 1) m ~-
Z -< I is the usual d o m i n a n c e order on p a r t i t i o n s D r(~)

(b)

is s e l f - a d j o i n t for the scalar product

Now fix a p o s i t i v e integer g o n a l i z a t i o n to the basis the~ bottom (Jl)

(i.e.,~with

w i t h each

:

Z

Jl

~

k _< n

(ml)iii= k

m(ik)

).

(loc. cit.)

i

and apply G r a m - S c h m i d t o r t h o of the space

ikn '

starting at

We shall obtain an orthogonal basis

of the form

(~)m

(I)

W i t h respect to the basis (Jl), the m a t r i x of each d i f f e r e n t i a l (~) operator D is s i m u l t a n e o u s l y t r i a n g u l a r and hermitian, by virtue r (e) of (a) and (b) above, hence is diagonal. It follows that the D r commute w i t h each other and that the Jl are s i m u l t a n e o u s eigenfunctions of each

Remark: ~ Jl

D (~) r

In fact the L a p l a c e - B e l t r a m i

are u n i q u e l y d e t e r m i n e d

p o l y n c ~ i a l s of the form

3.

Jack's

operator

DJ e)

(up to scalar factors)

(i) that are e i g e n f u n c t i o n s

In order to avoid e x t r a n e o u s scalar factors,

partition

for

Di s)

symmetric functions

r e n o r m a l i z e the symmetric p o l y n o m i a l s

m u l t i p l e of

1

of Jl

k ~ n

we define

Jl

ek)

it is c o n v e n i e n t to

d e f i n e d in §2.

Jl(Xl,...,Xn;~)

in which the c o e f f i c i e n t of

e l e m e n t a r y symmetric f u n c t i o n

is

m(ik)

For a

to be the scalar (i.e. of the

= Jl (Xl ' " " " 'Xn;~)

(which w o u l d not be the case for the we have a well defined e l e m e n t = 4--lira An,JR Chapter I).

n + 1

we have

Jk (Xl .... 'XnO;~)

1

kth

k!

It is then not d i f f i c u l t to v e r i f y that, when we pass to variables,

the

is decisive:

as the symmetric

Jl) . Jl(x;~)

Hence for each p a r t i t i o n of the ring

of s y m m e t r i c functions with real c o e f f i c i e n t s ( j 1 0 ] , The

Jl(x;~)

is h o m o g e n o u s of degree

are Jack's symmetric functions;

J/(x;~)

II]

There is another and m o r e c o m b i n a t o r i a l way of d e f i n i n g these

193

s y m m e t r i c functions. the

rth

For each integer

pl = pllPl2 . . . . ~ - b a s i s of

~

The p r o d u c t s .

Pl

I, ~

Pr = Ex[

for all p a r t i t i o n s A,

1

be let

from an

for w h i c h the Schur

is such that for any two p a r t i -

we have

where

let

I = (11,1 2 .... )

The usual scalar product on

functions form an o r t h o n o r m a l basis, tions

r ~ i,

power sum, and for each p a r t i t i o n

~I~

= 61~zl' is the K r o n e c k e r delta,

and

z I is the order of the

c e n t r a l i z e r of a p e r m u t a t i o n of cycle-type

1

in the s y m m e t r i c g r o u p

Slxl We m o d i f y this scalar p r o d u c t as follows:

define

< P I ' P ~ > ~ = @IzzI ~£(~) where

~(I)

is the length of the p a r t i t i o n

nonzero parts

li).

I

(i.e. the number of

Then it can be shown that the

Jl(x;~)

p a i r w i s e o r t h o g o n a l with respect to this scalar product, w i t h respect to the scalar product = i,

the two scalar p r o d u c t s

In other words, orthogonalization the m o n o m i a l

the

'

defined

coincide;

Jl(x;~)

equal to

k!

(where

[7].

k =

m

Ill)

they don't.)

(x) ,



on

~R

from

the scalar factors being

adjusted so as to ensure that the c o e f f i c i e n t of

definition

(when

can be c o n s t r u c t e d by G r a m - S c h m i d t

relative to the scalar product

symmetric functions

as well as

in §2.

when ~ ~ i,

are

m(ik)

in

Jl is

This is e s s e n t i a l l y J a c k ' s o r i g i n a l

To show that this d e f i n i £ i o n agrees w i t h the

p r e v i o u s one it is e n o u g h to v e r i f y that the f u n c t i o n s of the L a p l a c e - B e l t r a m i o p e r a t o r

Jl

so d e f i n e d are e i g e n -

D~ ~)

w i t h the appropriate

eigenvalues. In g e n e r a l the formal p r o p e r t i e s of the m i m i c those of the Schur f u n c t i o n s

sl (x)

not to

sl (x)

but to

h(1)sl(x),

where

~

on the algebra

~R

Jl(x;~)

reduces

is the product of the

[10] that there is an

which m a y be defined by

(r ~ I)

This involution has the p r o p e r t y that where as before

h(1)

Recall

~ ( P r ) = (-l)r-I Pr

1 ,

~ = I,

I. )

One example is the following. involution

appear to

in a v e r y s a t i s f a c t o r y way.

(We should observe at this p o i n t t h a t when

h o o k lengths of the diagram of

Jl(x;~)

I'

e(s I) = sl,

is the conjugate of

I.

for any p a r t i t i o n This p r o p e r t y now

194

~eneralizes

as f o l l o w s :

~6(Pr ) =

~ 6 ( J i (x;6))

6

Suppose define

1

£(s)

s

The u p p e r

s

lies in the

the

= lj'

T h i s can be p r o v e d

-

ith r o w and

a(s)

= £(s)

hl.(s) = £(s)

£'

£(s)

(s)

=

i

-

(i + a ( s ) ) 6

+ 1 + a(s)~

h~ (s) ,

are d u e

(i)

+ I.

~ h~(s), s~l ~ results,

to R. S t a n l e y

<Jx ,Jl>

We

I. s

by

[14]:

= h* (1)h. (I)

i' (s)

of

s

by

h, (s)

are then d e f i n e d

by

.

+ a(s)

The f o l l o w i n g

of of

I.

,

hl(s)

=

column a'(s)

of

1

b o t h of t h e s e are e q u a l

h*(1)

jth

and l e g - c o l e n g t h

W h e n 6 = i, = £(s)

with

D~ 6)

a' (s) = j - 1 ,

,

+

in the d i a g r a m

and a r m - c o l e n @ t h

leg-length

i

a square

and l o w e r h o o k - l e n g t h s

h~(s)

a g a i n by v e r i f y -

of the o p e r a t o r

and some c o n j e c t u r e s

= I i - j,

and l i k e w i s e

by

eigenvalues.

the a r m - l e n g t h

a(s)

I.

are e i g e n f u n c t i o n s

be a p a r t i t i o n ,

that

~R

= 61 llJl, (x;6-1)

Some theorems Let

of

theorem

~6(Jl)

the a p p r o p r i a t e

w6

(r a i)

and a n y p a r t i t i o n

ing t h a t the

4.

an a u t o m o r p h i s m

(-l)r-i ~ P r

we have the d u a l i t y

f o r any

if we d e f i n e

to the hook

Finally

define

h.(1)

~ h~(s) sel

which

=

confirm earlier

length

conjectures

of m i n e ,

195

(2)

The

(3)

Let

coefficient X

~x(Pr ) = X

X

value

of

mI

for all

r a i.

=

H s~l

is s p e c i a l i z e d

it is e q u a l n!en(

where

and d e f i n e

to

h,(1)

~X

: ~

÷ ~[X]

by

Then

at has

c o e f f i c i e n t of

Jl

is a h o r i z o n t a l

is e q u a l

to a p o s i t i v e

Jl(Xl,...,Xn;~)

The

- ~

Jl

(X + a' (s)~ - £'(s))

In a d d i t i o n , S t a n l e y (4)

in

be an i n d e t e r m i n a t e

~X(Jl) (When

of

xI = proved

in

strip

.

integer

~X(JI)

is the

.°. = x n = i.) a Pieri

formula

for the

is z e r o u n l e s s

J~J(n)

([103

n,

Chapter

i) of

Jl:

I ~ ~

length

n,

and and

then

to

h (s))( ~ hl(s)) -I s¢~ ~ s~l

(for

~ = I

or

f = the(s)

ha(s)

~)

if

I - ~

as [h~(s)

From

this

in the

same

column

sr

otherwise.

result

Hall-Littlewood

a square

contains

it f o l l o w s

functions)

(as in the

that

Jl(x;~)

strict)

tableaux

case

of

Schur

can be w r i t t e n

functions

or

explicitly

as

a sam of m o n o m i a l s :

(5)

Jl(x;~)

summed

over

x T = x~

= Z WT(~)x T all

where

(column ~

is the w e i g h t

[10])

and

WT(~)

which

both

numerator

namely

upper

defined

and

T

of

is an e x p l i c i t l y and

lower

b y the t a b l e a u

given

denominator

hook-lengths T.

T

of

rational

for

It s h o u l d

shall

conclude

this

these

are d u e

to R.

Stanley,

First

of

tables

of the

suggest

all, that:

section others Jl,

I.

Here

the

some

WT(~),

have

~,

in

factors,

in g e n e r a l ,

such

that

WT,(~)-

conjectures.

as t h e y

of

partitions

that,

and

of

linear

(i.e.,

to K. K a d e l l

as far

of

intermediate

be r e m a r k e d

with

is that

function

are p r o d u c t s

different tableaux T, T' of the same w i e g h t T T' x = x ) g i v e rise to d i f f e r e n t c o e f f i c i e n t s We

shape

(the t e r m i n o l o g y

T

the

Some

of

author.

been

computed,

196

(Cl)

The

with

coefficient

non-negative We m a y

above, (C2) the

remark

which When

In o t h e r (C2)

would

(C3)

(i)

does

only

that

in

each

~

C mXU (~)

is a p o l y n o m i a l

in

obviously

follow

rational

of the p o w e r ~

with

from

(5)

functions

of

sum p r o d u c t s

integer

~.

pz,

coefficients.

We m a y

remark

(~) £ Z[[~], with each m m ~ = (i 12 2...).

that

coefficient

of

if

be p a r t i t i o n s

([10]

JX

as a sum of

and

> ,

C~

is a n o n - n e g a t i v e

rule

not

in terms

i~l~m !~ ,

v

in

£ ~ [ ~ ' P I ' P 2 "''']"

(~) = < J i J D , J

C XU v

Richardson

Jl

by

~,

C~

m~

are p o l y n o m i a l s

imply

X,

(so t h a t

(Cl)

~XZ(~)

words,

divisible Let

of

coefficients.

is e x p r e s s e d

coefficients

~i~(~)

that

gives Jl

vX~(e)

integral

Chapter

let

= <s~s

integer,

i)).

is a p o l y n o m i a l

,s >

given

by the

Littlewood-

Then

in

~

with

non-negative

ingegral

coefficients; (ii)

CvX~ (a) ~ 0

(iii)

If

where

(for

CV

= I,

o = I, ~,

ho = and e a c h

Z)

there

compares

the

scalar _ ~

for

conjecture

n

6L

~

xi(1

i=l

is of the

form

h(X)h(z)h(~

,

at any

rate

relate say

products

shape

to the

to the

x = on

of

author.

situation

(x I ..... x n) A

n ,~

C XU ~ > 1 (i.e., ~ - X and w e i g h t

where

the

number

Conjecture

(C4)

:

n+a' (s)e-i(s) n+(a' ( s ) + l ) ~ - ( Z ' ( s ) + l )

~ = i

integrals.

=

(~)

LR-tableau

is finite,

two

Finally,

w(x)

one

unclear,

xi

is t r u e

Selberg

than

l a s t two c o n j e c t u r e s

<JI'Jx>~ <ji,jl >

(C4)

Cv

v)

(s) is e i t h e r h*(s) or hT(s) o o s h o u l d be e x p e c t e d to h a p p e n w h e n

is m o r e

of v a r i a b l e s

This

h

what

is at p r e s e n t The

then

I{ h a (s) s{o

factor

Exactly when

iff C mlZ ~ 0;

(obviously) (C5)

connects

Let b

- x i)

and

2C

IA(x) I

~ = 2 .

Jack's

symmetric

functions

with

197

where for the sake of p r u d e n c e A(x)

b,

a,

is the V a n d e r m o n d e determinant,

be a p a r t i t i o n of length

(C5)

I

]n J l ( x ; c - l ) w ( x ) d X l ' ' ' d x

means

when

5.

~0,

and

I = (11,...,I n)

=

n (li+a+c(b-i)) ! (b+c(n-i)) ! (ci) ! H i:l (li+~+c(2n-i-l)+l) ~c!

F(x + i)

This is true when I = (i k)

Aomoto

Let

n

Jl(l,..,l;c-l)

x!

are real numbers

~n:-

[0,I

where

c

as in §2.

c = 0

(0 s k ~ n)

or

i,

for all

for all a,

b,

a, c

b

and

(Selberg

also

I; [13],

[i])

Zonal p o l y n o m i a l s When

~ = 2,

the symmetric functions L - K.

coincide with the

Z1

[8] around 1960.

There is a large literature on these, m o s t l y due to

statisticians

i n t r o d u c e d by

Jl(x;~)

zonal p o l y n o m i a l s

Hua [63 and A.T. James

(see [43 and the b i b l i o g r a p h y there)

W h e n the Schur functions are e x p r e s s e d in terms of the power sums, the c o e f f i c i e n t s involve the c h a r a c t e r s of the symmetric groups. runs through the p a r t i t i o n s of Sl = ~ z-i I/

l

Xp

where as before partition

~;

cycle-type character

is the product of power

sums c o r r e s p o n d i n g

to the

is the order of the centralizer of an element of

~ X~

If

we have

Pp

pp z

n,

in of

Sn; and Sn

Correspondingly,

X~

is the value of the irreducible

at such an element. in the case of the zonal p o l y n o m i a l s

Zl

we

have Zl = Z z-I 2p ~ 1 P~ where ~2~ the

X~)

(i)

is the p a r t i t i o n

(2~1,2~2,...)

of

2n,

and the

w

{like

are integers, which arise from zonal spherical functions

place of characters. In detail,

let

S2n

be the group of all p e r m u t a t i o n s of

symbols

(l,2,...,n,l',2',...,n'),

in

of the e l e m e n t

S2n

and let

(ii') (22')...(nn').

H

n

2n

be the c e n t r a l i z e r (H n

is the hyper-

in

198

octahedral

group,

is a G e l f a n d

or the W e y l

pair,

that

multiplicity-free; S2n 1H = n Let

~ be

W

1

group

is to

of t y p e

say the

Bn.)

induced

The p a i r I SHn 2n

character

(S2n,Hn) is

in fact, 21

~ X [lI=n

the

characteristic

function

of

Hn,

and

let

~X21

=

X the p r o d u c t is the and

being

zonal

spherical

is c o n s t a n t

double

cosets each

vertices

I,

2,

... , n,

and o n l y

The

....

may verify

double

that

lie

orthogonal

~

are even,

zonal this

to e a c h

space may

matrices Sn

function zonal

of

is a s i m i l a r sign

the

set-up,

with

X

[kI=n

the

of

of

l

~n

~n

space

may

(up to a s c a l a r

and one

Then

D

of

has

of the

G = GLn~R)

a fixed v e c t o r

if all

there

if

the

at e l e m e n t s

coset

the p a r t s

I.

Corres-

is t h e r e f o r e space

of p o s i t i v e

action

~I

n,

Hn

some p a r t i t i o n

the

Thus

~

same p a r t i t i o n of

if and o n l y

the

modulo .

and

case

e;

namely (2 I) '

=

(wr,wr j) lengths

coset

length

2n

(rr') ,

on the d o u b l e

X

~ = ½

but with

character

S2n n

of

as with

of e v e n

length

for

n,

r(w)

and e d g e s cycles

,

of the

factor)

Now

diagonal

symmetric

be r e g a r d e d

a

K\G/K.

group

as a s y m m e t r i c

it is just

the

Z I.

in the

multiplicity-free,

¢H

1 ~X

xi)

x I ..... x n,

Finally,

~ = 21

(Xl,...,Xn)

polynomial

by the

i.e.

be i d e n t i f i e d

diag

of

X 21

Now these

of

a graph

representation

K = 0(n)

.

are

is the v a l u e

~

group

function

(by p e r m u t i n g

double

Then

component

S2n

a partition

the p o l y n o m i a l

partition

spherical

n Ptw)

determine

same

to a p a r t i t i o n

the

ponding

in the

hand,

defines

determines

in (i) above U i n d e x e d by ~.

coset

of

w

of

in

S2n.

b y the p a r t i t i o n s

2' , ...

I

corresponding

~i

i',

of

to the

Hn x Hn

e S2n

two p e r m u t a t i o n s

On the o t h e r

under

indexed w

algebra

corresponding coset

components

Thus

if t h e y

coefficient

function

permutation

(i ~ r ~ n) 2~2,

in the g r o u p

on e a c h d o u b l e

are n a t u r a l l y

follows:

2~I,

convolution

of J a c k ' s

the

here

trivial

again

the

symmetric

character induced

of

functions, Hn

character

there

replaced of

S2n

is

199

In this

situation

we c a n d e f i n e

(x) = e (x)~ ~ ' (x),

Jl(x;½)

obtained

from

which

= 2 -n ~ z

(I) by use

~

occur

'twisted' in the

p~(x)

of d u a l i t y

(§3)

zonal

formula

spherical

functions

200

References [i]

K. Aomoto,

Jacobi p o l y n o m i a l s a s s o c i a t e d with Selberg integrals,

SIAM J. Math. Analysis, [2]

A° Debiard,

to appear.

P o l y n ~ m e s de T c h 4 b y c h e v et de Jacobi dans un espace

e u c l i d i e n de d i m e n s i o n

p, C.R. Acad.

Sc. Paris 296

(1983)

S@rie I, 529-532. [3]

F.J° Dyson,

Statistical theory of the e n e r g y levels of complex

systems I, J. Math.

Phys.,

3(1962)

140-156.

[43

R.H. Farrell, M u l t i v a r i a t e calculation,

[53

S. Helgason,

Springer-Verlag

(1985).

D i f f e r e n t i a l geometry, Lie groups and s y m m e t r i c

spaces, A c a d e m i c Press

(1978).

[6]

L.-K. Hua, H a r m o n i c analysis of functions of several complex

[7]

H. Jack, A class of symmetric p o l y n o m i a l s w i t h a parameter,

[8]

A.T. James,

[9]

K. Kadell, A proof of some

v a r i a b l e s in the classical domains, AMS T r a n s l a t i o n s

Proc. R.S. E d i n b u r g h 69A

(1970)

1-18.

Zonal p o l y n o m i a l s of the real p o s i t i v e d e f i n i t e

symmetric matrices,

Ann. Math.

I.G. Macdonald,

74

(1961)

456-469.

q - a n a l o g s of S e l b e r g ~ s integral for

k = i, SIAM J. Math. Analysis, [10]

6 (1963).

to appear.

Symmetric functions and Hall polynomials,

U n i v e r s i t y Press

Oxford

(1979).

[Ii]

I.G° M a c d o n a l d ,

[12]

J. Sekiguchi,

[13]

A. Selberg, B e m e r k n i n g e r om et M u l t i p e l t Integral, Norsk. Mat.

Analysis,

Publ°

13

(1982)

SIAM J. Math.

988-1007.

Zonal spherical functions on some symmetric spaces,

RIMS, Kyoto U n i v e r s i t y 12

T i d s s k r i f t 26 [14]

Some c o n j e c t u r e s for root systems,

(1944)

71-78.

R. Stanley, private communication.

(1977)

455-459.

Some Surfaces Covered by the Ball and A Problem in Finite Groups G.D. Mostow

§i.

and Stephen S.T. Yau

Lattices in PU(I,n) defined by monodromy Consider the multivalued function on

(~l)n

Fst(X2 .... ,Xn+l) = /~z-~0 (z-l)-~I (z-x 2) -~2 .. . (Z-Xn+ I) -~n+idz where s,t ~ {O,l,=,x 2, .... Xn+ I} x2,...,Xn+ I

are distinct elements in ~i_ {0,i,~ }

0 < ~i < i

I <

,

i = 0,i ..... n+l

n+l Z ~i < 2. 0

Locally, in a neighborhood of each point, many determinations.

However, at any

determinations is an

n+l

is holomorphic and globally it has

Fst

x2,...,Xn+ I

dimensional space

the linear span of all its

W -- for topological reasons (cf. [2],

§I, §3.8). An isotopy

~

~I

of

S = {O,l,~,x 2 ..... Xn+ l}

which returns the

n+3

points

to their initial positions effects a linear transformation

on the integrand and on the homology class in the punctured line ~i _ {O,l,~,x 2 .... ,Xn+l} transformation line

~i

~W

of

of the path of integration and hence effects a linear W.

The group of equivalence classes of isotopies of the

which return each point of

(or pure) braid group on

n+3

S

to its initial position is just the colored

strings in

F 1 : Cn+3(e I)

or alternatively

~I((FI) n+3 - all diagonals). For each automorphism automorphism induced by spaces of

A

A

of the vector space

on the projective space

W,

let

Proj W

Proj A

of 1-dimensional sub-

W.

The maps

~ ~ ~W

and

~ -+ Proj ~W

Supported in part by NSF Grant DMS-8506130

denote the

define homomorphisms

202 '

i

@ : Cn+ 3(~ ) -+ Aut W % : Cn+3(FI ) -~ Aut Proj W which we call the monodrom~ actions of the colored braid group. F' = Im @' , F

Set

= Im @

Set n+l ~, = 2 -

Z

~i"

0 In [2] we prove i.

(Corollary 2.21).

signature

(l,n)

Let ball in

~s

on

preserves a hermitian form n

W

of

minus signs].

B+ = Proj{w 6 W; ~(w,w) > 0}.

B+

may be identified with the unit

¢ . n

+ ~t < I, one has

of

F'

[one plus sign and

(Theorem 10.19).

2.

The group

Assume condition (i - (~s + ~t ))-I

a)

F

is a lattice subgroup in

b)

If

~s + ~t < 1

c)

r \B+

for all

INT:

for all

an integer.

s,t E S

with

Then

Aut B+(~ PU(I,n)).

s,t ~ S, then

F \B+

is compact.

is the set of ~-stable points in the Mumford

Aut ~i \ (~l)n+3,

~-quotient variety

and coincides with the quotient variety if

F\B+

is

compact. These results were proved by H.A. Schwarz for but only partially proved by E, Picard for The integrality condition

INT

n = 2

is satisfied only for

torsion-free subgroup of finite index in the lattice The computation of the one dimensional n = 2.

Morse theory.

and essentially stated

([6], [5]).

fore of some interest to understand the varieties

for

n = 1

F0\ B+

n ~ 5. where

It is thereF0

is a

F .

~I(F0\ B+)

poses a challenge even

In this paper we describe an attempt to compute

~ l ( F 0 k B+)

via

203

~2.

The Surface

Y

For eonvenience~ we restrict ourselves to the case s,t E S.

We assume

n = 2, we write

F

for

F

, B

~s + ~t < i

for

for all

B+, and set

M = F\B. In this case, and

(-,-).

2-space

M arises from ~l × pl Alternatively,

~2

by blowing up

intersection

-i,

M 4

by blowing up the three points

(0,0), (I,i)

may he described as arising from complex projective points.

M

has

i0

exceptional lines of self

and schematically one can depict them with the diagram x3=x~

x

2

~

~

Xl=X0

x3=x 0

That is,

if in

~2

we blow up

Pl

z3=0

4

points

P2

{pl,P2,P3,P4}

colinear, we obtain a DelPezzo surface whose 4

points and the Let

F0

6

i0

no t h r e e of w h i c h a r e

exceptional lines lie above the

lines of their complete quadrangle.

denote a normal subgroup of finite index in

F; set

Y : = Fo\B ~:

Y -~ F\B = M , ~':M _~ 2

[': = the set of

I0

exceptional lines of

Choosing non-homogeneous coordinates

M.

(z2,z 3)

on

~2, we can assume that

204

the six lines of the complete z2 =

0 {i

The i0

0 {i

z3 =

exceptional

image of the denotes

'

i0

quadrangle '

z2 = z3'

lines

diagonals

the line at

{L;L 6 ['}

xs = x t

the subset of points

are

in

~.

may be regarded

equally well as the

Aut Pl\(~l'S)stable

(Xs; s ~ S)

of

(~I)S

where

with

Z X

(~l)~table

~s < 1

for all

=Z S

z 6 eI

(i.e.

~s I + ~s 2 +...+ ~s k < 1

We have

whenever

=

xsl

Xs2 =...= xsk).

the dictionary

'-l(z 2 = O)

<->

x2 = x0

~'-l(z 2 = i)

<->

x2 = x1

n'-l(z 3 = O)

<->

x3 = x0

~'-l(z 3 -- i)

<->

x3 = x1

n,-l(z 2 = z3) < - >

x 2 = x3

~'-l(pl)

<->

xI = x

'-l(p2)

<->

x2 = x

~'-l(p3)

<->

x 3 = x=,

~'-l(p4)

<->

x0 = x

~:Y ~ M

The map

is a branched

(I - ~s - ~t )-I over the line ~I.S x s = x t of ( )stable where By hypothesis,

F0

L st

cover with ramification

in

L'

corresponding

s,t ~ {0,i,2,3, ~}

is a normal

subgroup

index

to the diagonal

.

of finite

F .

index in

Set

G = F/F 0 • Then

G

operates

on

Y

and G\Y = M.

As mentioned transformations Study metric Cs(~P I)

which

on

above,

the action of

preserving

the hermitian

B.

loops

point of a complex

F

form;

The effect of an element s

once around

line lying above

t

on the ball thus Y

st

F

Lst

arises

preserves

from linear

the Fubini-

of the colored braid group

is an isometry

the line

B

of

B

which

fixes each

6 i', and has order

205

(i - ~s - ~t )-I; thus each

acts as a complex-reflection

Yst

on

B

([2],

Prop. 9.2). For suitable choice of 5

lines of

generate

{ Tst; Lst E i"}

G

in its conjugacy class and for any set

st

no four of which are disjoint,

i'

We give

¥

Y

L"

of

the five complex reflections

F.

the metric induced from

B.

Correspondingly,

on

Y, the group

acts by isometries and is generated by complex reflections in the connected

one-dimensional

complex subvarieties lying over the lines of

If two elements

Tst

and

Yuv

of the braid group

follows at once that their fixed point sets in the ball are orthogonal.

For any point

q E Y,

C5(P I) B

if

~(q) ~

Us - ~t )-I

if

w(q) E Lst

(i - ~s - ~t )-I(I - ~u - ~v )-I

if

w(q) E Lst N Luv ,

-

commute, it

and in the surface

the order of its stabilizer

1 (i

i".

G

q

in

~ L LEL' and no other line (st) # (uv) .

G

Y is

206

Mqr.se Theory on On

~2, let

f

Y denote the meromorphic function

f = Zl(Zl-l)z2(z2-1)(Zl-Z2) and define the real valued function

~

by

= log f~, where Set

(Zl,Z2)

denote non-homogeneous coordinates.

z i = x i + ~-ly i

gradient of

(i = 1,2).

The critical points of

~ , at which the

vanishes, are obtained by solving the simultaneous equations

@

Ii = 8Zl@

i Zl

=

+

i + 1 Zl-i Zl-Z 2

i,+ z2

Oz2~

i z2-1

i Zl-Z 2

One finds that there are exactly two critical points p = (a,a')

where

5+~

a

i0

and

a'

1 a 2 - a + ~ = O.

equation

5-~

and

p' = (a',a)

are the two solutions of the quadratic

i0

We note also that

1 1 1 qo(p) =
The computation of the Hessian at

)2

2 ~Xl~

= (~Zl + ~ i

p

is straightforward.

2 ~ = (~Zl +

2_ Zl)~

I 1 i . . . . . . zI~PIP = _--~zI (Zl-l) 2 (Zl-Z2)21p

/32

= -

[5 + a + a '

1 2/5]

a

= _

[5 +

3/5 1/25

= -20

-20

= -40

25

Hence

~2 2

OXlIp Similarly

-5 log 5

C

i 2

• = -

1 1 - [5 + a2+a'2 - a,2 - y = a2--~S, 2 ]

[5 +

15]

= -20.

207

a2 = 2Re 8x I x21 p

%Zl 8 z2~Ip

(%z I + 8{i) (Sz2 + ~-2)~I p

= 2 ~i =

= 2- 1

(a-a') 2

Thus the Hessian at

p,

and at

p'

-4

0

0

0

4

-i

0

-i

4

i0

i0.

too, is

which is conjugate to

In particular, the critical points are non-degenerate. Let

V

denote the tangent space to V = V+ + V_

where

V+

and

V_

Then

+~

We have

, v+=~

Xll p

v

X21p

and an identical decomposition of the tangent space to point

p.

are the eigenspaces of the Hessian corresponding to positive

=~a

-

at the point

(direct)

and negative eigenvalues respectively.

v

F2

~2

at the other critical

p'. We shall apply the following result from Morse theory ([4], Theorem 5.3) :: Let

For any

b

M

be a smooth manifold and

with

- ® ~ b ~ ®,

~:M ~ ~ U -- U ~

a continuous function.

set

M b = ~-l(b),

M b] = ~ - l [ ~ , b ] .

Assume that (i)

~

is smooth on

(2)

c

is a critical value such that all critical points in

non-degenerate.

M - (M-~ U M~). ~-!(c)

are

208

(3)

c

is the only critical value of

(4)

~-l[c-~,c+~]

in

[c-~,c+~].

is compact.

Then homotopically

M c+~] ~ M c-~] where in the disjoint union critical point of index

_~

D k,

U II

D~

one has one k-dimensional disc for each

k(: = the sum of the dimensions of the eigenspaces of

the Hessian with negative eigenvalues). We apply this result to the function for any

Ek

yC+~] ~ y C - ~ ]

Ek

U

I] g Ek g6G

is a connected component in

(resp.

E 'k)

o n

U

II gEG

V

way with the tangent space at

n'-l(Dk)

to p

~2

at

Y

y~ ;

-

and similarly for

p

p

it yields

~

flows down to fill out the triangle

in

&

on

E 'k .

may be identified in a natural ~

2.

~2, a small disc about

~2

Thus

(resp. p').

to the real projective subspace

downward gradient flow of the function

- -

lying between

p

Under in

V

x I = 0, x 2 = i,

xI = x2 in

•2

in

x2=l ...................

Correspondingly in gradient flow of Y - Y~

~

mapping into

M

m3

x2=0

~

on

g E'k

is a 2-disc containing a point over

The tangent subspace

and

o~,

~ > 0:

(MT)

where

=~

~ - i

M,

the 2-disc

Dk

flows down to a pentagon

and similarly the 2-disc A

containing the point

edges of their boundaries

under p'. a~ M

~,o~ .

Ek

~

~M

under the

flows down to a pentagon

Similarly,

We orient the 2-cells and

m_~/

D 'k OM

so as to have

~

in

flows down to a pentagon and

~

and label the

209

8~ M = m 0 + m I + m 2 + m 3 + m 4 8o~ = m 0 + m_l + m_2 + m_3 + m_4 and we index the -4 ~ i ~ 4. ~'

of

7

lines of

(Note that

-i(~)

so that

L_4 = L 4

and

Li

is the line containing

L_I = Ll. )

the edge

m i,

We choose a connected component

so as to satisfy n(support

~ N support a') = support ~M N support ~

We label the oriented edge of -4 ~ i ~ 4.

L

8~

and

8~'

which lie over

mi

. by

Yi'

Thus 4 8~ =

Z

y~

0 4 8~'

=

Y-i

Z

0 Inasmuch as the complex conjugation of can be lifted to 2-cells

~

and

M ~'

~2

which fixes each point of

and in turn to a complex conjugation

of the ball

can be taken to lie below a real geodesic

and indeed to be geodesic pentagons

in

Y.

2-plane

B,

the

~2 N B

~F 2

210

~4.

The small cell complex

As above

U ~-I(L): =Tr-I(M-~

y=

L(L with

L = {Li; -3 ~ i ! 4, i # -i}

M" =

and

U

L

LEL'-L with

n'(M ~)

a line in example,

lying at

L' - L

~

in

~2

We note that each line

meets a unique such line; we denote this line by

Y~

the structure of a cell complex

of the following cell complex structure The vertices of

L'

(for

~i~

on

mY -~ which is the pull-back

M -=

are the points of intersection

M

with

L i N Lj i.

which meets

L I = ej3).

We give

0.

L E i

L i, Lj ~ L.

L E L

The i cell of a line

of the seven lines

consist of a single sllt joining the two

vertices of

L

if

L N M~

is not empty; and consists of 2 slits joining the

vertices of

L

if

L N M"

is empty, i.e. if

2.

The open 2-cells of

L ~ L

L

has three vertices.

consist of the complement of the union of its

1-cells. We take as open k-cells

~

k

my-~

in

the connected components

inverse images of open k-cells in

mM-~, 0 _< k _< 2.

homeomorphism

ok

if

~(a 2)

with

on each open k-cell

lies in a line

L N M

L

which

L

for

k = 0,i,

M ".

However,

~

is a

and also for if

k = 2

~(a 2) c L

not empty, then 2

~ Tf(

2)

c', the order of the complex reflection

in the line

L' E L ~ - L

meets.

It is clear that the one cells in one-cells

mY -~

which does not meet

T~: a

has degree

of

It is clear that

of

mM-~

can be taken to coincide with the

211

{mi; -4 ! i ~ 4} described

in §3.

Thus the 1-cells of -4 ~ i E 4,

and as above

quadrilaterals where of

c'

my--

if

of

L n M~

Y

-4 mM-~.

and

G

4, we let

2c' - gons if

index of the line

L'.

L.

l

±i = i, 4,

G

Moreover,

are

of a 2-cell in

is not empty, the closed cells

Y

L

my-~

with respect

to

{L_3,L_2,L0,LI,L2,L3,L4} sometimes ~-l(Li)

c.c~ ii

if

For each

i

which contains the 1-cell

to write

L5

with for

L 0.

is a cyclic group of order

since it fixes each point of the 2-cell.

product of two cyclic groups of order

consist of

B.

It will be convenient in

L N M"

subsets in

denote the line in L

~(yi ) = m i,

~-I(L)

the cells of the cell complex

The seven lines of

The stabilizer if

is empty and of

permutes

where

The closed 2-cells in

induced from the ball

L_4 = L 4, L_I = L I.

ci

{G Yi; -4 ~ i ~ 4}

may be taken to be closed convex geodesic

The group

mi

are

G = F/F 0.

is the ramification

the metric on

between

my-~

ii = 0, 2, 3.

However,

it is a

212

§ 5.

HI(Y)

as a quotient o_f H I ( Y - ~ .

We continue the notation of Lemma 5.1. Proof.

Let

-l(p)

HI(Y Y+

(resp.

- (Y

NY

(resp.

Y_)

-l(p~))

) ~HI(Y)

2-dimensional set

is surjective.

denote the set of points in

Y - Y

which flow into

under the downward flow of the function

Then each connected component of representing an element

~.

h

in

Y+

is a 2-cell.

Given any path in

HI(Y) , it can be deformed in

Y~ U Y+ U Y ,

and thus

h

~ o~ ' o

Y

Y

so as to avoid the

can be represented by a path which

flows downward under the gradient flow into an arbitrary small neighborhood of Y-" - (Y-~N

Y~).

Since the latter is an absolute neighborhood retract, the Lemma

follows. Lemma 5.2.

In the exact homology sequence with coefficients in

H2(Y - Y~, Y-~- (Y~ N y_~)) _ _ 8 the image of

8

~[Goa,

G ~ ~'].

Proof.

By

>Hi(Y_®_

(Y~ N Y-~))

~, > HI(Y - Y~)

is the subgroup represented by the group of 1-cycles

of §3 and the fact that gradient flow moves

(MT)

arbitrarily close to

Y

Y-

- (Y

N Y

Y®~(Y-~-

yC-8]

downward

), we see that homotopically

(Y~N

Y-B)@

Go

U G~'

where, by abuse of notation, we denote the support of

~, ~'

by

~, ~'

respectively. From this, the ler~ma follows at once. Lemma 5.3. Proof.

H2(Y, (Y - Y ~

Any 2-chain in

Y

U Y-~) = 0.

can be deformed in

at only a finite number of points. can be deformed into a disc in can deform

Y-~

Y

Since any small disc meeting meeting

Y~

~ , near each intersection point with

only at points of

Y= N Y-~

i:(Y-Y ®) U Y-~ ~ Y and a surjection for

and to lie in

induces an injection i = 2.

so as to meet

in a point of Y~

Y=

transversally

Y

transversally y® N Y

, we

in turn, so as to meet

(Y-Y~) U Y-"

Thus the inclusion

i,:Hi((Y-Y ~) U Y-~) ~ Hi(Y)

From this the Lemma follows.

Y"

for

i = 1

213

P r@position 5.4.

HI(Y) = HI(Y-~)/Im i,8

inclusion

Y

-

(Y~ n Y

where

8

is as in Lemma 5.2 and

denotes

the

Proof.

Consider the diagram of exact homology sequences

i

) ~ Y

H2(Y , (y-y') U Y-~)

H2 ((Y-Y~) H2(

Y

U Y-',Y-') ,

>

y-')

Hl(i

-=)

> HI(Y

)

--> HI( ( -Y ) U Y ) i ~ -~

>0

--> H 1 (Y)

>0

H2(Y , (Y-Y) U Y-')

0

the zero on the third row following from Lemma 5.1. From Lemma 5.3, we infer

Hl(Y) = HI(Y

deformation retraction of a neighborhood

)/Im ~'.

U(Y =)

onto

On the other hand, using a y"

followed by excision, we

have ((Y-Y~) U Y-',Y-') ~ ((Y-U(Y~))

U y-®,Y-')

((Y-U(Y~)) U Y-®- [Y-= n u(Y')], Y-~ = (Y-U(Y'), Y-= (y-Y',y-~ where

~

denotes homotopy equivalence.

identified with

Im D.

-

[y

N U(Y ~) ]

[Y-" N u(Y®)]) [Y-" N Y=]) From this we deduce that

Proof of the lemma is now complete.

Im 8'

can be

214

~6.

).

HI(Y

In view of Proposition

5.4, we take a closer look at

Nor- -~ H1 (Y)

where

i

Z

denotes the inclusion map

Lemma 6.1. intersect

Let

L i E L'

L i.

Set

and let

+

Relabelling

Lj, L k, Lg

+

We define

Next we define the quotient

L'

be the other 3 lines of

index of

,l,

Ck

L i.

which

Then

= i _ _ ! _2

cg

ci

the indices of the set

S, we can assume that

Since the ramification

Lj = L12, Lk = L23, L~ = L31. (i - ~s - ~t )-I'

~-IL ~ Y.

i

Cj

).

i, HI(~-IL)

c i = the ramification

_!_l

Proof.

=

HI(Y

index of

L

L i = L04, is

st

we find -i 1 - cj = ~i + ~2 -i 1 - c k = ~2 + ~3 -i 1 - c~ = ~3 + ~i 2(1 - c: I) =

4 5 - ( i _ ~ _ + _ ~ i +_i_i +_2_2 ) = 2 Z ~i = 4, cj ck c~ ci 0

Adding yields Notation.

For any finite set

g, h, ...

in a group

Lemma 6.2. cL

2(~0 + ~4 )

Let

G,

G,

IGI = cardinal of

G.

which implies the lemma. For any elements

denotes the group generated by

L ~ L', let

denote the ramification

X

denote a connected component of

index of

L.

g,h,

...

~-I(L), and let

Then the first Betti number of

X

is

given by

(2)

21GxI 2 cL

~i (X)

where Proof.

GX

denotes the stabilizer GX

acts on

X

having three vertices

of

X

with fundamental lying above

in

+

2

G.

domain a convex geodesic quadrilateral

L n Lj, L N L k, L n Lg.

Computing

the Euler

215

characteristic

X(X)

from the cell complex on

~2(X) - ~l(X) + ~o(X) =

xnxj

where

XN

Cj, C k, Cg)

Xj, X k, XZ)

~0(X) = ~2(X) = i,

with 2-cells

1 (i - 2 + - - +

1 + c---[

cj

GX ~, we find

i c~ )

Xk

CLCj = IGx n X. I ' GX N X. 3 3

CX (resp. (resp.

IGxI -~L

X

= < Cx'Cj >

etc.

denoting the complex reflection in the subvarieties

lying over

L

(resp.

By Lemma 6.1,

L i, Lj, Lk).

using

we get

~I(X) - 2

.2

IGxI CL

=

(

CL )

as required. Lemma 6.3.

Let

C*

denote the one-dimensional simplicial complex whose vertices

correspond to the connected components of lines

L,

L i, Lj ~ L.

~-I(L)

as

varies over the set of 7

L

- I ( L i n Lj)

and whose one cells correspond to points of

with

Then blOOp (y-®) = HI(C* ) 1 "

Proof.

Modulo

H~°r(Y-~), each closed path in

• Lll' Lil

D

. . LI2' Li2 , LI2

D

Li3

wifh homotopy corresponding to homotopy in Lemma 6.4.

y-~

....

C*.

,

is determined by a sequence

L i

n

=

Lil

This implies the result.

~o(C*) = i.

The lemma is equivalent to the assertion that

If ~-l(L )

is connected

LEL in

Y.

This will follow at once from the stronger assertion: Let

v:B -~ M = F\B

be two distinct lines of

denote the natural projection, and let Li, Lj -i 6' with L i N Lj not empty. Then v (Li U Lj)

is

216 connected. Proof.

Let

containing FX.

and

q ( ~-I(L i n Lj), and let q

FX ,

1

v - l ( L i U Lj), Li, Lj

of

the stabilizers in

F

denote the connected component

Z, X i, Xj

respectively. of

Xi

and

Clearly

Xj.

Z is stable under

Moreover,

FX.

j

the complex reflections in all the complex lines over elements of it orthogonally.

Hence

FX, U FX. i j

Z = F Z = v-l(Li U Lj).

Lemma 6.5.

where of

ci

4 Z i=-4 i#-i

L'

4 Z -3 i#-I

1 CiCi+l

is the ramification index of the line Xi

which meet

for which < F X, ,FX•> = i j Proof of the lemma is now complete.

dim H~°°P(Y-~>± = tGi <

IGx I, where

L'

contain the complex reflections in at least 6

complex lines lying over 6 distinct elements of Hence

contains

1

Li ( i

is a connected component of

iui-~--I~l )

and

~-l(Li),

F •

+ i

is the order

IGil

-3 ~ i ~ 4.

I

Proof.

x(C*)

The Euler characteristic of

=

c~}l

l{O-cells in

C*

is given by

l{l-cells in

Z Iconnected components of L ( L

C*}l

n-I(L)

-

E Iconn.comp. Li#L j

n-I(L in Lj) I

Li,Lj( L,L i ~ e j # 4

4

IGi ( Z

l~i~-lI _

-3 i#-i

Z

.

i

-4 i#-i

CiCi+l

Hence Pl(C ) = P0(c ) - ×(C )

= i - x(¢*) which implies the lemma. Corollary 6.6.

4 ~I(Y -~) = IGI [ Z -3 i~-i

2 ( --~-- + ci

1 ~il

) +

4 Z -4 i#-I

i

] + i

ciei+l

This follows immediately from Le~nas 6.2, 6.4, and definitions. We close this section with some additional identities that will be used below.

217

Lermma 6,7.

Proof.

i

Z L ( L' 1

l L 6 L'

= 2.

CL

=

Z 0 _< i < j ~ 4

CL

I - (~i + G j)

4 i0-

4

E

~i

0

L e m m a 6.8.

Let

c.

=

i0

=

2.

- 4-2

be the ramification

index of the line

L i ( L'

i

(-4 -< i _< 4).

Then 4 Z 0

Proof. with

i -ei

4 Z 0

= I=

We can assume that the five lines (st) = (01)(12)(23)(34)(40). 4

c-i

L O, L I,

Then writing

=

Z i=O

ci

are the lines

..., L 4

L

st

~5 = gO'

4

i

Z 0

i

4 ~i=

i - -(~i + ~±i+l ) = 5 - 2 Z 0

5-4=

i.

Similarly 4 Z 0 Lemma 6.9.

2

Z i < j

1 cicj

i ~ c-i Z L ( L'

= 1 -CL

i.

- 2

Z L ( L'

1 2 c_ L

n i N Lj # Li,L j ( L'

Proof.

2

Z i < j L i n Lj ~

by Lemma 6.1.

i cicj

Z L i n Lj# ~

I cicj

Z L ( f'

1 -(I CL

2 eL

)

218

~7.

The Boundary Operator Set

V = ~[G],

product on

the group algebra of

and

t ~ ~ ~ O

= ~1

Then for any

V

carrying each

v ~

of

E ~ g gEG g

and

g

for

On

V • V

g E G,

V

if

g # h

if

g = h

=

~ =

<~,~

% ~gg, gEG

> = % ~g~g

This inner product is positive definite and the induced inner product;

of the cell complex

8

~g, ~g E ~

for all

described

H

G

under

we take

G x G.

to the one dimensional

:

% gEG

~ g g

~

Z 8o gEG ~gg

8' :

Z gEG

~gg

~

Z ~ggS~' gEG

g ~ G, and the

at the end of §3.

of

V

bi-invariant.

chain group

my-~

D = For any subgroup

G

it is bi-invariant

We define two G-module maps of

pentagons

to

~v.

{~

~ =

g

g,h E G



where

Introduce the inner

-i is the involution of

p(~):

cI(mY -~)

~.

Tr O(a)p(~ t)

denotes the regular representation

Thus for

over

V <~,~>

where

G

2

Define

cells

~, ~'

are the geodesic

D:V ~ V ~ cI(mY-~)

as

8 e~ 8 '

set = {v E V; vH = v}

VH

is a left

~[G]-module.

sum of the nine G-modules

The G-module

Cl(mY -~)

is isomorphic

to the direct

219

G ~

V

Yi

-4_
where

G

denotes the stabilizer of the 1-cell Yi of §3. Yi C. d e n o t e a g e n e r a t o r The group G is cyclic (cf. §2); let 1 Yi -4 ~ i ~ 4 with C . = C. for i = 1,4. Set -l i

V.

=

V l

V0, 0

=

{v @ v

!

Lemma 7. i.

Proof.

(Ker D)~ =

V0, 0 +

The element ~ @ ~

4 l 1

; v

Z h,h' 6

V. • V I -i q

gYi

cI(mY -~)

Z ~gg(y0+Y_l+y 2+Y_3+Y 4 ) = 0 g~G

to zero, yields

egh +~gh' = 0

for all

g 6 G/

(i)

Z ~gh h E i

= 0

for all

g E G/__

(-i)

Z ~gh h E

= 0

for all

g 6 G/

Let

Yi

V0}

~

Z ~gg(y0+Yl+Y2+Y3+y 4) + gEG

(0)

G

- 4 -< i -< 4

6 Ker D i f and only if in

Equating the coefficient of each

of

6g = hE 8gh h,

where

5g h = 0

or

i

according as

l g#h

(0)

or

g = h,

-4 ~ i ~ 4.

~@

Then the conditions above are equivalent to

~iSg

@

6g

, for all

g

(i)

~J_~

, 1 _< i _< 4

, for all

g

(-i)

~6g
.>, 1 ~ i ! 4 -i

, for all

g

g l

From this the lemma follows. The following elementary observation is used repeatedly in the estimates we are about to make.

220

Lemma 7.2.

Let

A I, A, BI, B

be vector spaces with

dim A n B - dim A I N B I ~ dim(A/Al) Proof.

A N B AIA B I

~

A N B AIQ B

AI c A

and

B I c B I.

Then

+ dim(B/Bl).

× _ A_1 =n _ B+ A1 n BI

A x B A1 BI

Set (7.2.1)

O(G) =

(Note that

sup {dim V i N Vj, dim V_i n v_j } . OEi<j~4 [i-j[ =2 or 3

dim V i N Vj = IGI/I I .

We have therefore 4 dim Z i

V 0 N V i - dim(V 0 n v I + v 0 n v 4) E 20(G)

4 dim Z 1

V 0 n V_i - dim(V 0 N V I + V 0 n V 4) _< 20(G).

and

4 001

Define

-4 00_1

and

via

4 4 dim V 0 n Z V i - dim E i 1

4 (V 0 n V i) = 001

4 4 -4 dim V 0 n Z V_i - dim Z (V 0 n V_i) = O0_l 1 i 3 401

Similarly define

-3 40_1

and

via

3 3 3 dim V4 N Z Vi = dim Z (V 4 n v i ) + 401 1 i 3 3 dim V_4 N Z V_i = dim Z 1 1 Lemma 7.3. centralizing

Let

G

GI

be a finite group and and

G 3.

Set

V = ~[G]

-3 (V_4 n v_i) + 40_1

GI, G2, G 3 and

subgroups with G 2 G. V i = V i (i=1,2,3). Then

V 2 N (V I + V 3) = V 2 n v I + v 2 n v 3.

Proof.

Let

o

which stabilizes get

=

i

Z x ~ G 2 x.

~ VI

and

V 3.

Then

Given

v ~ vo

is a projection of

f2 = fl + f3

with

V

onto

V2

fi E V i (i=1,2,3), we

221

f2 = f2 "~ = fl "~ + f3 "~ E (V 2 O V I) + (V 2 N V3).

Lemma 7.4. 4 Z -4

dim ImD =

4 dimV, - Z l -4

dim V i N Vi+ 1 - g

i~-i 4 -4 3 -3 g ~ 001 + 00_1 + 401 + _40_1

where Proof.

dim ImD = dim(Ker D #

positive definite, D

maps

+ 80(G).

- in fact, since our inner product on

(Ker D)i

isomorphically onto

Im D.

V @ V

is

Thus

4 4 4 dim Im D = dim V0, 0 + dim Zl vl ~ V_l. - dim V0, 0 N El V.1 ~ El V_l 4 4 4 4 = dim V 0 + dim Z V i + dim Z V_i - dim(V 0 Q ~ V i) n (V0 N Z V_i) 1 1 i 1 We have, by Lemma 7.2 4 4 4 4 dim(V 0 N Z V i) N (V 0 N Z V_i) = dim[ Z (V0 N V i) N Z(V^ N V i) ] + gO 1 1 1 1 u 4 g0 -< 001

with

+

-4 00_1 .

Moreover,

4 3 3 dim Z V, = dim V 4 + dim Z V. - dim V 4 n Z V i. 1 l i i i 3 Substituting dim Z V i = dim V 2 + dim(V l + V 3) - dim V 2 N (V1 + V 3) 1 get, using Lemma 7,3,

we

4 4 dim Z V.l = Z dim V.l - dim(Vl n V 3) - dim(V 2 N V I) - dim(V 2 N V 3) 1 1 3 3 - Z dim(V 4 n V i) - 401 1 4 dim Z V .. 1 -i

and a similar expression for

dim Im D =

4 Z dim V. -4 z

3 Z dim(V i_ -4 i@-i, 0

3 -

Finally,

60(G)

-

401

-

-3 40_1

-

Therefore

4 N Vi+ I)_ - dim Z(V O_ 1

£0"

4 N V i) N Z(V^ N V i ) 1 u -

222 4 dim Z (V 0 N V I) = dim[(V 0 0 V I) + (V0 n V4)] + el' 1

el S 20(G)

4 dim Z (V 0 A V_i) = dim[(V 0 n V I) + (V 0 n V4)] + E_l, I Consequently

4 4 dim Z V0 N V i N Z V0 N V_t = dim V0 n v 1 + v 0 N V 4 + ~2 1

82 ~ Cl + ~-i

dim linD =

V5 = V0

and

the special case 4 Z dim V i -4

4 Z -4

8 ~ 4 01 +

A I = B I.

as seen from the

82 = ~i

This yields

dim(V i N Vi+ I) - e

+

+ 80(G)i.e.

8 ~

,

3 -3 + 401 + 40_1 + 80(G),

as required.

Set 4 PD =

Z -4

1 ei

4 Z -4

1 cici+l

i#-i We can restate Lemma 7.4 as: (7.4)'

where

i

by Lemma 7. 2, and indeed we can take

proof of Lemma 7.2 in

where

~-i ~ 20(G).

dfm Im D = IGI (PD - SD)"

'

gD =

O°l

+

223

§8.

~I(Y) Let

Q,

let

• C (my-~) I

ai

denote the boundary operator in

B i = Im 8i +i

eemma 8.1.

where Proof. of

Gi

denote the group of i-chains of

my--

ci(mY-~),

with coefficients in

Z i = Ker 8i,

(i=0'1'2)"

dim Z

4

3 = IGI [_~__i+__!_l + Z 1 Cl c4 -3 i#±l

is stabilizer in

G

4

]+1

i cici

-3 Z c . i#-i i

-4 cici+l i#-i

of a connected component of

The kernel of the boundary map

each connected component of

82

n-l(Li ),

n-l(Li).

is spanned by the fundamental 2-cycles -3 ~ i ~ 4, i # -i.

dim Im 8 2 = dim C2(mY ~)-

Hence

4 Z -3

-

i#-i i

IGI

by the result of the end of §2.

+

i

[ Cl

3 Z

+

c4

Inasmuch as

i=-3 i#±l

4

1 C l.c'.l-

-3Z i#-i

1 ~,~i,

]

dim Z I = dim HI(Y -~) + dim B I,

result follows at once from Corollary 6.6. Set PZ =

4 E -3 i#-i

2 -~-+ c. l

4 Z -4 i#-i

i + cici+l

3 Z -3 i#±l

1 , c l.c. l

+

i i +-cI c4

From Lemma 8.1, we have i

(8.1)'

dim Z I = PZ + SZ

where i ~Z = Lemma 8.2. Proof.

P7 = PD"

The identity

4 Z - - -i -4 ci

4 Z -4 i#-1

1 - - = cici+l

2

4 E -3 i~-i

i

4 z

C.

--4

-~-+ l

i#-i

- - + cici+l

3 Z -3 i#±l

i 1 + ---q3v + cic i cI

-1c4

the

224

is equivalent to 4

2

-3 i#-i

Set

c; : c 7,

4

1

z T+

c13

2

+

ci

cici+ 1

-4 i#-i

3 E -3 i#il

4

1

1 ci

Z

=

CoC~ i i

-3

i#-z

: c 6 : c~, c~ : c 8 = c12 .

We use the fact that the fifteen points {L n L'; L, L' ~ L ' }

L}

{L N L';L,L' ( of the form Thus

{L n L ' , L , L ' Z L i N Lj#+ i<

consist of eight of the form {L N L'; L ~ L, L' ~ L} and two

and five of the form

~ L}. 4 Z -4 i#-l

i C.C.

I 3

j

i

3 Z -3 i#±l

+

cici+l

i i -----T-+ c .zc i c6c 7

i

+

c7c 8

and the identity can be rewritten, using Lermna 6.7 4 2

Z -3 i#-i

3 I --~-- - Z c. -3 z i#±l

1 c.c' i i

I + 2(

Z LiNL i<

=

i c6c7

j#¢ cicj

...)

c7c 8

j

1

1

1

c6

c7

c8

2

By Lemma 6.9, this is equivalent to

2(-

l

2 c6

z3 -3

l -i--) -

1 2 c7

c8

~ i+ ' cici+l

2 ( - - - -i c6c 7

_11__)

I

i

i

c7c 8

c6

a7

c8

i#+l that is 1

1 c7

c6

1 c8

1 c_3c_ 6 l

2( ~

1 c_2c 8

1 c0c 7

1 c2c 6

1 c3c 8

+__i +_!+ z + 1 c27 c28 c6C7 c7C~8)

The left side is i (i c6 1

(

c6

-

1 _ c_ 3

2 + c6

1)+ c2 1 ) + c7

i

_!__l(

Hence the identity reduces to

(1 c8

c8

2

c8 +

1 c3

_!_z) c7

1 ) + c_ 2 +

i -~7

(i

-

i (ic7

z) Co

i )

:

cO b y Lermna 6 . 1 .

225 1

(i -

2

\ ) =

c7

i.e.

1

1 co

2 c7

+

i

1

c6c7

c7c 8

1 + - -i c6 c8

which follows from Lemma 6.1 , applied to the line

L7

in

M®

meeting

L 0.

Theorem 8.3.

1

i ~i (Y) -< s7

Proof.

+ ~D -< ~

4 -4 3 (80(G) + 001 + 00_I + 401 +

-3 40_I

+ i).

Combining (7.4)', (8.1)', and Lemma 8.2 yields

~I(Y)

= dim

ZI(Y ) Im D+Im ~2

<

dim

ZI(Y ) Im D

IGI (Sz + SD)"

The remaining inequality of Theorem 8.3 now follows from the definition of and

sD•

sZ

226

99.

A problem in finite groups. Let

Set

G

be a finite group and let

V = ~[G], the group algebra of

G

be subgroups of

G0,GI,G 2, .... G k over the field

G.

of rational numbers and

set V. = V

G. I

l

= {v ( V; v'G i = v} .

Set

k 00~(G) = dim V 0 N Z i Problem:

Estimate

If the group groups with of

H,

Vi -

dim

,Z V 0 N V i1

00~(G). G

is abelian, then

00~(G) # 0.

If

G c H

00~(G) = O.

However, one can find

and one considers

G0,GI,...,G k

as subgroups

then one has

l l-"

o o¢O k

Conjecture:

If

1

g0 - - ~

-< i,

and

G O ,...,G k

generate

G

and are cyclic

and no element ~ I of G. is conjugate to an element of G. if i ~ j, then i 3 (9.1)

00~(G) < IGI ~

In the situation of §8, the group

G

,

~<

1.

is a homomorphic image of an infinite group

with the presentation given by the Coxeter diagram: q03

4 ,

E

0

Each node represents a generator

(CiC j) CiC j

Each

Gi

qij

= i,

= CjC i ,

Ci

of order

ci

I - -

= i

ci

with

if the nodes are joined if the nodes are not joined.

is cyclic, generated by the image of

Ci

(i=0,i,2,3,4).

227

The lattice F 0'

When we take

of

U(2,1)

defined in §i has such a presentation.

to be a congruence subgroup modulo a prime ideal, then the

r~/r~

resulting

F'

becomes a subgroup of

coefficients

in the finite field

problem for

G

Fq

subgroup isomorphic to large subgroup of

SL2(F)

GL3(F q)

F~/F~.

matrices with

Thus we may take

where

F

is a subfield of

and we conjecture for the

O(G) _< IGI ~

Discussion.

3 × 3

~

C~ <

Fq.

0(G)

G, and the

G = F~/F$.

G i, Gj, the subgroup < Gi,G j >

(9 • 3) for the group

of

and is a central extension of

can be deduced from

each non-commuting pair of groups

GL3(F q)

For

contains a The group

G

defined in (7.2.1).

1

G = F /F 0. As a consequence of conjectures

(9.1) and (9.3), Theorem 8.3 would

imply ~I (Y) (9.4)

~

12 ''IGI-a

a > 0.

This conjecture can be compared with the known result of DeGeorge and Wallach [3]: Given a nested sequence of normal subgroups subgroup

F

and given

~ E 5,

lim j ~ N(F,~)

G

with

the set of all equivalence classes of

irreducible unitary representations vol(Fjk_G)-iN(Fj,e)

of

G, then

= 0, if

is the multiplicity of

e

in

m

is not square integrable;

L2(F\G).

Yj = Fj\B.

Set If

in a cocompact discrete

of a linear connected semi-simple group Lie group

n Fj = {i}, 1

here

Fj

as

j

~

~i(Yj) constant

then one could find an element and hence

~

o~ ~

A G

with

would be in the discrete series

N(Fj ,~) jV-2~, >\~)

vol(r

is a

c > 0

c > 0, as

228

(cf.

DeGeorge-Wallach

[3] Theorem 5.4, also [i], pg. 214).

the discrete series contributes to cohomology only in a maximal compact subgroup of

G;

in our case this is

0

as

½

But it is known that

dimension i ~ dim B = 2.

G/K,

K

being

Thus

j

Conjecture (9.4) offers a more precise estimate for

~l(~j)

- - ~

Bibliography

[l]

Borel, A., and Wallach, N., Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Ann. of Math. Studies, 94, (1980), Princeton Univ. Press.

[2]

Deligne, P., and Mostow, G. D., Monodromy of Hypergeometric Functions and Non-Lattice Integral Monodromy, Publ. I.H.E.S., 1986.

[3]

DeGeorge, D.L., and Wallach, N., Limit Formulas Annals of Math., 107, (1978), pp. 133-150.

[4]

Milnor, J., Morse Theory, Annals of Math. Studies, v. 51, (1951), Princeton Univ. Press.

[5]

Picard, E., Su~ les Fonctions Hyperfuchsiennes Provenant des Series Hypergeometriques de Deux Variables, Ann. ENS, III 2 (1885), pp. 357-384.

[6]

Schwarz, H.A., Uber DiejenigenFalle in Welchen die Gaussische Hypergeometrische Reihe Eine Algebraische Function Ihres Viertes Elementes Darstellt, J. Reine u. Angew. Math., 75, (1873), pp. 292-335.

for Multiplicities in

* Dept. of Math., Yale University, New Haven Conn. 06520, U.S.A.

L2(G/F),

INV~RIANT THEORY AND KLOOSTERMAN SUMS L

I.

Piatetski-Shapiro

Introduction. A. Selberg [!] intr0dnced, the following type of series. ^

s

P(z,s) = ~

.,

z~iK-

az+b

e cz~ d Icz~d] 2s

(z

in the upper half plane)

where the summation is o~er representatives of cosets of E D}.

k

is a positive integer. The poles of

sely~ if

~(z)

is a Maass wa~e form satisfying

then for

so

the residue of

P(z,s)

P(z~s)

correspond, to Maass wa~e forms.

so

is

More preci-

A~ = X~, (A . . . . +-, Ox there is a pole of P(z,s) at s = s o

X = s0(l - s o ) at

I x • FO = ((0 1 )ix

A. Selberg was able to pro~e the following remark-

able property~

such that

Fo\SL(2,~).

~(z).

and

Another remarkable property found by

A. Selberg was that the Fourier coefficient of

P(z,s)

is Dirichlet series

Z(s)

whose coefficients are Kloosterman sums oo

% S (k,n) 2s

Z(s) =

n=l n

S(k,n) =

% 2~ik xiy xyml(mod n) e n

The aim of this note is to generalize the Selberg construction to arbitrary split reducti~e group

G.

Different generalizations of Selbergts construction were considered by D. Bump, S. Friedberg, D. Goldfeld [3], [4], and G~ Stevens [5].

They are dealing with

Poincar~ series which, like Eisenstein series, depends on many complex variables. The Foincar~ series, which we introduce here, depends only on one complex parameter. Their main property is that they produce global L-functions which were introduced by R. P~ Langlands.

It is possible also to define Poincar~ series which produce

local Langlands L-factors. Denote by

LG

the

More precisely, we prove the following result. L

group of

G.

In our situation it is a reducti~e group

230

over

~.

Let

n = ® np

an irreducible places of

k

is a cuspidal automorphic

finite dimensional

representation

which includes all the archimedean

representation

of

LG.

Let

oY

%

places in case

Go

Let

p

be

be a finite set of k

is a number

field. We assume that

~p

is an unramified

introduced

L%(~,p,s)

representation

p ¢ %.

for

R. Langlands

E L (~p,p,s).

=

pCZ p We construct a Poincar~

series

P(g,s)

such that

fGk\GAP(g,s)~(g)dg where

and

c # 0,

~

is a cusp form with Whittaker model lying in automorphic

such that

representation

= cL%(~,p,s)

~

is right invariant under

K

for all

p ~ Z.

Kp

of course depends on

%.

P is the standard maximal compact subgroup of Our construction

can be modified

function at a given nonarchimedean verges absolutely

G . P

P(g,s)

in such a way that it gives a local L-

place.

In all cases our Poincar~ series con-

in some right half plane.

In the case where the series produces

a local L-factor at a given place, we can easily get the meromorphic of the series using spectral theory. L-functions

LE(~,p,s),

In the case of series which produce global

the problem of meromorphic

the problem of meromorphic

continuation

continuation

of

continuation

LE(~,p,s)

is equivalent

to

which is part of Langlands'

conjectures. It is interesting Poincar~-Selberg

to observe that the Whittaker

Fourier coefficient

of our

series is related to some sorts of generalized Kloosterman

We prove that in the general case these sums can be expressed

sums.

through sums of the

type S (n) =

and

~

is a character of

good results.

However,

n~.

Construction

Usually Weil's estimation does not produce very

one can always expect that some version of Linnik's

jecture implies Ramanujan's 2.

% ? (Xl+...+ x~) Xl---x~sa(n)

conjecture

in general.

of Poincar~-Selberg

Series.

First we recall some known facts from invariant reductive group defined over

¢,

and

con-

p

theory.

an irreducible

Let

H

be a

finite dimensional

repre-

231

sentation.

In our application

H = LG.

Consider

Sym%.

It is usual]y reducible.

Write symnp = @a(n,T)~ where

~

multiplicity of

~.

=

is the

(2.1)

% a(n,~) tn n=0

can be presented in the form M (t)=

where

a(n,T)

Consider the Poincar6-Molien series M(t)

M (t)

H.

are irreducible finite dimensional representations of

Pc(t)

and

PT(t) q(t)

are polynomials and

Q(t)

(2.2)

Q(t)

has the form

r d. Q(t) = ~ (I - t i). i=l One can choose Let

G

Q(t),

which does not depend on

[2]

be a split semisimple group defined over a global field

be the Cartan subgroup of group of

~c.

G(kp),

and

G(kp) , (p - a place of k).

Let

Kp - the maximal compact subgroup.

f

on

G

E = C n K

and

element of of

L

H = LG.

f L

Denote by

~p

Xp

~(c)

j(c).

qp

G(kp).

be the lattice

C/E.

Denote by

j

(2.3) C , P

and more:

the projection

It is easy to see that

Xp

to

Xp).

Let

f(c) = 0 Denote by f

for

c

A(c)

such that

j(c)

C ~ L.

Let An T

does not lie

the jacobian of the map

be as in i(2"3) such that

f(c,s) = P<(c)(qp s)A2(c) where

of

can be naturally interpreted as a character of the maximal torus

(from

P

the non-

Vx ~ Xp, k E Kp

is uniquely defined by its restriction to

in the corresponding Weyl chamber. x + cxc-i

C

satisfying

f(xgk) = '~p(X)f(g) Such a function

Let

be the Borel sub-

Bp

degenerate normalized character of the maximal unipotent subgroup Consider any function

k.

is an irreducible representation of

H

(2.4) with highest weight equal to

finite set of places containing all the Archimedean places.

k . Let % be a P For p # Z, let fp(g,s)

be a function on

let

is the number of elements in the residue field of

G

satisfying (2.3), (2.4). P

tion on

G

satisfying P

For

p ( Z,

fp(g)

be a func-

232

fp(Xg) = ~p(X)fp(g) Usually,

but not always, we assume that for

Now for

g £ GA

(2.5) fp(g)

p ~ Z,

does not depend on

s.

define f(g) = N fp(gp,S) P

We assume that

~/ = ~ p

is a nondegenerate

character of

f(Sg,s) = f(g,s)

~ \ X A.

Then

V6 6 X k

Put P(g,s)

=

Z

f (yg,s)

(2.6)

Y~ Xk\G k This series converges absolutely Denote sentation of on

K = N K . PP K

G(A)

such that

is of type

ates the representation, Let

THEOREM.

~

c~ = ®op

c~IKp = 1

(K,~),

~,

takes the value

on

i

and

(K,a)

be a decomposable for

p # %.

which is isomorphic to

of type ~,

finite dimensional

G(A)

(K,~),

Q)

K

of

@,

form

it gener-

o.

which generates an irreducible and assume that the Whittaker

then there exists a Poincar~

(and not on

repre-

We say that an automorphic

if under right translation by

be a cusp form on

phic representation

only on

Let

in some half plane.

series

automor-

functional

(2.6) which depends

such that

r

Here

~(diS)fG\G P(g,s)~(g)dg = c(@)L%(~,p,s) i=l k A is the restricted ~-function of k, d. appear in Q(t) l

~k

(2.7) in (2.2).

c(@)

#0. First we prove two lemmas. LEMMA i.

Let

~

be an unramified

representation

of

P Whittaker of

~ , P

function with respect to

~-i P

G . P

which corresponds

Let

W(g)

be the

to an unramified vector

then r

-ds (i - qp i ) -I f X ~ Gpfp(g,s)W(g)dg

= n(~p,p,s)

(2.8)

i=l PROOF:

It is well known that

~

corresponds

to a semisimple

P in

H = LG,

such that L(~p,p,s)

= det(I - P('~p)qp"-s,-ll

Let us recall the Shintani-Casselman-Shalika

formula for

W(g):

conjugacy class

A~p

233 1 W(c) = t r %(A,~p)A(c) 2

where j(c)

~

is the representation of

H

with highest weight

in the corresponding Weyl chamber).

dg = A-l(c)dxdcdk.

(2.9)

j(c), (c ~ C

P Recall that the Haar measure on

with G

is

P

Using this, it remains to prove fcMr(c) (qpS)W(c)dc = L(~p,p ,s)

(2.10)

Using (2.1) and (2.9) and changing the order of integration we get that (2.10) is equivalent to •

% tr(SymnP(A~p))qpnS = det(l - p ( ~ p m=l

)

-s.-i

qp )

which is an identity. Proof is easy. LEMMA 2.

Let

p £ o

and

Wp

in the Whittaker model of

~p

such that

Wp(1) # 0, ^

then there is a function

f (g) P

satisfying (2.5), which depends only on

(K,~p)

such that fX \G fp(g)Wp(g)dg # 0. P P f satisfying Lemma I and Lemma 2, then

Pick

PROOF OF THE THEOREM:

/Ck\Gf(g,s)~(g)ds = /~GAf(g,s)~(g)dg = fXAk GAf (g,s)W(g)dg = K f X \ G fp(g,s)Wp(g)dg P P P Now (2.7) it follows from Lemmas 1 and 2. 3.

Fourier Coefficients as Kloosterman Zeta Functions. Let

G

be as before.

Consider the Fourier coefficient (3.1)

Z(g,s) = /~\X~$'-l(x)P(xg,g)dx We have Z(g,s) =

where

X6 = X n 8-1X8

(since =

Z 8~gK/x

/.8-~ f(Sxs,s)~-l(x) dx ~k~AA

8 6 G ,k X8 k

i s an a l g e b r a i c group)

%

8E ~ \ G,K/Y~ 7X'6\X'K K ~ f (Sxg, s)q/-I(x) dx Choose representatives

6

in the normalizer

N

of the Cartan subgroup

C,

234

(Bruhat's lemma).

Let

W

be the Weyl group of

G,

then

C%N ~ W.

Put for

w E W

Zw(g's) = Z w'-A'/~6\XAf(6xg's)~-l(x)dx 5 projects to It is easy to show that Iff if

~

Zw(g,s) m 0

is a simple root then

G = GL(n)

then

w~

W ~ S n

unless

w

satisfies the following assumption:

is either negative or a simple root.

the permutations on

the subgroup of permutation matrices in

G,

n

W

is isomorphic to

i.e. those which have in each row and w

Then the elements

each column only one nonzero element which is one. ins the ab°ve assumpti°n are °f the f°rm

elements.

For instance,

IOi " " " Ill1

k]r

satisfy-

Put

"

15 (g, s) = fx~\xAf (6xg, s)~ -1 (x) dx Let

y5

> 0

be the subgroup of and

w~ < 0,

X

generated by all the root subgroups

X

such that

then

18(g,s ) = fy6f(6yg,s)~-l(y)dy = ~ fy6fp(Syg,S)~pl(y)dy -A p p Put -.-I (y)dy I~ p)(g,s) = fySfp(6yg ,s")~p P We now consider the case G = PGL(2), then we have the following result: Write I~P)(I,s),

0 -i0 )" 6 = (0 01) (i

Assume that

g = I,

Ip (~,s)

and write

for

then we have:

Assume that

l~Ip > qp

then

If Ip(~,s) =

-i ~ Ixl=q~* (7 + x)dx

if val (~) is odd P 2~ if l~Ip = qp

We see that the integral does not depend on

s

and that it equals a

Kloosterman sum. Assume that

lel = q

-m

, m >_ 0

Ip(~,s) = Assume that

I~I = q,

then

p ( -s,

m q

)qp

m 2

- Pm+2(q-S)q

then --S Ip(a,s) = -Pl(q )q

i

-7

m 2

i

235

We see that for

(p

I~Ip ~ qp

the integral

Ip(~,p)

enters in the definition of the polynomials

P (t)).

depends on

Analogous properties are true for a general semisimple group eral, the part of

16(g,p )

~(Xl+ x2+'''+ Xm_l+ XlX2...Xm_l)dXl'''dXm"

IXll ..... IXm_iI=q where

g

and on

is a positive integer.

P.

G, and in gen-

which depends on trigonometric sums can be expressed

through sums of the form Y l~l=qm

s

236

References (i)

A. Selberg. On estimation of Fourier coefficients of Sym. in Pure Math., vol. III (1915).

(2)

T.A. Springer.

(3)

D. Bump, S. Friedberg, D. Goldfeld. for SL(3,Z). Preprint.

(4)

S. Friedberg.

(5)

G. Stevens.

Invariant theory.

Poincare

Department of Mathematics Yale University 12 Hillhouse Avenue New Haven, Connecticut 06520 and School of Mathematical Sciences Tel-Aviv University Tel-Aviv, Israel

Proc.

Lecture Notes 585, Springer-Verlag.

series for

Poincare series on

of modular forms.

Poincare series and Kloosterman

GL(2).

GL(n)

sums

Preprinto

and Kloosterman

sums.

Preprint

ON ACTIONS OF ~ a O N A n

V.L. Popov MehMat, MGU Moscow, USSR

To T.A. Springer on his 60th birthday

I. Let k be an algebraically closed field of characteristic zero. We identify the k-algebra Fn = k[Xl,...,x n] of polynomials in the indeterminates x 1,...,x n with the algebra of regular functions on n-dimensional affine s p a c e ~ n by means of the isomorphism which sends x. to i-th standard coordinate function. If o :~n ÷ ~ n l an isomorphism, we denote by ~

the automorphism F

n

÷ F

n

f(~(a)), f C Fn, a E ~ n. The map A u t ~ n ÷ AUtkFn, o ~ ~ , We identify o with the set of polynomials

is

given by (o~f)(a) = is an anti-isomorphism.

(O~xl,...,o~x n) (which defines o by the

formula o a = ((o~Xl)(a),...,(a*Xn)(a)) , a c ~ n ) .

2. The group Aut ~ n

(or, which is the same, AUtkF n) has the structure of an

infinite dimensional algebraic group, [2]. At present a satisfactory description of its structure is known only for n ~ 2, see [2,3,4,5,...]. For n ~ 3 a number of key problems remains open, [1,2,5]. One of these is the problem of the structure of finite dimensional algebraic subgroups of Aut.~ n or, which is the same, the structure of (regular algebraic) actions of finite dimensional algebraic groups o n ~ n. The precise formulation of this problem is connected with the consideration of two subgroups

of A u t ~ n : the affine subgroup A fn = {~ = (f1'''''fn) E A u t ~ n l d e g

fi ~ I for each i}

238

and the triangular "Borel" subgroup Bn = {(fl,...fn) C A u t ~ n l f i

= cix i + hi, e i E k, c i # 0,

h i E k[xl,...,xi_ I] for each i } (we assume that h I = 0). If n ~ 2 then every finite dimensional algebraic subgroup G of Aut A n is conjugate to either a subgroup of A fn or to a subgroup of Bn, see [2,3,4,5,...]. The problem (posed in [2] by l.R.Shafarevich)

is:

can one generalize

this to the case of an arbitrary n ? If G is unipotent then this reduces (because of the Lie - Kolchin theorem) to the following question : is G conjugate group of B

n

to a sub-

?

3. In his recent paper [I] H. Bass constructed an example of a one parameter unipotent algebraic subgroup of Aut A 2 which is not conjugate to a subgroup of B 3 (or, in other words, an example of an action of the additive group ~ a o n ~ 3 which cannot be triangularized). This construction is based on the automorphism 2 T = (xl,x 2 + xIU , x 3 - 2x2U - xIU2 ) C A u t ~ 3, U = x|x 3 + x2, proposed earlier by M. Nagata [6] as a conjectural example of an element of Aurae 3 which is not a product of elements of A f3 and B3. H. Bass observed that one can include T into the one parameter unipotent algebraic subgroup {~t = (x1'x2 + txIU' x3 - 2tx2U - t2xl U2) C A u t ~ 3 ] t

C k};

this subgroup furnishes the desired example (the latter is proved by means of an investigation of the ideal in F 3 defining the variety Fixo t of fixed points of the automorphism Or' t # 0). I shall show here that this example of H. Bass is a special case of a simple general construction which furnishes in a unified way examples of non-triangular actions of ~ a on A n for arbitrary n.

4. Let D be a locally nilpotent k-derivation of F exists an s such that DSf = 0 ) .

E m~0

(i.e. for each f 6 F

Then, f o r t E k, one h a s a w e l l - d e f l n e d

exp tD of the k-algebra F n given by the formula (exp tD)(f) =

n

m t ~.t Dmf,

f 6 F n'

n

there

endomorphism

239

(the sum is finite because D is locally nilpotent, (exp tD)(f).(exp tD)(g) follows from Leibniz'

and the property (exp tD)(fg) =

formula). We write

m

exp tD =

Z m>~O

~t

Dm .

It follows from formally verifiable propertieS of exponentials, (exp sD) = exp ( t + s ) D

to wit (exp tD)

for each t,s C k, and exp OD = Id, that in fact exp tD is an

automorphism of the k-algebra Fn and that {exp tDit C k} is a subgroup of AUtkFn. This subgroup is a one parameter algebraic unipotent subgroup,

i.e. for each f C F

n

the linear span of {(exp tD)(f)It ~ k} over k is finite dimensional and the action of @a on this span inducedby {exp tDlt C k} c AUtkF n is given by a rational linear representation of ~a" Or, in other words, if OtD C Aut ~ n

is an element such that

O~D = exp tD then the formula t(a) = otDa , t C k, a C ~ n, defines a regular algebraic action of ~a on ~ n

(i.e. ~ a × ~ n

~n,

(t,a) ÷ t(a), is a morphism)

LEMMA I. If {exp tDit E k} is conjugate in AUtkF FiXOtD is for each t E K a cylindrical variety,

n

to a subgroup of B

then n --

i.e. is isomorphic t o ~ I × Z for a

certain variety Z (depending on t). PROOF. Let y C Aut ~ n he an element such that ~t = (y~)-1. exp tD. y~ E B

for n

each t ~ k. We have 6tx i = citx i + hit , c i t

C k, cit # 0, hit C k[x|,...,xi_1] ,

hlt = 0 for each t C k and i = 1,...,n. Since {6tit E k} is an algebraic subgroup of AUtkFn, the map t ~+ cit is a regular

(polynomial)

function o n ~ Io It follows from

this and from the conditions cit# 0 and Cio = I that cit = I for each i and t. Using the equality 6 t = (YOtD ¥ -1)~ we see now that F i x ~ t D Y-1 = {a c~nlhit(a)

= 0

for each i}. Therefore FixY~tD Y-1 •"s a cylindrical variety in the coordinates x 1,...,x n (because hit C k[x 1,...,xn_ I] for each i and t). The assertion of the •

lemma follows now from the equality y FiX~tD = FlxTOtD Y

--I

m

5. Let us point out now two simple ways to construct locally nilpotent k-derivations of F . n Each k-derivation A of F n is completely defined by the elements Axi, i = |,.°.,n, and for an arbitrary set of elements fl,...,fn C Fn there exists a k-derivation A such that Ax i = fi' i = 1,...,n. It follows from Leibniz'

formula that A is locally

240

nilpotent iff there exists an s such that ASx. = O, i = 1,...,n. l Let V be the linear span of xl,...,x n over k. We shall say that A is linearized (in the coordinates x|,...,x n) if V is invariant with respect to A. In this case one can consider the restriction of A to V; this is a linear operator AIV on V which completely defines A. The k-derivation A is locally nilpotent iff gIv is nilpotent. If AIv is nilpotent we can, without loss of generality, assume that Xl,...,x n is a Jordan basis for AIV , i.e.

i

xi+ I for i # pl,...,ps,

Ax i =

(M) 0 for i = pl,...,p s

for a certain set of integers I ~ Pl < "''< Ps ~ n. Therefore with each such set of integers pl,...,p s we can associate a locally nilpotent

k-derivation of Fn defined

by (W); we denote this derivation by

A n pl,..ps. Another way to construct (a lot of) locally nilpotent k-derivations of F

follows. Let D be such a k-derivation and h 6 F

n

n

is as

be one of its invariants, i.e. Dh = 0.

Then it is easy to see by induction that (hD) m =hmD TM for an arbitrary integer m > 0. Hence hD is also a locally nilpotent k-derivation of F

{exp thD =

l m>0

tm ~ hmD m It 6 k }

n

and

is a one parameter unipotent algebraic

subgroup of Aut F . n LEMMA 2. Consider a nonzero linearized k-derivation A = nAP1...ps o f Fn, a nonconstant invariant h of A and the

k-derivation D = hA. Then for each t # 0 the

hypersurface Fh = {a 6 Anlh(a) = 0} is a union of certain irreducible components of the variety Fix a tD. PROOF. Let i be an integer, I < i ~ n. If i is equal to one of pl,...,p s then (exp tD)x i = xi; if not - and, say, Pr-1 < i < Pr - then

(exp tD)x i =

Pr - i ~ j =0

tj ~ t hJxi+j.

It follows from A # 0 that there exists at least one i of the second kind. But FiXeD

is defined by the system of equations -xi+(ex p tD)x i = O, i = 1,...,n.

241

Hence we have that:

I) h divides each polynomial

2) if t # o then at least one of these polynomials from I) that F h c F i X e D ,

(exp tD)xi-xi,

i = 1,...,n;

is not equal to zero.

and:from 2) that dim F i X e D

lemma follows now from the fact that each irreducible

It follows

~ n-1. The assertion of the component Of F h has dimension

n-1.

D REMARK.

It follows from the proof that F i X e D , t # O, is the union of F h and

of several linear subspaces which are defined by the vanishing of somexi's.

6. Since the union of some irreducible

components

of a cylindrical

variety is

itself cylindrical, Ler~mas I and 2 imply a way to construct non-triangular ~a ° n ~ n

: if (using the notations

of Lemma 2) the hypersurface

actions of

F h is not cylindki -

cal then t t+ @tD is an action of such type. Therefore we come to the problem: can one construct

such invariants h that the hypersurface

I do not know a general criterion for F h to be cylindrical; F h is not cylindrical.

Nevertheless,

how

F h is not cylindrical apparently,

?

"in general"

having in mind the furnishing of examples of

non-triangular

actions of ~ a o n ~ n the following observation will be sufficient for us : if h is a nondegerate quadratic form in x 1,...,x n then Fh is not a cylindrical hypersurface

(indeed,

such F h has only one aingularpoint;

clear that the dimension of the singular positive).

examples

locus of a singular cylindrical variety is

There are in principle no difficulties

nondegenerate

quadratic

invariant.

on the other hand it is

in solving when

A has a n PI"" "Ps We shall show that one can furnish the desired

in this way for an arbitrary n.

It is convenient

to use some facts of the representation

be a finite dimensional a nondegenerate

sl2-module.

sl2-invariant

On the other hand it is known, sl2-invariant

(automatically

This module is selfdual,

in the symmetric

theory of sl 2. Let L [7], hence there exists

square of the sl2-module L • L.

[7], that if L is simple then there exists a non-zero

nondegenerate)

in the symmetric

square of L iff dim L

is odd. Let now A be a linearized A]V being nilpotent,

locally nilpotent k-derivation of F . The operator n it follows from the Jacobson - Morozov theorem that one can

include A IV into a sl2-triple. clear that each sl2-invariant A. This sl2-module modules

So one has a structure of sl2-module

on V and it is

in the symmetric algebra of V is also an invariant of

is simple if A = nAn and is the sum of two isomorphic

simple sl 2-

if n is even and A =nAn/2,n . It follows from this that A definitely has a

nondegenerate

quadratic

invariant h if either n is odd and A =

and A = nAn/2,n . It is not difficult

h =

Therefore

d E (-I) I XiXn+ i- i , i = I

A or n is even n n to point out this invariant explicitly:

where d = n if n is odd and d = n/2 if n is even.

the above proves the following

242

THEOREM. The action of ~a °--n/%n' t ~+ (fl,...,fn), given by the formulas d-s fs = i O =E

t ~

i

•

r

Xi+shl'

h = i =E l(-t)iXiXn+l-i

where d = r = n if n is odd, and d = n/2 for I ~ s ~ n/2, d = n for n/2 < s < n, r = n/2 if n is even ~ is non-triangular. It is easy to see that for n = 3 the action given in this theorem is conjugate by the automorphism (-x3/2 , x2, xl) with the action of ~a ° n ~ 3

given in the example

of H.Bass [I].

REFERENCES

[I]

H.Bass, A non-triangular action of ~a °n~%3' Journal of Pure and Applied Algebra 33, 1984, I - 5.

[2]

l.R.Shafarevich, On some infinite dimensional groups, Rendiconti di Matematica e della sue applicazioni, Ser.5, Voi.25, 1966, 208 - 212.

[3]

D.Wright, Abelian subgroups of AUtk(k[X,Y]) and applications to actions on the affine plane, Illinois J.Math. 23, 1979, 579 - 634.

[4]

R.Rentschler, Operations du groupe additif sur le plan affine, C.R.Acad. Sci. Paris, Series A, t. 267, 384 - 387.

[5]

T. Kambayashi, Automorphism group of a polynomial ring and algebraic group action on an affine space, J.Algebra 60, 1979, 439 - 451.

[6]

M.Nagata, On automorphism group of k[x,y], Lectures in Math., Kyoto Univ. n.5, 1972, Kinokuniya - Tokio.

[7]

T.A. Springer, Invariant Theory, Lect.Notes Math. v. 585, 1977.

NORMALITY OF G-STABLE SUBVARIETIES OF A SEMISIMPLE'LIE ALGEBRA

R. W. Richardson Department of Mathematics Research School of Physical Sciences Australian National University Canberra ACT Australia

To T. A. Springer

§0. Introduction Let

g

be a semisimple Lie algebra over an algebraically closed field

characteristic zero and let subalgebra of

g

and let

G W

be the adjoint group of

be the Weyl group of

be a closed G-stable subvariety of iant regular functions on X N ~ that

and let D

X .

Let

W 0 = Nw(D)/Zw(D)

is a normal variety.

.

Let

W-invariant polynomial functions D).

g

and let D

g

k[X] G

g .

Let

~

k

of

be a Cartan

with respect to

~ .

Let

X

denote the algebra of G-invar-

be an irreducible component of the intersection

For the purposes of this Introduction, we assume k[t] W

(resp. k[D] W0 )

denote the k-algebra of

(resp. W0-invariant regular functions) on

~

(resp.

In this paper we will prove the following result, which gives an elementary

necessary condition for

X

to be a normal variety:

Let the notation be as above. (NI) the homomorphism (N2) k[X] G

Then the following two conditions are equivalent: given by restriction is surjective; and

k[~] W ÷ k[D] W0

is an integrally closed k-algebra.

In particular, if

×

is a normal

variety then condition (NI) holds. The condition (NI) is our elementary necessary condition for the normality of

X .

This seems to be a very useful condition since, in a number of concrete examples, it can be easily checked by using the detailed information available on Weyl group invariants. We were led to formulate and prove the above result by the following question of De Concini and Procesi [i0, p.8]:

K

and let

the orbit subalgebra

~

8: g ÷ g

denote the -1 eigenspace of

G.~ . ~

Let

Is

Z

a normal variety?

such that

a "Caftan subspaoe" of

p n t = a [

and

8

on

be an involutive automorphism of K •

Let

Z

denote the closure of

In this case, one can choose the Cartan

is an irreducible component of

W 0 = NW(~)/Zw(~)

Z n t;

is the "little Weyl group".

a

is It is

244

known that

k[t] W

and

k[a] W0

are graded polynomial algebras and one has explicit

information on the degrees of the homogeneous generators of these polynomial algebras. Using this information and condition (NI), we give several examples of pairs such that the corresponding variety

Z

tive answer to De Concini and Procesi's question. exceptional simple Lie algebra of type

(g, 8)

is not a normal variety, thus giving a negaIn all of our examples,

[

is an

El .

We also consider the situation in which

X

is the closure of a "decomposition

class" ("Zerlegungsklasse" in the terminology of Borho and Kraft [3]) in

g .

case, easy computations allow us to show that, in a larKe number of cases, X normal variety.

A number of examples are given in §7 - §9.

family of examples: and let

X

let

[

is not a

We mention one particular

be a simple Lie algebra of type

A l (I>2)

be either (i) the closure of the subregular sheet of

complement of the set of regular semisimple elements of

In this

g ;

g

then

, D l , or

or (ii) X

El

the

is not a

normal variety. In §5 we prove a theorem which states that, for certain G-stable cones

X

condition (NI) is a necessary and sufficient condition for the normality of

in

g ,

X .

§1. Preliminaries Our basic reference for algebraic groups is [i] and our basic reference for algebraic geometry is [8]. closed field Let

G

k

All algebraic varieties are taken over an algebraically

of characteristic zero.

be a group and let

denotes the result of

g

X

be a G-set.

acting on

x , G.x

stabilizer, or isotropy subgroup, of G-Y

is the G-orbit of

centralizer G;

ZG(Y)

of

Y

in

Y . G

and

X

reductive algebraic group and let the algebra k-algebra. and let

k[X] G We let

of X/G

~X : X ÷ X/G

inclusion homomorphism G

and

~X,G

~X

NG(Y)

Let

G

G

X

and

x E X , then

x .

Let

Y

x

and

be a subset of

G

x

X .

ZG(Y) = {g 6 G I g'Y = Y(Y E Y)} is the

g-x is the

~r~r~lizer

Then

is the of

Y

in

.

acts morphically on the affine algebraic

is an

affine G-variety.

Let

be an affine G-variety.

G-invariant regular functions on

X

G

be a linearly

By Hilbert's theorem, is a finitely generated

denote the affine algebraic variety such that

k[X/G] = k[X] G

be the morphism of algebraic varieties corresponding to the k[X] G ÷ k[X]

;

is the "quotient morphism",

instead of

g 6 G

NG(Y) = {g E G I g'Y = Y}

is a normal subgroup of

X , then we say that

at

The subgroup

If the (affine) algebraic group variety

G

If

is the G-orbit of

we say that

X/G

If reference to

is the "quotient" of G

X

by

is necessary, we write

~X "

be a linearly reductive algebraic group and let

X

be an affine G-variety.

245

Then the f o l l o w i n g r e s u l t s are known: i.i.

is a surjective map and each fibre

~X

~x-l(y)

contains a unique

, y ( X/G ~

closed G-orbit. Let

1.2.

Y

be a closed G-stable subset of

the homomorphism of

~X

1.3.

to

Y

G is finite.

x ( X , the fibre

~-l(~(x))

It follows from l.l that on

X

to the points of

points of

X/G

Then

onto

Y/G

~X

is closed in

~X(Y)

~X

~x(Y)

The restz~Sction

.

G-x .

determines a b i j e c t i o n from the set of closed G-orbits If

G

is finite, then every G-orbit is closed and the

c o r r e s p o n d to the orbits of

G

on

X .

The proof of 1.2 uses the

R e y n o l d ' s o p e r a t o r and r e q u i r e s the h y p o t h e s i s that c h a r a c t e r i s t i c ( k ) Let let

X

be a n i r r e d u c i b l e affine G - v a r i e t y and let

m = S U P x 6 Y dim G-x

and we set

n o n - e m p t y r e l a t i v e l y o p e n subset of If (resp.

a

Y

= 0 .

be a s u b v a r i e t y of

yreg : {y ( y idi m G-y = m}

.

Then

X .

yreg

We

is a

Y .

is a Lie s u b a l g e b r a of a Lie a l g e b r a

z (x)) --g

and

X/G

is a finite morphism and, for every

is just the orbit

X/G .

Then

given by restriction is suz~ective.

k[X] G ÷ k[Y] G

induces an isomorphism of

Assume that

X .

denotes the c e n t r a l i z e r of

a

g

(resp.

(resp.

x (g)

x) in

g

, then

.

z (a)

We sometimes w r i t e

a

g--

(resp.

x)

i n s t e a d of

z (a) -Z_-

(resp.

z (x)) --K

.

The f o l l o w i n g n o t a t i o n w i l l be u s e d for the rest of the paper: denote a s e m i s i m p l e Lie a l g e b r a w i t h adjoint group and

~ = Lie (T)

Weyl group of and in

B

B

G ;

G

w i t h respect to

T ; R

g

R ; if

g

will always

; W = NG(T)/T

is the set of roots of

is a base of the root system

~

is a m a x i m a l torus of

is the c o r r e s p o n d i n g Cartan s u b a l g e b r a of

g

G

is the

w i t h respect to

is simple then we label the roots

as in Bourbaki [4, Planches I-IX].

§2. A c t i o n o f a f i n i t e In this section

group on an a f f i n e H

variety

denotes a finite group and

We let

z = ~V

Lemma 2 . 1 .

Let

be a closed H-stable subvariety of

E

: V + V/H

(closed) irreducible subvariety of irreducible components of By 1.3, ~

closed in

V/H .

.

V

H-variety.

Proof.

T

V/H .

Then

H

V

such that

~(E)

is a

acts transitively on the set of

E .

is a finite m o r p h i s m and Let

is an i r r e d u c i b l e affine

First we prove an e l e m e n t a r y lemma.

E 1 . . . . , Er

E = ~-I(~(E))

.

In particular, ~(E)

be the i r r e d u c i b l e components o f

is the union of the closed irreducible subsets

~(E.) i

, i = i,

E .

..., r .

Then

Since

is ~(E)

~(E)

246

is irreducible,

we see that

E

It follows

meets

E. . ]

just the translates

~(E) : ~(Ej)

for some index

j .

Hence each H-orbit on

easily from this that the irreducible

components

of

E

are

h.E. , h ( H . ]

The following lem~m is the key to the proof of our main theorem:

Let

Lemma 2.2.

D

be a closed irreducible subvariety of

Consider the following tloo conditions on

V

and let

D: (i) the homomor~hism

K : NH(D)/ZH(D).

9 : k[V] H ÷ k[D] K

given by rest~ction is surjective; and (ii) ~(D)

is a normal subvariety of

Then condition (ii) implies condition (i).

is a normal variety, then conditions

If

D

V/H .

(i) and (ii) are equivalent.

Proof. of

~

Let to

T : D/K ÷ ~(D) D

and let

be the surjective

i : ~(D) ÷ V/H

morphism

determined by the restriction

be the inclusion map.

The h o m o m o r p h i s m

admits the factorization

K[V]Hkk[~(n)]~k[n] ~ where

i

~(D)

and

Y

are the comorphisms

is closed in

surjective 2.2.1.

V/H

and

if and only if

T

Y

of

i

and

is injective

T .

since

is an isomorphism.

Condition (i) holds if and only if

Now

T

i

is surjective

is dominant.

Consequently

T: D/K ÷ z(D)

Thus

since

~

is

we have proved:

is an isomorphism of

varieties. We need the following result: 2.2.2.

T

Proof. and

is a birational morphism.

Let

E : H-D : ~-I(~(D))

z(D) = z(E)

.

Let

E0

to exactly one irreducible results are immediate: a dense, open K-stable open subset of Now

then it follows ~D(X) = zD(y) bijectively

(a)

.

E0

subset of

.

If

immediately

E

is a closed H-stable

of

D ; (c)

; and (e) x

E .

Let

of

and

~D(D0) y

T

z ( E 0)

E

subset of

of

DO

DO

maps the open subset

T is birational.

z(E 0)

such that

that

z(E)

.

of x

D belongs

Then the following E ; (b)

y ( K.x ZD(Do)

DO

~(x)

D/K . = ~(y)

, hence that of

D/K

Since we are in characteristic

This proves

2.2.2.

is

is a (dense)

is a (dense) open subset of

are points of

of

subvariety such that

DO = D n E0 .

z(D O) = z ( E 0) ; (d)

from the definition

onto the open subset

x

is a dense, open H-stable

But this shows that

zero, this implies that

Then

component

z(D) = z(E)

~(D 0) = T(~D(D0))

.

be the set of all points

,

247

Assume now that are finite. that

T

~(D)

Therefore

is an isomorphism of varieties.

Assume now that and

T

is a normal variety.

It is clear that all fibres of

it follows from Zariski's Main Theorem

D

Thus condition

is normal and that condition

is an isomorphism of varieties.

(ii) implies condition

(i) holds.

Thus condition

T

[8, p. 137, Cor. 2]

Then

(ii) holds.

D/K

(i).

is normal

This proves

Lemma 2.2.

§3. Proof of the necessary condition f o r normality Let

~ : ~ ÷ g_/G and

W I : t ÷ t/W

denote the quotient morphisms,

The following

theorem is the main result of this paper: Theorem A.

Let

X

be a closed irreducible G-stable subvariety of

irreducible component of the intersection

X n !

and let

D

! , let

W 0 = Nw(D)/Zw(D)

.

be an Consider

the following three conditions: (NI)

The homomorphism

(N2)

k[X] G

is integrally closed.

(N3)

Wl(D)

is a normal 8ubvariety of

kit] W ÷ k[D] W0

given by restriction is surjective.

t_/w .

Then (N2) is equivalent to (N3) and (N3) implies (NI).

If, in addition, D

is a

r~rma~ variety, then (NI) implies (N3) and the three conditions are equivalent. Proof. 3.1.

The following two results are well known:

Let

× E ~

. 3..2.

If

.

then

The homomorphism

k[g] G ÷ k[t_]W

if and only if

G'x

meets

given by restriction is an isomorphism.

Hence

V : k/W ÷ g_/G of affine varieties is an isomorphism.

~(x) = ~(E) .

This follows from 3.1 and i.i.

Now we can prove Theorem A.

It follows immediately

(N3) of Theorem A implies condition variety,

~

E = X n t .

Lemma 3.3.

Proof.

is closed in

G,x n ~ = W.x .

the corresponding morphism Let

G'x

Then the orbit

x E ~,

then conditions

and, by 1.2, ~(X) ~ X/G . are equivalent.

(NI).

(NI) and (N3) are equivalent. Thus

from Lemma 2.2 that condition

It also follows that, if

~I(D) ~ X/G .

D

is a normal

By Lemma 2.1, zI(D) : zI(E)

Therefore we see that (N2) and (N3)

248

§4. A lemma on graded polynomial algebras. A graded commutative

k-algebra

A = @

algebra if there exists a finite family which are algebraically Let

A

be a graded polynomial

called a Hilbe~ of homogeneous of

b I,

...

independent

basis of

elements

b

in

A .

of

A+

A+/(A+) 2

n ~ 0 (al,

A

with

n

..., a ) q

and generate algebra.

Let

of homogeneous

elements of

A

A .

Then a family

A + : ~n > 0 An

(al,

..., a ) q

Then a family

is a Hilbert basis of form a basis of

is a graded polynomial

A0 = k

A

(bl,

as above is ..., br )

if and only if the images

A+/(A+) 2

r

Lemma 4.1.

Let

~

:

A

polynomial algebras. integer

s ~ q

÷

be a surjective homomorphism of degree zero of graded

B

Then there exists a Hilbert basis

such that

(~(a I) . . . . .

(a I . . . . , aq)

is a Hilbert basis of

~(as) )

of B

A

and an

and

~(aj) = 0 , j > s .

Proof.

Let

such that

J : Kernel(~) 12 + J = I 2 @ E

I : D @ 12 ~ E . (resp.

E)

Let

(al,

which consists

and let

I : A+ .

and let

D

..., a ) s

Let

E

be a graded subspace of

be a graded subspaee of (resp.

of homogeneous

ation of Hilbert bases that the family

(as+l,

..., a )) q

elements.

(al,

I

J

such that

be a basis of

D

It follows from the characteriz-

..., aq)

satisfies

the conclusions

of

Lemma 4.1.

§5. A necessary and s u f f i c i e n t condition f o r normality Let

~ : q

.

The following theorem gives a necessary

for the normality of certain G-stable Theorem B.

Let

!

cones in

be a linear subspace of

k[~] W0

is a graded polynomial aZgebra.

is surjective; (N3) ~(S)

W 0 : NW(S)/Zw([)

~-l(~(c))

itions are equivalent: (NI) the homomorphism

condition

.

~ , let

denote the closed irreducible G-stable cone invariants

g

and sufficient

.

and let

X

Jsswne that the algebra of

~hen the following three condgiven by restriction

k[t] W ÷ k[c] W0

is a normal variety; and (N4) X

is a normal Cohen-Macaulay

variety. Proof.

Assume that condition

be the morphisms

(NI) holds.

n

: ~ / W 0 ÷ t__/W and

determined by the inclusion maps

: ~ / W 0 + g__/G be the composition varieties.

Let

It is known that

k[g] G

of

~

and

~

c ÷ t .

and

By 3.2, ~

is a graded polynomial

t ÷ g

~

: t/W ÷ g_/G and let

is an isomorphism

algebra.

Thus the

of

249

comorphism

~*

algebras.

: k [ g ] G + k[e_]

Let

which

is a s u r j e c t i v e

s = dim ! = dim !/W 0 .

of algebraically and

W0

independent

satisfy

the

By Lemma

homogeneous

following

homorphism

of graded

4.1 there

elements

of

two conditions:

(i)

exists

k[g] G

a family

which

v * ( P I )"

polynomial PI ' " ' ' ' P l

generate

k[g] G

"'" ' D * ( P s )

are alge-

W0 braically

independent

and

generate

k[c]

; and

(ii)

w*(P.)

--

Let

P

: g + kI

G-orbits

be d e f i n e d

and determines

(ii) a b o v e

We n e e d

Let

the

P ( x ) : (Pl(X),

an i s o m o r p h i s m

= 0

T

...,

Pl(X))

: g/G + k I

a ( [/G .

Let

(i : s + I . . . .

following

g_ of codimension

1 .

C

results

l)}

of Kostant

Then the ~ b r e

.

.

Then

P

It f o l l o w s

is c o n s t a n t

easily

from

on

(i) a n d

.

(5.i)

[12]:

is an irreducible normal subvariety of

~-l(a)

There exists a dense G-orbit

be the complement of

z-l(a)

.

in

is at least two.

0

, ..., 1 .

that

X = {x 6 g I P . ( x )

5.2,

by

: 0 , i = s + 1

i

Let

x 6 g F eg

0

in

~-l(a)

0

in

.

Then the codimension of

Then the differentials

.

and

~-l(a)

0 = g_reg n

C

(dPi) x , i = i . . . . . 1

are linearly independent. We a l s o

need

Lemma 5.3.

Proof.

x

Let

closed

the

following

elementary

lemma:

is an irreducible subvariety of XI,

"'" ' r X

irreducible

be t h e

G-st~le

irreducible

subvariety

of

g

g_

.

components

of

.

each

By 1,2,

X

.

Then

z(X.)

--

irreducible

s~set

of

z(X)

~(X.) i

= z(c)

f o r at l e a s t

~(X.) l

: ~(e) --

for

d

denote

the

d. = d i m X. i i i : i,

..., m

then

z(c)

dimension z. i

there

each

that

i .

of the

fibre

exists

of

an i n d e x

..., m

dim z.-l(a)

<

.

Let

~-l(a) Thus

for

.

= U. x . - l ( a ) ii ass~e

~(X.) 1

~ ~(c) --

that

Then

U

of

the restriction

~

.

For

j ( {i,

..., m}

exists

a 6 U.

such

i : re+l,

of

such

that

that open

d I : d i m X 1 : d+s

open

than .

d

Since

~i

For each

subset

U. i

) = ~

.

If

of for

p

of

, which

Let

.

U. n z ( X

s~set

.

, let X. . i

d. = d + s

z

is a n o n - e m p t y less

to

that

that

..., r

..., m ~

d. - s ~ d i

a non-empty

and

it f o l l o w s

i = i,

is of d i m e n s i o n

is of d i m e n s i o n

is

is a c l o s e d

we may assume

for

fibres

, there

d

is i r r e d u c i b l e , ten.bering,

z

U = n. U. ii

we m a y

z(c)

denote 7. i

X.i

i

After

and that

•

p > m then

Since

: X. ÷ ~ ( c ) l --

i = i,

--

.

index

..., m

, the generic

that

for

such

co~on a n d let

I claim not,

i : i,

= ~(c) one

each

~(c)

gives

.

If

a ( U

a contradiction.

is s u r j e c t i v e

and the

,

250

and the generic fibre of

~i

least

d .

.

~-l(a)

, w h i c h is irreducible

Thus

Let

a E ~(c)

Then

X

is a prime ideal.

miX

It follows

meets

(dP.) , i = s+l, i x in commutative

is an R-sequence Cohen-Macaulay

..., I

in

d .

is irreducible.

X

X

zl-l(a)

Thus

~-l(a)

in

= zll(a) c X I .

This proves Lemma 5.3.

independent.

X

in

g

in

X

and that each

from a standard result

I(X)

is generated by

, the sequence

Ps+l'

Cohen-Macaulay

variety.

X

is n o n - s i n g u l a r

the complement

of

X reg

Thus the set of singular points of

Therefore

X

is a

in

of 5.2(b) and (5.1) that each point of

and, as noted above,

(NI) of Theorem

"''' P ~

..., Pl) = k[g]/m(x) ~ k[X]

it will suffice to show that

at least two.

at least two.

shown that condition

X

2

It follows

~ - s

k[g_]/(Ps+l,

It is an easy consequence

in

{

[17, p. 345]) that the ideal

is an irreducible

is normal~

is a dense open subset

x E X reg , then by 5.2(b) the differentials

Therefore X

X reg = X N g reg

has codimension

is of codimension

is a smooth point of

has eodimension

in

, are linearly

k[g_] .

X

X reg If

algebra and

one.

codimension

is contained

from 5.2(a) that

(see e.g.

Since

To show that

~eg

X

of

Xr e g .

algebra

..., P~ .

codimension

= d

and

has dimension at

] f(x) : 0 (x E X)}

X , that the complement

Ps+l'

Zl

is irreducible,

I(X) : {f ( k[g]

fibre of

dim zl-l(a)

d , each fibre of

and of dimension

X : ~-l(~(c_)) = X I , and Since

of

is of dimension

is a normal variety.

B implies condition

Consequently

(N4).

in

X

X

is of we have

The other conclusions

of Theorem B follow from Theorem A.

6. Application to the De Concini-Procesi question Let of

e

8 : g ~ g on

g .

be an involutive

A linear subspace

a maximal abelian subalgebra be a Caftan subspace of . Let

Let

W 0 = NG(~)/ZG(~)

K = {g ( G I ge = @g}

the closure of the orbit 6.1.(a)

Then

WI

K.~

is dense in

of ~

automorphism

~ ~

of

and let

;

the group

•

Then

G.~

in

[ g

~

~

The following

G.~

subspace of

if (i) ~

of

g

is Let

which contains

(c)

W0

g .

Let

Z

denote

results are well known [9, 13, 22]:

is dense in

W0 .

~

are semisimple.

is often called "the little Weyl group".

is a K-stable

k[a] W0

denote the -i eigenspace

be a Caftan subalgebra W0

.

~ . Hence

Hence

~

and (ii) all elements of

is canonically isomorphic to

generated by reflections.

and let

is a Cartan subspace of

~

Z .

(b)

Let W I : NW(~)/Zw(~).

is a finite subgroup of

is graded polynomial algebra.

GL(~)

251

Let

w : g + g/G

be the isomorphism Lemma 6 . 2 .

Proof.

and

of varieties

z(~) c z(Z)

t_/W and consequently

G,a c z-l(z(a))

Therefore

Z(Z) c z(a)

LeiTIIla 6 . 3 .

a

contains

a .

Since

is a finite morphism,

~ : t_/W + g_/G

t ÷ g .

.

Zl

.

Since

is closed

G-a

in

g_/G .

is dense in

if

Now

is closed

~(G-a)

Z , we have

= ~(~)

in and

Z c ~-l(z(a))

.

a c Z n t .

Let

Z n t

.

be an irreducible

c

component

of

Z @ t

which

Then

= dim z(~)

, we see that

.

dim £ = dim ! , thus that

c = a .

6.3. Z

is a normal variety,

~omomorphism

it follows

k[t] ÷ k[a] maps k[t] W

k[~] W0

are graded polynomial

denote the homogeneous easy consequence Lemma 6 . 4 .

Wl(~)

.

dim ~ = dim ~i(~)

This proves

and

by the inclusion

= dim ~l(C)_ = dim w(c)_ ~ dim ~(Z) = dim ~(a)_

£

Now

Let

morphisms.

determined

is an irreducible component of

Clearly

Since

be the quotient

w(a) = Z(~I(~))

therefore

dim

: t ÷ t/W

~T(Z) = 7T(a) .

Clearly

Proof.

Wl

If

component

of the remarks Z

onto

k[a]

algebras.

of

W0

For

k[t] W

above,

from Theorem

(resp.

A that the restriction

The invariant n { 0 , let k[a]

W0

)

algebras

k[~]~

(resp.

of degree n .

k[t] W k[a]~ 0)

Then as an

we have:

is a normal variety, then for every

dim kit] W > dim k[a]~ 0

(6.4.1)

n

-- n

A classification algebras

of (conjugacy

is given in Helgason's

For each pair (~,W 0) .

(g,6)

and

of) involutive

automorphisms

[9] (see in particular

of simple

the tables on pp.

, he also gives the type of the root system corresponding

Thus for each pair

dim k[t] W

classes

book

dim k[~]~ 0 .

(~,@)

, with

For exactly

g

simple,

four classes

we have precise of pairs

Lie

518-520). to

information

(g,e) the condition

-- n

above

(6.4.1)

(a) B2 .

Thus (b)

A2 .

(g,e)

of type EIII.

dim k[~]~ = i

(g,@)

Thus

is not satisfied.

and

of type EIV.

dim k[~]~ = 0

and

These are: Here

(~,W)

is of type

E6

and

(~,W 0)

is of type

dim k[!][ 0 = 2 . Then

(~,W)

dim k[a]~ 0

is of type .

E6

and

on

(~,W 0)

is of type

252

(c) C3 .

(g,8)

of type EVIl.

Therefore (d)

dim k[~]~ = i

(~,8)

In this case

of type EIX. dim k[t]

Z

and

([,8)

(~,W)

and Here

= i

Thus we see that if variety

Then

is of type

and

(~,W 0)

is of type

dim k[~][ 0 = 2 . (~,W)

is of type

wo k[~] 6 =

dim

E8

and

(~,W 0)

is of type

F 4.

2 .

is of type EIII, EIV, EVIl, or EIX, the corresponding

is not normal.

For all of the other classes of involution (6.4.1)

E7

is satisfied.

surjective

In these cases, one can probably

and hence, by Theorem A, that

not checked the details. the normality

of

of simple Lie algebras,

Z/G

show that

k[t] W ÷ k[a] W0

is a normal variety.

In any case, the normality

of

Z/G

the condition

However,

is

we have

does not directly imply

Z .

17. Decomposition classes and sheets The concept of a "decomposition algebra

g

("Schichten")

in

~ .

classes and sheets in of

~

7.1.

semisimple

and

xI

and

that, letting conditions

classes

x ( g D(x)

irreducible Let of h .

gh

Lie

g

, we refer the reader

discussion

to [2].

Roughly

of decomposition speaking,

class if they have "similar"

two elements

Jordan decompositions.

Let Yl x2

xI ~ g

have Jordan

nilpotent)

and let

decomposition

x2 E g

(with

have Jordan decomposition

are in the same decomposition

g-x 2 = h 3 + Y3

Xl = hl + Yl

class if there exists

be the Jordan decomposition

of

hI

x2 = h2 + Y2 "

g E G

such

g.x 2 , the following

hold:

Ghl = Gh3

If

in a semisimple

we have:

Definiti0D

Then

("Zerlegungsklasse")

For a very clear and detailed

are in the same decomposition

More precisely,

(i)

class"

was introduced by Borho and Kraft [3] in their study of "sheets"

;

and (ii)

, we let , x E g

~(x)

M = Ghl = Gh3

of

g

M.y I : M-y 3 .

subalgebra

class of

into disjoint,

The set of decomposition x = h + y

have Jordan decomposition

; the (commutative)

, then

denote the decomposition

, give a partition

subvarieties.

x E g

if

x .

The decomposition

G-stable

locally closed

classes is finite. and let

z = z(g h)

is the "double centralizer"

be the centre

subalgebra

of

It is easy to see that D(X) = G'(z reg + M'y) = G'(z reg + y)

We let

z

denote the quotient map

.

(7.2)

253

For each root

~ E R , we let

If

J

is a subset of the base

Wj

be the subgroup of

Nj

is the normalizer

subgroup of

GL(~j)

The following semisimple

of

in

h E ~

h E ~eg

(c)

Let

J

h E g

g

= ~(x)

Proof.

G

Since

For each

K

We set

j c B

and if

We let

Mj : Nj/Wj

and we let

Nj = N W ( ~ )

and consider

double centralizer

z = z(g h)

such that

B .

;

Mj

then

as a

~

Each sheet

g

.

g

w E W

Hence

~

such that

is closed in

~(~) = ~(~a) •

and

~K

are

w(J) = K . of

h , then it

g_/G .

x = h + y , let

(n)

this follows

and let

z : z(g h)

:from (7.2).

: {x E ~ I dim G.x = n} ~

.

is an irreducible

is a finite union of decomposition

class in

.

.

A sheet in

class.

S , then clearly

We wish to apply the necessary closures of decomposition

of

be the double centralizer of g.~ = ~

is the double centralizer ~(~)

each sheet contains a d e n s e decomposition dense decomposition

subalgebras

Then the subalgebras

have Jordan decomposition

is closed,

closed subsets of

~

1.3, and 3.2 that

n ~ 0 , let

of

•

a .

~j = z(g h) .

be subsets o f

~(~) : ~(z)

~(z)

S

l a E J}

if and only if there exists

x E g

Then

{s

characterizes

and

Then

is semisimple

Let

.

W .

to

.

g E G .

and

from 7.3(a),

Lemma 7.4.

by

corresponding

~i = {h E ~ I a(h) : 0 (a E J)}

be semisi~rple and let

Let

If

in

standard result

(b)

follows

be the reflection

we set

generated Wj

Then there exists

conjugate under

B

E W

.

elements

Let

7.3.(a)

h .

W

s

classes

condition in

g .

If

S

~ = ~

The sets

g

(n)

are locally

(n)

component of some classes

in

~

.

is a sheet and

In particular, ~

is the

.

for normality

given by Theorem A to the

The following proposition

is an easy conse-

quence of 7.3, Lemma 7.4 and Theorems A and B . Proposition

and let

7.5.(a)

j c B

position class

D(x)

z

have Jordan decomposition is G-conjugate to

~

.

x : h + y , let

z = z(g h)

If the closure of the decom-

is a normal variety, then the following condition holds: pj : k[t] W ~ k [ ~ ] MJ

given by restriction is surjective.

Let the notation be as above and assume further that (i)

element of ~(x)

x E g

the homomorphism

(NI)j

(b)

Let

be such that

g_ and (ii)

k[~/] MJ

is a graded polynomial algebra.

is a normal variety if and only if condition

(NI)j

x

is a regular

Then the closure of

above holds.

254 In [Ii], Howlett has given an explicit description in

W

and of the representation

of

matter to check whether condition subset of

Mj = Nj/Wj (NI)j

check condition

(NI)j

A l , Bl , C 1

subregular

above holds.

decomposition

for all subsets

or

D1

J

of

B

Nj

If

J

Wj

is a proper non-empty

(NI)j

does not hold, so

classes is not normal. when

of

In most cases, it is an easy

g

In §8, we

is a simple Lie algebra of

and in §9 we check the condition

in cases related to the

sheet.

Remark 7.6.

Perhaps the most interesting case of the closure of a decomposition

is the closure of a nilpotent conjugacy class. is trivially detailed

~j

B , it turns out that, in most cases, condition

that the closure of the corresponding

type

on

of the normalizer

satisfied and Proposition

information

7.5 gives no information.

on the closures of nilpotent

has been obtained by Kraft and Procesi

§8. Condition

In this case, J = @ , condition

(N1)j

class (NI)j

A great deal of

classes in the classical Lie algebras

[14,15].

f o r the classical Lie algebras

In §8 we shall use the results of Howlett [ii] without explicit reference. 8.1.

g

(l+l)

of type

A1 .

Let

m~trices and let

manner.

Thus

~

g = s/~+l(k)

t = g n d .

, let

~

be the space of diagonal

We shall identify

is identified with

{(x I . . . . .

is identified with the symmetric group

Sl+ I

d

with

k l+l

Xl+l) I Ex i : O} .

acting on

k l+l

(l+l) ×

in the obvious

The Weyl group

by permutation

W

of the

coordinates. Let J

has

Then

J c B . n. l

Assume that (the subdiagram of the Dynkin diagram corresponding

components

Mj = Nj/Wj Each root

Lena

8.1.1.

tion.

Then

Let

PJ

Adi_l

is isomorphic to

~ E R

d~ : {x ( ~ [ a ( x )

of type

'

Sj : Sn0

can be considered

= 0 (~ ( J)}

.

i = i, ×

"''' .

..

s, and let

n O=l+l-Es

i=l

to)

nidi

"

× Sns

as a linear function on

d .

Let

We will need the following elementary lemma:

~j : k[d] _ W ÷ k[~]] MJ

be the homomorphism dete~nined by restric-

is surjective if and only if

~a

is subjective.

We omit the proof, which is easy. For each positive integer

m , let

A

be the algebra

k[Xl,

S ..., Xm ] m

Then

k[~] W

of

m

symmetric polynomials isomorphic

in the indeterminates

(as a graded algebra)

to

Al+ I .

XI .... , X

is canonically

It follows from Howlett's result

(or by

255

Mj an easy direct argument) A

~ ... ® A nO

that

k[d~]

By comparing the dimensions

to the tensor product

of the graded components

of degree two

ns

of

A£+ I

and

and

An0 ®

n O > 0 , then

surjective. Adl_l

If

, where

Proposition

B .

... ® Ans ~j

s : i

, one sees that if either

cannot be surjective.

If

and

has

n O : 0 , then

nld I = £ + i .

is surjective.

Let

g

Then the condition

As a consequence

be simple of type

(NI)j

Let

8.1.3.

If

x

the conjugacy

g

Ad_ I , where

Let

J

and

~j

is

each of type shows that

~j

be a non-empty subset of J

has

m

md = £ + I .

the number of non-empty

be simple of type

is nilpotent, class of

then the closure of

x

A£

subsets

of Proposition

and assume that

J

of

B

£ + I . 8.1.2.

Z + i

is prime.

is a normal variety if and only if (i)

D(x)

then the decomposition

in

×

is

class

~(x) = D

is just

C (x) ,

g , and it follows from a result of Kraft and Procesi

C (x)

~(x)

is a proper non-empty D

components,

s : i

is a regular semisimple element.

x

[14] that the closure of

that

A£ .

is an amusing consequence

Then the closure of

nilpotent or (ii) Proof.

nI

!j = !

holds is equal to the number of divisors of

The following proposition

x ( g ,

s = 0 , then

or (ii)

of Proposition 7.5 holds if and only if

(NI)j

of this, we see tha

such that condition

Let

s > I

Thus we obtain:

8.1.2.

Proposition

J

(i)

In this case, an easy direct argument

connected components, each of type

J

is isomorphic

is a normal variety.

is equal to subset of

g

.

If

x

is regular semisimple,

In all other cases

~(~)

= ~(zj)

B , and it follows from Propositions

, where

8.1.2 and 7.8

is not a normal variety.

Remark 8.1.4. Then the set

Let

[

~reg

be of type

is a "Dixmier sheet" in

that every Dixmier sheet in ment of

~reg

points of

~

in

_g of type

type

B

in

D

g

B£

and

g

a 2 .

.

be semisimple

P : ~(x)

It has been shown by Peterson

For

Hence the set

m > 0 , let

vector

space

E

~(sing)

= k[E*] W . m

[16]

of singular

over

R = R(Bm ) k

, let

be a root system of W = W(B )

by the

m

B

Then

B

is a graded p o l y n o m i a l m

.

and it is easy to show that the comple-

m

group and let

and let

~ 2 .

C£ .

an m-dimensional

x 6 g

is n o n - s i n g u l a r

is of codimension

has codimension

8.2.

A£ , let

algebra with

Weyl

256 algebraically

independent

2i , i : i , ..., m Now let

homogeneous

generators

Pl'

B

of the root system

R .

Let

B1 J

r : 1 - j - Fs din i i=l "

B n l X ... x Bns x B r

Hence

Then

M

k[zj] MJ

acts on

J

If

dim k[~6]~J

= c .

Proposition

Let

8.2.1.

-empty subset of

is of degree

B .

g Then

g

pj

(set AI

j = 0

if there

by the root length.

as a reflection

group of type

to the tensor product

if

r : 0 .

Bnl®...®

pj

pj

(resp.

is not surjective.

is surjective.

is simple of type

B£

factoriz-

Then an easy argument

c > 1 , then that

subset

of type

terms in this tensor product

if

be simple of type

be a non-empty

B. J

from type

c = i , then one can show by a direct argument Exactly the same arguments work if

J

components

of type

is isomorphic

In particular

n. l

k[z_~1]

B n ~gB Let c be the number of non-trivial r s ation; thus c = s + i if r # 0 and c : s shows that

and let

have

Ad _i , i : i, ..., s , and possibly one component l is no such component). Type B I is distinguished Set

P,z

.

g_ be a simple Lie algebra of type

of the basis

"''' Pm , where

C1

.

Thus we obtain:

C£) and let

J

be a non-

is a surjective homomorphism if and only if one of

the following three conditions holds: (a)

J

has only one component, which is of type

(b)

J

has

m

(c)

J

has

m + i

where

B.3 (resp.

components, each of which is of type components, m

of type

Cj) , j = l, ..., I ,"

Ad_ I , where

and one of type

Ad_ I

£ ; or

md=

B. 3

(resp. Cj)

md + j = 1 .

As particular

cases of Proposition

8.2.1, we record the following results,

which

we will need later:

Let

8.2.2.

g

be a simple Lie algebra of type

B1

or

C1 , 1 > 2 .

Let

B = {al, .... ~l } , where the roots are numbered as in [4]. (a)

Let

J = {al} .

Then

pj

is not surjective.

Let

J = {a£} .

Then

pj

is surjective and consequently

is a

-I(~(~))

normal variety.

8.2.3.

Let

Then

pj

8.3.

~

g

be simple of type

B2

and let

is surjective and consequently of type

D1 (1 ~ 4) .

J

denote either

~-l(~(zj))

{~1 }

or

la2]

.

is a normal variety.

In this case the situation

is slightly more complicated

257 since, in a number of cases, Mj

does not act on

tions.

shows that if

However an easy argument

~j J

as a group generated by reflec-

has components

of

type

A.

more than one value of

i , then

dim k[zj]~ J ~ 2 , so that

Since one has a simple description

remaining

of the generators of

cases can be checked directly.

Proposition

subset of

8.3.1.

B .

Then

pj

k[t] W

cannot be surjeetive. (see [4, Chap. 6]) the

We state the results without proof.

be simple of type

Let

for

1

M

D£ , £ ~ 4 .

Let

J

be a non-empty

i8 surjective if and only if one of the following two cond-

pj

itions i8 satisfied: (a)

has

J

m + i

of type (b)

J

components

Ad_ 1 , where

has

(m ~ 0) , one component of type

In the above proposition i f J

, it

is

considered

Remark 8.4. G-stable

S

sheet of

g -In

g .

g .

(resp.

(resp.

£ = md

d

{el_2,~£_l,~Z})

m

is even.

is a component o f

D3). is closed

In [i0, p. 15], Procesi suggests that the properties If

S

is a sheet of

g

Moreover every irreducible

component of

of

and if the G-orbits on

m , then it is easy to see that the closure of g-m

S

is an irreducible

is the closure of some

It follows from lhoopo&ition 7.5 and the results of §8 that, in a large

number of eases, the closure of a sheet is not normal. properties

and

g_~ = {x E g lrank ad(x) ~ m} ; ~m

should be studied.

have dimension

component of

D2

m ~ 0 , let

subvariety of

these varieties

Ad_ 1 , where

{~l_I,~£}

to be of type

For each

and

£ = md + j ; or

components, each of type

m

Dj (j > I)

of the varieties

g-m

Thus we see that the geometric

are not as nice as one might have hoped.

§9. Non-normality of the closure of the subregular sheet Let

~

be simple of rank

of the centralizer elements of

£

is

£ .

An element

£ + 2 .

there are two subregular

sheets.

is the closure of a subregular Proposition

The irreducible

are the subregular sheets of

~

length there is only one subregular If

g .

components

of the set of subregular

If all roots of

R

are of the same

sheet and if there are two rooth lengths, a ~ B

sheet of

7.5 is not satisfied,

is subregular if the dimension

x E ~

g

and

then

J = {~} , then the closure of

(see [19]).

If condition

(NI)j

then this closure is not a normal variety.

the results of §8 and, for the exceptional

groups, the tables of Howlett

an easy matter to check whether condition

(NI)j

is satisfied.

G-~j

of Using

[ii], it is

The results are as

follows: 9.1. (a)

Let

g

be simple of type

A£ (£ a 3) , B£ (£ a 3) , C£ (£ ~ 3) , D£ (£ ~ 4),

258

or

E6 , E 7 , E 8

and let

F4

Then condition

J : {al } .

of Proposition 7.5

(NI)j

is not satisfied. In the following cases, condition

(b)

rank

(i)

If

Bl

g_ is of type

As a consequence

or

J = {a 2} ;

C1 (l ~ 2)

F 4 , and

and

Let

a : {a 4} , then

Let

g_ be simple o f type

g

be simple of type

B1

or

Let

g

be simple of type

the closure of Remark 9.3.

S

If

C1

X/G

and

X

S

g

is not a normal variety.

in both cases).

Then the closure

is not a normal variety.

a : {a I}

and let

F4

be a subregular sheet of

g_ .

Then

is not a normal variety. g

is simple and if

X

of the cases not covered by Proposition or

is not satisfied.

(NI)j

A£ (1 >= 3) , nZ (1 >_--4) , E6 , E7 ,

C1 (Z >-_ 3

of the subregular sheet corresponding to (c)

J = {~/} .

Then the clsoure of the subregular sheet of

E8 .

(b)

or

J = {a I}

of 9.1, we obtain:

Proposition 9.2.(a)

or

and

of type

(ii) g (c)

g_= 2

is satisfied:

(NI)j

corresponding

is normal, but not that

to X

is the closure of a subregular 9.2 (i.e.

J = {~l})

g_ of rank two or

, then we can conclude

is normal.

sheet in one

g

of type

Bl

from Theorem A that

In certain of these cases,

X

is not a

normal variety. Now let semisimple of

X

g

be simple and let

elements

of

are of the form

one element. irreducible

If

.

Let

be the complement

in

g_ of the set o f regular

Then it is easy to see that the irreducible

z-l(~(z6))

g_ has one (resp.

components.

Proposition 9.3.

g

X

, where

J

is a subset of

two) root lengths,

As a consequence

then

B X

containing has one

of Theorem B and the results

g_ be simple and let

X

components exactly

(resp. two) of 9.1, we have:

denote the complement in

g_ of the set

of regular semisimple elements. (a)

If

g_ is of rank two, then each irreducible component of

(b)

If

g

is of type

A1 (Z >= 3) , D Z (1 >_- 4) , E 6 , E 7

or

X

is a normal variety.

E 8 , then

X

is irred-

ucible and is not a normal variety. (c)

X

If

Z_ is of type

corresponding to

corresponding to

B1 (1 ~ 3)

J : {a£}

J = {(~i}

or

C1 (1 >-_ 3) , then the irreducible component of

is a normal variety and the irreducible component

is not a normal variety.

259 If

(d)

g

is of type

is a normal

F 4 , then neither irreducible component of

variety.

§10. Normality of the G-orbit of a l i n e . As a last example of applications

of Theorems

A and B, we consider the following

and let

denote the closure of the orbit

question:

Let

L

G.L .

be a line (through Is

X

t

it seems to be relatively

A holds in this situation.

reasonable

X

a normal variety?

In concrete examples Theorem

O) in

easy to check w h e t h e r condition

(NI) of

However we have not been able to formulate

sort of general theorem.

We prove below a few easy results

any

on the above

question.

Proposition 10.1. group

w .

Let

Let

Then

Nw(kx) = {1} .

denotes the line Proof.

Let

g

does not belong to the Weyl

-i

P(!)

be the projective

set of all points

x

~' is a non-empty open subset of

kx , then the closure of the orbit

be the canonical map.

non-empty

be simple and assume that

~' denote the set of non-zero elements

Since

~

If

x ~ ~' and if

to

acts faithfully

!

on

and let P(!) .

P ( t- -)

such that stabilizer

Wa

open subset of

P(t)

and it is clear that

t r = p-l(u)

and hence

k[L]

Let

x 6 !, and let

W0

- = k[L]

L = kx .

Then

is a graded polynomial k[t] W + k[L]

such that L

is not a normal variety.

of

-empty and open.

a

~ . G.L

space corresponding

-i ~ W , W

of

is trivial.

U

Then

•

W 0 = Nw(L)

p : i- {0}+P(!)

Let

Thus

be the U

Zt

is a is non-

is equal to

{i}

algebra generated by an element of W0

degree one.

Thus the homomorhpism

surjeetive.

It follows from Theorem A that the closure of the orbit

given by restriction G.L

is not is not a

normal variety.

Proposition 10.2.

Let

non-empty open subset in

g

~oof.

of the orbit Let

VI

g

be simple and asswne that

t~' of G.L

~

such that if

-i

E W .

x E t~J and if

L = kx , then the closure

is a normal Cohen-Macaulay variety.

denote the set of

x ( ~

such that

Nw(kX) = {±i}

similar to the one given in the previous proof shows that

VI

subset of

~

defined by Then

!

.

Let

B

denote the Cartan Killing form of

F2(Y) = ~(y,y)

.

Let

V 2 = {x ( ! l F2(x) # 0}

t" is a non-empty open subset of

X = ~-I(~(L))

.

Since

Then there exists a

x

is a r e g u l a r

t .

Let

semisimple

;

is a non-empty and let

and let

x ( t", let element of

an argument

L = kx g , X

open

F 2 E k [~]2G

be

t~r= V i N V 2 n g f

~eg

and let is the closure

260

of the orbit and that

G.L .

Let

W 0 = Nw(L)/Zw(L)

W 0 = Nw(L) = {±i} .

.

Since

x E V I , we see that

It follows easily from this t h a t

polynomial algebra generated by an element of degree two. ction of

F2

to

surjective.

L

is non-trivial.

Proposition

Let

10.3.

is normal in

~(L)

See Bourbaki

and

g/G

[5, Chap.

Proof.

We may assume that

further

assume that

~(h)

X

k[t] _ W

t r i p l e in

.

8, §ll) for the definition of an

F2(h) # 0 . if

Let

w E Nw(L)

Thus, if

0 , 1 , or

, then

: k[!] w ÷ k[L] w0 invariants

Let .

[

w'h = ±h . k[~] ~

maps

k[L] W0

surjective

F 2 E k[g] G

W 0 = Nw(L)/Zw(L)

.

2

~

is

Consequently

9

k[L] W0 2 . Since

onto

spanned by

{x,h,y}

a

s£2(k)

for every

W

triple.

{H

e E B .

Let

l e E R} ; 8

then

to

~

is

stable, it follows easily that

.

Then

L

is a line, the algebra of

In order to prove that w E W L

such that

(~,L)

w E W

The proof of Proposition

9

is

w'h = -h .

is a Cartan subalgebra of

of the Weyl group of

this implies that there exists

is surjective.

Then

F2(h) @ O , we see that the homomorphism

is a graded polynomial algebra.

By a standard theorem,

variety.

is defined as in the previous proof, then

it will suffice to show that there exists

be the subalgebra

is

L = kh .

of the Cartan Killing form

Since

Since

Thus there exists an element

k[L] WO

It follows from [5, Chap. 8, §ii] that we may

is equal to

It is known that the restriction

positive definite.

÷

g and let

be the rational vector space spanned by the set of co-roots h 6 ~

x E V 2 , the restri-

is a normal Cohen-Macaulay variety.

~-I(~(L))

h ~ t .

is a graded

is a normal Cohen-Macaulay

be an jZ_2(k)

(x,h,y)

Since

Thus the homomorphism

It follows from Theorem B that

k[L] WO

Zw(L) = {i}

such that

such that

a.h = -h .

w.h = -h .

10.3 now follows from

Theorem B.

§11. A generalization of Theorem A The proof of Theorem A carries over to a number of similar situations. speaking,

one has an analogue of Theorem A wherever an appropriate

Chevalley isomorphism

k[~] G ~ k[[] W

holds.

Roughly

analogue of the

In order to make this precise,

we make

the following definition: ii.i.

Definition

irreducible let

Let

K

be a linearly reductive

affine K-variety.

F = NK(M)/ZK(M)

.

four conditions hold: (i)

F

Let

We say that

is a finite group;

M

algebraic group and let

be a closed irreducible M

V

subvariety of

is a Cartan subvariety of

V

be an V

and

if the following

261

(ii)

if

x E M , the orbit

K.x

(iii)

every closed K-orbit on

(iv)

the homomorphism

is closed;

V

meets

M ; and

k[V] G + k[M] F

given by restriction

We have the following generalization Theorem C.

Let

K

V

Let

V

F = NK(M)/ZK(M)

and let

F 0 = NF(D)/ZF(D)

V

Ass~ne that there exists a Cartan 8u]pvariety

and let

subvariety of

of Theorem A:

be a linearly reduetive algebraic group and let

ible affine K-variety. K-variety

is an isomorphism.

D

.

Let

be an irreducM

of the affine

be a closed irreducible K-stable

X

be an irreducible co,rponent of the intersection

X n M .

Consider the following three conditions:

.

(NI)

The homomorphism

(N2)

The algebra of invariants

(N3)

~M,F(D)

given by restriction is surjeotive.

k[M] F + k[D] FO

is integrally closed.

k[X] G

is a normal 8ubvariety of

M/F .

Then conditions (N2) and (N3) are equivalent and (N3) implies (Ni).

If

D

is a normal

variety, then (NI) implies (N3) 8o that the three conditions are equivalent. The proof of Theorem C follows from Lemmas 2.1 and 2.2 in exactly the same way as the proof of Theorem A.

We omit further details.

We list below several examples of affine K-varieties

which contain Caftan

Subvarieties: 11.2. t

Let

k

be a reductive

be a Cartan subalgebra of

Lie algebra, k .

Then

t

let

K

be the adjoint group of

is a Caftan subvariety of

semisimple,

then Theorem C applied to this case gives Theorem A.

11.3.

K

Let

Let K

be a reductive algebraic

act on

K

affine K-variety

group and let

by inner automorphisms. K .

Then

11.4. let

NK(T)/ZK(T) Let

k

be a maximal torus of

is

K .

[20].

In this case one has

k[T] F , where

F

denotes the Weyl

.

~ , @ , [

and

~

be as in §6.

K = {g ( G I g o e = @ o g} .

is a Cartan s u b v ~ i e t y f o r is the "little Weyl group" R0

T

and let

If

is a Caftan subvariety of the

This result is due to Steinberg

precise information on the algebra of invariants group

T

k

k .

is a (not necessarily

Let

the affine K-variety ~ . W0

G

be the adjoint group of

~

and

Then it follows from the results of [13] that

Let

R0

In this case the group

be the set of roots of

reduced) root system and

W0

~

on

F=NK(a)/ZK(a) ~ .

is the corresponding

In this situation we have analogues of most of the results of §7 and §8.

Then

Weyl group.

262 11.5.

(k = C) .

work.

Let

K

The "polar representations"

p : K + GL(V)

and let

E

subvariety

be a polar representation

be a "Caftan subspace" of

of the affine K-variety

and 11.4 above.

V .

Let G

(See [18]). P :

V

of the reductive

in the sense of [7].

These examples

They also include the r e p r e s e n t a t i o n s

Other examples of polar representations 11.6.

of Dadok and Kac [7] fit into our frame-

Let

e : G + G

{ g e ( g )-I I g ( G} .

and is K-isomorphic

[18] for definition).

to

G/K .

Then

A

P

group

is a Caftan

include the examples of 11.2 considered by Vinberg

automorphism

is a closed irreducible

Let

E

in

[91].

are given in [6] and [7].

be an involutive

Then

algebraic

Then

A

be a maximal

is a Cartan subvariety

and let

K-stable

-anisotropic

K = G .

subvariety

torus of

of

G

of the affine K-variety

(see P .

§12. More on the De Concini-Procesi problem After the manuscript C. Procesi concerning of M. Kashiwara strengthening Let

e

concerning

Let

involution on

g

e : G ÷ G

X

be an involution g

by

0 •

of

e

to

g-o

Lie subgroup of G0/K 0

G

is a Caftan involution with Lie algebra

is a Riemannian

Let

symmetric

g~

space.

of

~

and let Let

D(G0/K 0)

enveloping

T

y : Z(g) ÷ D(G0/K 0) .

is not surjective.

T

is equivalent

Proposition 12.1. and let

~

in

U(g)

g

of

X .

~

eigenspace Let

be the connected real Then the coset space

denote the algebra of all space

.

Go/K 0 .

Let

The homomorphism

(N2)

~i(~)

(NI).

denote

(G,e)

such

that the surjectivity

of

We have the following proposition:

Let

~

which contains

w 0 = NW(~)/Zw(~)

.

be a Cartan subspace of ~ . Let

~i

: t ÷ t/W

t/w .

be

Then the following four conditions

C [ t ] W ÷ C[a] WO given by restriction is surjective.

is a normal subvariety of

Z(g)

C

Then there exists a canonical

are equivalent: (NI)

K = Ge .

In fact, it turns out that the surject-

Let the notation be as above.

be a Cartan subalgebra of

-i .

such that the restrie-

It is known that there are pairs

to our condition

the quotient morphism and let

g GO

It was suggested by Kashiwara

might be a test for the normality

ivity of

algebra

denote the G'p

K0 = GO n K .

the centre of the universal

that

led to a

We also denote the corres-

of

Let

operators on the symmetric

%

~

g~

linear partial differential

homomorphism

a suggestion

This suggestion

G •

denote the closure of the orbit

It is known that there exists a e-stable real form tion of

Procesi mentioned

6 w h i c h we indicate below.

of the Lie algebra

and let

In this letter,

the problem posed in [i0, p. 8].

of the results of

k = C .

ponding of

for this paper had been typed, we received a letter from

the results of §6.

288

is integrally closed,

(N3)

C[X] G

(N4)

T : Z(K) + D(G0/K 0)

Proof.

is surjective.

The equivalence of (NI), (N2), and (N3) is given by Theorem A.

The equivalence

of (NI) and (N4) is an easy consequence of a theorem of Helgason [23, p. 590, Prop. 7.4].

REFERENCES i.

Borel, A.:

Linear Algebraic Groups.

2.

Borho, W.: (1981).

Uber Schichten halbeinfacher Lie-Algebren.

3.

Borho, W., Kraft~ H.: Uber Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen. Comment. Math. Helv. 54, 61-104 (1979).

4.

Bourbaki, N.: 1968.

Groupes et alg@bres de Lie, Chapitres 4, 5, et 6. Groupes et alg~bres de Lie, Chapitres 7 et 8.

5.

Bourbaki, N.:

6.

Dadok, J.:

New York:

Benjamin 1969. Invent Math. 65, 283-317

Paris:

Paris:

Hermann 1975.

Polar coordinates induced by actions of compact Lie groups. Polar representations.

Hermann

To appear.

7.

Dadok, J., Kac, V.:

8.

Dieudonn6, J.: France 1974.

To appear.

9.

Helgason, S.: Differential Geomtry, Lie Groups and Symmetric Spaces. San Francisco - London: Academic Press 1978.

i0.

Hotta, R., Kawanaka, N. (ed.): Open Problems in Algebraic Groups, Proceedings of the Twelfth International Symposium, Division of Mathematics, The Taniguchi Founction. Conference on "Algebraic Groups and their Representations," Kotata, Japan, Aug. 29 - Sept. 3, 1983. (Copies available from R. Hotta, Mathematical Institute, Tohuku University).

ii.

Howlett, R.: Normalizers of parabolic subgroups of reflection groups. Math. Soc. (2) 21, 62-80 (1980).

12.

Kostant, B.: Lie group representations on polynomial rings. 327-404 (1963).

13.

Kostant, B., Rallis, S.: Orbits and representations associated with symmetric spaces. Amer. J. Math. 93, 753-809 (1971).

14.

Kraft, H., Procesi, C.: Closures of conjugacy classes of matrices are normal. Invent. Math. 53, 227-247 (1979).

15.

Kraft, H., Procesi, C.: On the geometry of conjugacy classes in classical groups. Comment. Math. Helv. 57, 539-602 (1982).

16.

Peterson, D.: Geometry of the Adjoint Representation of a Complex Semisimple Lie Algebra. Ph.D. Thesis, Harvard University 1978.

17.

Richardson, R.: An application of the Serre conjecture to semisimple algebraic groups. In: Algebra, Carbondale, 1980. Lecture Notes in Math. 848~ 141-151 (1981).

18.

Richardson, R.: Orbits, invariants and representations associated to involutions of reductive groups. Invent. Math. 66, 287-312 (1982).

19.

Slodowy, P.: Simple singularities and simple algebraic groups. Math. 815 (1980).

20.

Steinberg, R.: Regular elements of semisimple algebraic groups. I.H.E.S. 25, 49-80 (1965).

Cours de g@om~trie alg6brique, 2.

Presses universitaires de New York -

J. London

Amer. J. Math. 85,

Lecture Notes in Publ. Math.

264

21.

Vinberg, E.: The Weyl group of a graded Lie algebra. 463-495 (1976).

Math. U.S.S.R. - Izv. i0,

22.

Vust, T.: Op~ratlon de groupes r~ductifs dans un type de cSnes presque homog~nes. Bull. Soc. Math. France 102, 317-334 (1974).

23.

Helgason, S.: Fundamental solutions of invariant differential operators on symmetric spaces. Amer. J. Math. 86, 565-601 (1964).

UNIPOTENT ELEMENTS AND PARABOLIC SUBGROUPS OF REDUCTIVE GROUPS. II Jacques TITS I. Introduction

Let K be a field of characteristic

p and G a reductive group over K. In [3],

A. Borel and the author showed that if K is perfect,

then

(U) every unipotent subgroup (i.eo subgroup consisting of unipotent elements) of G(K) is contained in the unipotent radical of a K-parabolic

subgroup

of G. Furthermore, ~ame a a ~ t l o n

we conjectured

that, if G is ~uasi-simple

and simply connected,

the

holds when~p is not a torsion prime for G (i.e. p is any prime if G has

type A n or Cn; p ~ 2 if G has type Bn, D n or G2; p ~ 2, 3 if G has type F4, E 6 or E7; p ~ 2, 3, 5 if G has type ES). That conjecture will be proved in Section 2 (cf. Corollary 2.6).

Pairs (K,G) for which (U) is false are dealt with in the remaining sections, where we go a long way towards determining all of them in the case where G is split.

(This restriction is less serious than it may seem; indeed, if (U) is false

for G over K, it remains false over the separable closure of K, over which G splits: cf. [3], 3.6). Let us be more specific. We say that an element u of a reductive K-group H is anisotropic

(in that group) if it is contained in no proper K-parabolic

subgroup of H; by [2], (2.20), this is so if and only if the projections of Ad u in all K-simple factors of the adjoint group Ad H are anisotropic.

It is not difficult

to see that (U) is false if and only if there is a K-split torus in G whose centralizer possesses a K-rational anisotropic element of order p (cf. Corollary 3.3). Thus, the problem of determining all pairs (K,G) for which (U) does not hold is roughly equivalent to that of finding all (K,G) where G is K-simple and has a K-rational anisotropic element of order p. Until the end of this introduction, we

266

shall assume that G is quasi-simple and K-split. Under these conditions, we conjecture that any anisotropic element of order p in G(K) normalizes a maximal K-split torus of G. Observe that if an element of the normalizer N of a maximal K-split torus T is anisotropic, it fixes no nontrivial rational character of T. Elements of N with that property have been studied by T.A. Springer [8]; we call them special. If the above conjecture is true, it reduces our problem (in the case of split groups) to that of determining, for all G and K, which special K-rational elements of order p are anisotropic. This question seems to be of the level of an exercise, which we solve here in the case where G is a classical group (cf. 3.5, 4.2, 4.3, 4.4). As for the conjecture itself, we are able to assert that it is indeed true, except possibly if G has type E 8 and p = 3 or 5. If one forgets about the easy case of type A n (cf. 3 . 5 ) a n d

the difficult (and unsolved) one of type E 8

in characteristic 5, the only characteristics to be considered are p = 2 and 3. For p = 2, the conjecture follows from a short and rather standard argument (cf. 4.1). The geometric proof I can propose for p = 3 (and, so far, G ~ E 8) is a case analysis and requires the knowledge of specific properties of the buildings of exceptional groups and special features of the triality in characteristic 3; it is too long to be given here and will be published later (unless a better proof is found meanwhile).

When dealing with anisotropic elements, it appears more natural to take a slightly more general viewpoint and to consider anisotrgpic automorphisms, that is, K-automorphisms which stabilize no proper K-parabolic subgroup. Our conjecture (and ~II we have said about it) extends to that situation. Similar generalizations could be contemplated for other results presented below, but they do not seem to bring much improvement. Furthermore, when our arguments do remain valid for arbitrary automorphisms (instead of inner ones), the prerequisite necessary to carry them out is not always readily available in the literature. So, we prefer to leave those generalizations aside.

The research which led to the results presented in this paper was motivated by a joint work with W. Kantor and R. Liebler [6]..1 thank A. Borel who agreed to my

267

using here the title of a common entreprise initiated in [3].

Throughout the paper, K, p have the same meaning as above, we suppose p ~ O, G is a reductive group defined over K and K (resp. K s ) denotes an algebraic (resp. a separable) closure of K.

2. Good and very good unipotent elements 2.1 We say that a unipotent element of G is very good if its schematic centralizer is smooth; in simple terms, this means that its centralizer in Lie G is the Lie algebra of its reduced (= group-theoretical)

centralizer in G. A K-rational

unipotent element is called good if it is contained in the unipotent radical of a parabolic K-subgroup. By [3], 3.6, G possesses property (U) of the introduction if and only if all unipotent elements of G(K) are good.

2.2. The following easy proposition describes the behaviour of those notions under central isogenies. Let ~ : ~ + G be a central isogeny, Let ~ be a maximal torus of ~, set T = ~(~) and let X, ~ be the (absolute) character groups of T, ~. The cokernel of the homomorphismX ÷ ~ induced by ~ is a finite group whose order c is called the degree of 7. For any integer d and any algebraic group H, let ~(d,H) denote the d-th power morphism x ~ x d of H into itself.

PROPOSITION.

(i) If n is an integer divisible by c, there exists a unique

K-morphism ~n: G + ~ such that ~nO~ = ~(n,~); one has ~n(IG) = I~, ~o~ n = ~(n,G) and, for all d E ~ , ~n°~(d,G) = ~(d,~)O~n.

(ii) The map ~ injects the set ~(K) ....

set G(K) .....

of all unipotent elements of ~(K) in the U

of a l l u n i p o t e n t e l e m e n t s of G(K) and b i j e c t s

t h e s e t o f a l l good

U

unipotent elements of ~(K) onto the set of all good unipotent elements of G(K).

(iii) I f c is prime to p, ~(~(K) u) = G(K) u and z maps the set of all very good unipotent elements of ~ bijectively onto the set of all very good unipotent elements of G.

268

(i) The first ass e=t£on follows from the fact that ~(n,~) is constant on the (schematic) fibres~of W, which is surjective. The relation ~n(1) = I is obvious. Finally, applying the first and last terms of the following two sequences of equalities ~O~nO~ = ~o~(n,~) = ~(n,G) o~ and ~no~(d,G)ow

=

~nO~O~(d,~)

=

~(nd,~)

=

~(d,~)o~no~

to a generic point of ~, whose image by ~ is a generic point of G, we get the two last assertions of (i).

(ii) The first aSSertion is clear since the kernel of w

in ~(K) has order

prime to p, and the second foll0ws from [2], 2.15 and 2.20.

(iii) Suppose c prime to p, choose an integer n divisible by c and congruent I modulo the largest order of a unipotent element of G, and let ~n be as in (i). By (i), the orders of the elements of ~n(G(K) u) are powers of p; in other words, ~n(G(K)u) = ~(K) u. Since ~O~n = ~(n,G) is the identity on G(K)u, it follows, again by (i), that G(K) u = w(~(K)u). The opposite inclusion being obvious,the first assertion of (iii) is proved. The second one readily ensues since, as a consequence of the assumption made on c, (d~) I is an isomorphism of Lie ~ onto Lie G.

2.3. We recall that p is said to be good for a quasi-simple K-group H if it does not divide the coefficients of the basic roots in the dominant root of the root system of H; this means that p # 2 if H is not of type An, p # 3 if H is of exceptional type (G2, F 4 or E i) and p ~ 5 if H is of type E 8.

THEOREM (Richardson-Springer-Steinberg: cf. [9], 35). Suppose p is good for all ~uasi-simple normal subgroups of G and no such subgroup has type Akp_| for some k E ~.

Then, all unipotent elements of G are very ~ood.

2.4. PROPOSITION. Let u be a unipotent element of G(K) and let P be a K-parabolic subgroup of G. Suppose P contains the reduced centralizer ZG(U) of u in

269

G and its Lie al~ebra Lie P contains the centralizer ZLi e G(U) of u in Lie G. Then, for g E G(K), if gu is K-rational, the subgroup gP is defined over K.

Let U be the unipotent radical of a K-parabolic subgroup opposite to both P and gP, so that gP = Vp for some v 6 U(K), and let X be the conjugacy class of u in P. The set UPu is an open subvariety of the conjugacy class Gu of u in G, and its contains gu since g 6 vP. Let us show that the map ~: (y,x) ~ Yx of U × X into UP

u is an isomorphism of algebraic varieties. The group U x p operates transitively

on U × X by (u',p).(u",x) = (u'u", x p

-1

) and on

Up u

by (u',p).z -

U'zp-1

, and those

actions are compatible with ~. Therefore, we only have to show that ~-1(u) = {(1,u)}, which simply amounts to our hypthesis ZG(U) c P, and that the differential of ~ at the point (l,u) is injeetive. But if ~ E Lie U and p £ Lie P are such that u p is tangent to X at u and that d~(1,u)(~,u p) = (Ad u)M + p -

U

- 0,

we have p = 0 and ~ £ ZLi e u(U) c Lie U N Lie P = {0}. This establishes our assertion on ~. Now, it is clear that ~, hence also ~

-I

, is a K-isomorphism and

that v is the projection of ~-|(gu) in the factor U of U x X. Therefore, if gu C G(K), we have v 6 U(K) and the group gP = Vp is defined over K, q. e. d.

Remark. It seems plausible that the hypotheses made on u in the above proposition imply that the schematic centralizer S of u in G is contained in P. If it is so, the conclusion of the proposition itmnediately follows. Indeed, the inclusion of g in P induces a K-morphism of G/B, hence of the conjugacy class of u, onto G/P, hence onto the conjugacy class of P, and that morphism is nothing else but gu ~ qp.

2.5. THEOREM. Let u E G(K) be a unipotent element. Suppose that one of the following conditions is satisfied: (1)

u is very good;

(ii)

G is simply connected of type A;

(iii)

G is simply connected of type C;

270

(iv)

p ~ 2 and G is of type G 2.

Then, there exists a K-parabolic subgroup P o f G, stable under all automorphisms of G(K) fixing u (the group Aut G(K) acts on the set of all K-parabolic subgroups of G by the main theorem of [4]), and whose unipotent radical contains u. In particular, u is good. Any P with the above properties contains ZG(U).

The last assertion is obvious since, for any element z of ZG(U), the inner automorphism Inn z fixes u, therefore Zp = p, hence z E P.

If there exists a Ks-parabolic subgroup P having the desired properties, it is defined over K; indeed, Aut (Ks/K) , which operates on the whole situation, fixes u, hence P by hypothesis. Therefore, we may, and shall, assume that K = Ks, which implies that G is split. Without loss of generality, we also assume that G is defined and split over the prime field F of K, whose algebraic closure in K is denoted by ~; this is an algebraically closed field. Since G has only finitely many conjugacy classes of unipotent elements (cf. [7]), each one of them meets G(~). In particular, u is conjugate (in G(K)) to an element u' of G(~). Set u = gu' with g C G(K). By [3], 2.5, and the main theorem of [4], there exists a parabolic subgroup P' of G stable under all automorphisms of G(K) which fix u' and whose unlpotent radical contains u', Being stable by Aut (K/F), P' is defined over F. We now distinguish cases.

Case (i). By hypothesis, ZG(U') is contained in P'. Since u, hence also u', is very good, it follows that ZLi e G(U') = Lie ZG(U') c Lie P'. Proposition 2.4 now implies that P = gP' is defined over K and meets all our requirements.

Case (ii) o The group G is F-isomorphic to SL

n

for some n. By the Jordan normal

form theorem, u and u' are conjugate in G(K). In other words, we can take g in G(K) and, again, P = gP' has all the desired properties.

Case (iii) (resp. (iv)). In this case, it is well-known that G can be embedded in a simply connected K-group H of type A (resp. D 4) as the fixed-point group of an

271

outer automorphism o of order 2 (resp. 3) such that the K-parabolic subgroups of G are precisely the

intersections

with G of the O-stable K-parabolic subgroups of H

and that every automorphism of G(K) extends to an automorphism of H(K) fixing o. Now, the assertion follows from case (ii) (resp. case (i) and Theorem 2.3, which implies that u is very good in H) applied to H.

2.6. COROLLARY. Suppose G is semi-simple and simply connected. Then~ if p is a torsion prime for no quasi-simple direct factor of G, all unipotent elements of G(K) are gpod.

This is an immediate consequence of Theorem 2.5, in view of Theorem 2.3.

2.7. Remark. If u is a good unipotent element of G(K), there exists a K-parabolic subgroup of G whose unipotent radical contains u and which is stable under all automorphisms of G(K s) fixing u (cf. [3], 2.5), but it may happen that no such K-parabolic subgroup is normalized by (i.e. contains) the centralizer of u in G(K): an example will be seen in 4.3.5.

3. Bad and anisotropic unipotent elements

3.1. We say that a K-automorphism or a K-rational element of G is anisotropic if it normalizes no proper K-parabolic subgroup of G, and that a unipotent element of G(K) is bad if it is not good. We shall see (Corollary 3.3) that the existence of bad unipotent elements and the existence of anisotropic elements of order p are closely related phenomena. Clearly, any nontrivial anisotropic unipotent element is bad (and even especially bad !).

3.2. PROPOSITION. Let u be a unipotent element of G(K), let P be a K-paraboli! subgrou~ containing u, let L be a Levi subgrou p of P defined over K, so that P = R u(P) ~ L, and let u' be the projection of u in L with respect to that product decomposition.

(A) The following properties are equivalent:

272

(i)

u is good in G;

(ii)

u' is ~ood in G;

(iii)

u' is good in L.

(B) l_~fP is minimal among

all

K-parab0.1ic subgroups of G containing u,

then u' is anisotropic and its order is the smallest power q o__~fp such that u q i~s

~99d. (A) If Q is a K-parabolic subgroup of G whose unipotent radical contains u, u' is contained in the unipotent radical of ((Q n P).Ru(P)) A L, which is a K-parabolic subgroup of L (cf. [I], 4.4, 4.7), hence the implication (i)~(iii) of which the implication (ii) ~ (iii) is a special case (taking u = u'). Conversely, (iii) implies (i) and (ii) because if u' is contained in the unipotent radical of a K-parabolic subgroup PI of L, both u and u' are contained in the unipotent radical of the K-parabolic subgroup PI.Ru(P) of G.

(B) Suppose the hypothesis of (B) satisfied. If PI is any K-parabolic subgroup of L containing u', the parabolic subgroup PI.Ru(P) of G contains u, and the minimality assumption implies that PI = L, hence the first assertion. Let q be any power of p. By (A), u q is good if and only if U 'q is good in L, which happens only if u 'q = I. Indeed, if u 'q was good and different from I, its centralizer in L and, in particular u', would be contained in a proper K-parabolic subgroup of L (cf. [3], 3.1). This finishes the proof.

3.3. COROLLARY. A necessary and sufficient condition for the ~roup G(K) t__oo contain a bad unipotent element is the existence of a split K-torus in G whose centralizer possesses an anisotropic element of order p.

The condition is necessary by 3.2 (B), applied to any bad element of G(K) whose p-th power is good. The converse readily follows from 3.2 (A).

3.4. Remark. Since bad elements remain bad after separable extensions of the ground field and since G splits over such an extension, the investigation of bad

273

unipotent elements in arbitrary reduetive groups is, to a large extent, reduced by Corollary 3.3 to the investigation of anisotropie elements of order p in semisimple groups.

3.5. Example: split groups of type A.

Suppose that the group G is split and quasi-simple of type A and that G(K) possesses an anisotropic element u of order p. The adjoint group of G is the group PGL(V) for some K-vector space V. Let ~ be a representative in GL(V) of the canonical image of u in PGL(V). It is an anisotropic element of GL(V) whose p-th × power ~ is an element k of K (considered as a subgroup of GL(V)). Clearly, k does not belong to K p, otherwise, dividing ~ by ~k, we could assume that ~ has order p and the stabilizer of the space of all fixed points of ~ in V would be a k-parabolic subgroup of GL(V) containing ~. The same argument shows that G cannot be simply connected (which also follows from 2.5 (ii)). Now, V has a structure of K(~k)vector space defined by ~k.v = ~(v) for v 6 V. This vector space must have dimension I, otherwise the stabilizer in GL(V) of any nontrivial proper subspace of it would be a proper K-parabolic subgroup of GL(V) containing ~. Therefore, dim V = p and G = PGL(V). Conversely, for any k C K - K p, the canonical image in P PGLK(K(~k)) of the multiplication by ~k is an anisotropic element of order p. We conclude that

a split quasi-simple group of type A

possesses anisotropic elements of order

p if and only if n = p-l, G is adjoint and K is not perfect.

3.6. The next lemma and the proposition which follows remain valid in characteristic zero.

LEMMA. Let H be a reductive subgroup of G defined over K.

(i)

For every K-parabolic subgrou p P of H, there exists a K-para-

bolic subgroup Q of G whose unipotent radical contains that of P and such that eye r~ K-automorphism of G stabilizing P also stabilizes Q.

274

(ii) The automorphism

of H induced by any anisotropic

K-automorphism

of G stabilizing H is also anisotropic.

(i) Let S be a maximal the set of all roots of G relative coefficients

of the weights

split torus of the radical of P and let ~ be to S which are linear combinations

of S in the Lie algebra of Ru(P). Then,

with positive

the group

Q = G , with the notation of [I], 3.8, clearly has the desired properties.

(ii) is an immediate consequence

4. Anisotropic

involutions

4.1. PROPOSITION.

Let ~ be an anisotropic

Then, for any K-parabolic

there exist maximal

involutory K-automorphism

subgroup P of G, P and ~(P) are opposite.

torus stable by ~ is contained

in a maximal

In the spherical building

into a Euclidean

(loc.cit.,

sphere of radius

invariant I (cf.

3.1, 8.1). To each facet of I, let us assign

If P and ~(P) were not opposite

(a process which is usually

(which implies that P ~ G), the middle

point of the geodesic joining the centers of gravity of the corresponding would belong to the facet corresponding

[5],

strictly smaller than ~ are

a "center of gravity" defined by some covariant process not unique).

the automorphism

-I

any two points of the building at distance

joined by a unique geodesic

Every split

I of G over K, we introduce a distance

under G(K) and making each apartment

of G.

split torus stable by ~; in particular,

split tori stable by ~. If G is semi-simple,

induced by ~ on any such torus is t ~ t

§8). Then,

of (i).

to a parabolic

subgroup

facets

stable by ~. Hence

the first assertion.

If S is any split torus stable by ~, and if P denotes a minimal K-parabolic subgroup containing

S, the intersection

P N ~(P) is a Levi subgroup of P stable by

~, whose center contains a unique maximal contains

S.

split torus which is stable by ~ and

275

Finally, if G is semi-simple and if S is a maximal split torus stable by ~, the fact that ~ transforms each K-parabolic subgroup containing S into an opposite parabolic subgroup implies that ~ multiplies by -I all relative roots of G with respect to S, hence all characters of S. This means that ~ transforms each element of S into its inverse. The proof is complete.

Throughout the remainder of this section, we suppose G quasi-simple and split, and p = 2, and we denote by T a maximal split torus of G. The above proposition suggests to study the involutory K-automorphisms of G stabilizing T and inducing the automorphism t ~ t-| on it, and to find out which one of them are anisotropic. We shall do that for all groups of classical type.

4.2. Groups of type A.

Suppose G of type An_ I and set I = {I, ..., n}. We identify the adjoint group of G with the group PGL(V) of some vector space V in which we choose a coordinate system (xi)iE I such that the canonical image of T in PGL(V) is also the canonical image of the group of all invertible diagonal matrices in GL(V). Any involutory K-automorphism ~ of G stabilizing T and inverting its elements is "represented by" a nondegenerate symmetric bilinear form a: V x V ÷ K of the shape

a: ((xi)iEl,(Yi)iC I) ~ I aixiY i

(all a i ~ 0)

in the following sense: the form a defines an isomorphism of V onto its dual, hence an automorphism of PGL(V) which lifts uniquely to the automorphism ~ of G; proportional forms a define the same automorphism ~. Now, consider a flag V I ~ V 2 ~ ... ~ V r in V, with V I ~ {0} and V r ~ V. The K-parabolic subgroup of G defined by that flag is stable by ~ if and only if, for j C {I, ..., r}, the space Vr+1_ j is the orthogonal V~ of V. with respect to the form a, in which case the J J 2 quadratic form I a.x. vanishes on all V. for 2j ~ r+1. Conversely, if that form iI J vanishes on some nontrivial proper subspace Y of V, the K-parabolic subgroup of G corresponding to the flag {Y, Y±} is stable by ~. Consequently:

276

the involution ~ defined by the form a is anisotropic coefficients

a i are linearly independent

if and only if the

over K 2.

4.3. Grou~s of types B and D.

4.3.1. nonzero

Suppose G of type B

integers

i with

with a group PGO°(V,q),

or D

m

m

and let I be the set of all integers or all

lil <~ m accordingly. where the exponent

° means "connected component of the

identity",

V is a (2m+I)- or 2m-dimensional

degenerate

quadratic

form of m ~ i m a l

system (xi)iE I with respect

similitudes

space over K and q is a non-

to which q = I x_l.x., i where i runs from 0 or I to m,

diagonal matrices

stabilizing T and inverting

vector

Witt index in V. In V, we choose a coordinate

and such that the image of T in PGO°(V,q) all invertible

We identify the adjoint group of G

coincides with the iraage of the group of

preserving q. The K-automorphisms

its elements are induced,

in an obvious sense, by the

of the form

~:

(xi)iE I ~+ (Yi)iCl

where the a. are nonvanishing

with Yi = a-ix-i'

constants

such that the product c = a .a. (the ratio

I

of similitude)

--i

does not depend on i. Let such ~ and ~ be given.

we assume, without

loss of generality,

element u of the ad~oint group~

I

If 0 E I (case B m),

that a 0 = c = I.

4.3.2. The automorphism ~ is "algebraically

condition

c~ of G

inner",

that is, it comes from an

if and onl>r if dim V ~ 2 (mod 4). Assume that

satisfied and suppose that G = Spin (V,q)

(resp. O°(V,q));

image of an element of G(K) - that is, ~ is the inner automorphism

then, u is the

corresponding

to such an element - if and only if c and the product of all a i belong to K 2 (resp, if c f K2).

The first assertion

simply recalls under which condition -I belongs

to the

Weyl group.

Assume dim V ~ 0 (mod 4) and G = Spin (V,q)

(resp. O°(V,q)).

The involution

277

~I: (xi)i£1 ~ (Yi)iEI

with Yi

=

x-i

is the image in O°(V,q) of an element of Spin (V,q)(K) since this is already true over the prime field. Therefore, u "comes from" G(K) if and only if the image t of

~I e: (xi)iEI

(aixi)iEl

in PGO°(V,q) does. Observe that t belongs to the canoni6al image ~ of T in that group. Now, our assertion follows from the known fact that the character group of T is generated by the character group of ~ and two (resp. one) additional m character(s) ½ X , ~ X'" (resp. ½ X) such that X(t) = c and X'(t) = ~ a.. i=I l The case where dim V m

I (mod 2) is similar but simpler.

4.3.3. The following properties are equivalent:

(i)

the automorphism ~ is isotropic (i.e. not anisotropic);

(ii)

there exists a nonzero vector ~ 6 V which is singular (i.e...

q(~) - O) and orthogonal to ~(~) with respec..t,to the symmetric bilinear form associated to q;

(iii) the equat.ions

(I)

l (~ i£I i>O

+ c~)a i =

E ~i~iai = 0 i61 i>O

have a nontrivial solution with ~0 = nO (if 0 6 I). They imply

(iii') the ai, fo__~_ri 6 1 and i ~ O, are linearly dependent over the field K2(c).

Suppose now that c = I (i.e. that ~ belongs to the ortho~onal ~roup). Then~ the above conditions are also equivalent.......to:

278

(ii I) ~ fixes

a

nonvanishin$ singular vector;

(iii I) the ai, for i C I and i ~ O, are linearly dependent over the field K 2 .

If ~ is any e l e ~ n t by ~, it satisfies

of V - {O} contained in a totally singular subspace stable

(ii). Conversely,

if ~ satisfies

(ii), the linear span of ~ and

~(~) is a totally singular subspace stable by ~. This proves the equivalence (i)~-e(ii). Assertion

(iii) is just a reformulation of (ii), setting ~ = (~i)iEl

and H i = ai1~ i, and the implication all ~2i + crli 2 vanish,

(iii) ~ (iii') is clear: just observe that if

then c is a square, say c = c' 2, and the second equality (i)

2 becomes c'.l~ia i = 0.

~2 Now, suppose c = I, that is, ~ = I. Then, if ~ satisfies fixed by ~ , or ~ + ~(~) is a nonvanishing

singular vector fixed by ~; therefore,

(ii) implies (iil) , and the converse is obvious. Clearly, equivalent;

in particular,

(ii), either it is

(iii') and (iii I) are now

(iii) implies (iiil). To prove the converse,

suppose

2 that l~ia i = 0 for ~ome ~i C K, not all zero, with i C I, i ~ O. Then (iii) is satisfied by setting H i = ~i for all i. The proof is complete.

4.3.4. LE~VIA. S__u22ose dim V m 0 (mod 4) and c = I. l__n_nV @ K, let Y be the space of all singular vectors fixed by ~ and let Y0 be the space of all y + ~(y) for y E Y±. Then dim Y = m-l, dim Y0 = I, Y0 c Y and the only totally singular subspaces stable by the centralizer of ~ in G are ~0}, Y0' Y' and the two maxlmal

totally

singular subspaces containing Y.

We assume, without loss of generality, appropriate change of coordinates,

that K = K, that G = 0°(V,q) and, by an

that all a. are equal to I. Then, Y is defined by l m

the equations x . = x. (i E {I .... ,m}) and I x. = 0, its orthogonal "f~ is defined --i 1 i i=I by x_i + x i = x_j + xj for all i, j, and Y0 is the space of all vectors all of whose coordinates are equal. Hence the three first assertions of the lemma° For the remainder of the proof, we content ourselves with two simple observations which the last assertion follows by routine arguments,

from

the detail of which is left

279

to the reader. Let Z denote the centralizer of ~ (or ~) in G.

I) The group H of all linear transformations of V expressed by an arbitrary invertible self-contragredient substitution of the x. (i > O) and "the same" i substitution of the x . is contained in Z; the only nontrivial proper linear -i subspace of Y stable by H is YO"

2) Let V' be a maximal totally singular subspace such that V' N Y = {0} and let b: V x V + K be the symmetric bilinear form defined by b(v,v') = q(v + ~(v')) for v,v' E V'. Then, the system (V',b) entirely determines the system (V,q,~,V') up to unique isomorphism. Since all pairs consisting of a vector space of dimension m and a nondegenerate, nonalternating symmetric bilinear form on that space are isomorphic,

it follows that Z permutes transitively each one of the two classes of

maximal totally singular subspaces intersecting Y only at {0}.

4.3.5. An example. Suppose dim V = 4m' and, for j E {I, ..., m'}, let V. be J the 4-dimensional subspace of V on which all coordinates vanish except x 2j, x_2j+1, x2j_1 and x2j. Thus, V is the direct sum of the Vj's. Suppose further that c = I and that, for all j E ~I,

..., m'}, a2j_1 = a2j. By 4.3.2, this implies that ~ is

the inner automorphism of G corresponding to an element u of G(K) which belongs to the canonical image of Spin(V,q)(K)

in G(K) (if G = O°(V,q), u = ~). In Vj, the

space of all singular vectors fixed by ~ is the one-dimensional

subspace Z. defined J

by the equations x 2 j z x 2j+i = a2jx2j_1 = a2jx2j ° Let Z be the sum of all Zj's. It is easily checked that if v f V.j is orthogonal to Zj, then ~(v) E Z.j + v. Therefore, if v C V is orthogonal to Z, then ~(v) E Z + v. This shows that all subspaces of V orthogonal to Z and containing Z are stable by ~. Consequently,

there exist

maximal flags of totally singular subspaces stable by ~ (just take the union of a ~ximal

flag of Z and a maximal flag of totally singular subspaces orthogonal to Z

and containing Z). In other words, u is good, which implies that the centralizer of u in G(K) is contained in a proper K-parabolic subgroup of G. On the other

hand,

if not all a i belong to a1.K2 , the spaces Y0 and Y of Lemma 4.3.4 and the two maximal totally singular subspaces containing Y are not "defined over K", therefore

280 the Lemma implies that no proper K-parabolic subgroup contains the centralizer of u in G. In particular, u cannot be very good, b~ Theorem 2.5; this could of course also be checked by direct computation, or deduced from Proposition 5.3 below.

4.4, Grbups of type C.

Now, suppose G of type C . We can repeat the preliminaries of 4.3.1, replacing m m PGO°(V,q) by PSp(V,a), where a is the alternating form I x . A x.. Let ~ be i=I -l i defined by the same equations as in 4.3 and let ~ be the automorphism of G that it induces. The proofs of the following assertions are similar to the proofs in 4,3.2 and 4.3.3 but simpler.

The involution ~ is an inner automorphism corresponding to an element of Sp(V,a)(K) if and only if c 6 K 2. In order that e be anisotropic,

it is necessarY

and sufficient that c ~ K 2 and that the a. be linearly independent over K2(c). I

4.5. The case [K:K 2] = 2.

PROPOSITION. Suppose [K:K 2] = 2. Then G possesses an anisotropi 9 K-automQrphism of order 2 if and only if it is adjoint of type A I. If the group G is simply connected i all unipotent elements of G(K) are good.

Suppose G possesses an anisotropic K-automorphism ~ of order 2. We may and shall assume that ~ stabilizes the torus T and inverts its elements. For any closed symmetric subset ~ of the root system ~ of G relative to T, let G~ denote the corresponding semisimple subgroup of G normalized by T (this notation is not that of [I] used in 3.6 above). By 3.6 (ii), e induces an anisotropic automorphism of G~; therefore, ~ cannot be of type A 2 or B 2 (here, we use 4.2, 4.3.3 (iii I) and the hypothesis made on K). Since the root system ~ has no subsystem of type A 2 or B2, it must be of type A I and, by 3.5, G must be adjoint. Conversely, 3.5 also implies that "the" split adjoint group of type A I over K does have anisotropic automorphisms of order 2.

281

Now, suppose that G is.simply connected and let L be~the centralizer split torus in G. It is w e l l - k n o ~

that the derived group of L is simply connected,

hence a direct product of simply connected quasi-simple of the proposition, anisotropic

of a

already proved,

groups. From the first part

it follows that L contains no K-rational

element of order 2, and Corollary

3.3 implies our~second assertion.

5. Special elements

Until the end of th e paper, we assume G quasi-simple split over the prime field ~

of K; as before,

as well as defined and

T denotes a maximal split torus of

P G and X is the character

group of T.

5.1. Let W be the group of all automorphisms

of X preserving

the root system

of G; it contains

the Weyl group W. We say that an element of W is special if it

has no eigenvalue

I (that is, if it fixes no nontrivial

automorphism

of G normalizing

T or an element of the normalizer

(with respect to T) if its canonical

Then,

of T in G is special

image in W is special.

For any w 6 W, let N w denote the corresponding Suppose w is special.

element of X) and that an

the endomorphism

coset of T in its normalizer.

t ~ w(t).t -I Of T is surjective,

therefore all elements of Nw(K) are conjugate under T(K). It follows that any prime dividing the order of an element n of N

w

also divides the order of w, otherwise

there would exist a multiple k of I congruent have w k = w, hence n k 6 Nw, contradicting different.

I modulo the order of w, we would

the fact that the orders of n and n k are

If z is any central element of G, we have z.N w = Nw, therefore

of z divides that of the elements of N . In view of the preceding remark, w

the order this

implies that any prime number dividing the order of the center of G - or of any group having the same Weyl group - divides the order of any special element of W. In particular,

if w 6 W is a special element of order a power of p, the center of G (and that of the universal

covering of G) is purely infinitesimal

and the elements of N

have W

the same order as w.

-

-

28,2 We observe that all examples of anisotropic

unipotent elements we have met

thus far (cf. 3.5 and § 4) turn out to be special elements. by the following

two propositions

of bad unipotent

elements.

5.2. PROPOSITION.

Two more facts expressed

also point towards special elements

Let ~: ~ ÷ G be a central

isogeny,

let w be a special element

o f W of order a power of p and let u be an element of Nw(~p) , T I = ~(~(K))

- T(K). Then, the elements

as a source

set ~ = ~-I(T) a n d

of uT I (which are unipotent)

are all bad.

The set T I is empty if and only if K is perfect or ~ is an isomorphism.

Since ~

is perfect,

u is good (cf. [3], 3.1), hence contained

in ~(~(K))

P (cf. Proposition

2.2.

(ii)). Therefore uT I N ~(~(K))

= ~ and loc.cit,

implies that

no element of uT I is good.

Let ~ be the character the cokernel

group of ~. From the discussion

of the canonical map X ÷ ~ is a p-group,

only if ~0 is an isomorphism;

in 5. I, it follows that

and this group is trivial

the second assertion of the proposition

readily

X

follows,

since T(K) = Hom (X,K ×) and ~(K) = Hom(X,K ).

5.3. PROPOSITION,

The centralizer

in G of a special unipotent

automorphism

is

not reduced.

We omit the proof, observation,

except for making the following,

which can be considered

as its first step: if a unipotent

of G is special with respect to T, its group-theoretical whereas

its centralizer

schematic centralizer

5.4. CONJECTURE.

By Corollary

centralizer

in Lie T is at least one-dimensional,

automorphism

in T is finite

therefore

its

in the torus T is not reduced.

All anisotropic K-automorphisms

2.6, Theorem 4.1, the discussion

in the Introduction

trivial but suggestive

concerning

of G are special.

in 3.5 and the results announced

the case p = 3, the above conjecture

all cases except when G = E 8 and p = 3 or 5.

is proved in

283

Remembering Proposition3.2,

we may conclude in heuristic terms, and assuming

the truth of the above conjecture, that special elements of order p are "essentially the only source" of bad unipotent elements.

Coll~ge

de France, 11 Place Marcelin-Berthelot,

75231 Paris Cedex 05.

284

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