Conjugacy classes in algebraic groups (Lecture notes in mathematics 366)

Lecture Notes in Mathematics Edited by A. Dold, Heidelberg and B. Eckmann, ZCirich Series: Tata Institute of Fundamental Research, Bombay Adviser: M. S. Narasimhan

366 Robert Steinberg University of California, Los Angeles, CA/USA

Conjugacy Classes in Algebraic Groups Notes by Vinay V. Deodhar IIIIII

L¢

Springer-Verlag Berlin.Heidelberg. New York 1974

AMS Subject Classifications (1970): 14 Lxx, 20-02, 20-G-xx

ISBN 3-540-06657-8 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06657-8 Springer-Verlag New York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1974. Library of Congress Catalog Card Number 73-21212. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

INTRODUCTION

T h e f o l l o w i n g i s the s u b s t a n c e of a s e t of l e c t u r e s g i v e n a t the T a r a I n s t i t u t e of F u n d a m e n t a l R e s e a r c h d u r i n g N o v e m b e r and D e c e m b e r of 1972. T h e n o t e s a r e d i v i d e d r o u g h l y into two p a r t s .

T h e f i r s t p a r t a t t e m p t s an a p r i o r i d e v e l o p m e n t

of the b a s i c p r o p e r t i e s of affine a l g e b r a i c g r o u p s with e m p h a s i s on t h o s e n e e d e d in the s t u d y of c o n j u g a c y c l a s s e s of e l e m e n t s of r e d u c t i v e g r o u p s : the s e m i s i m p l e - u n i p o t e n t d e c o m p o s i t i o n , c o n j u g a c y of B o r e l s u b g r o u p s and of m a x i m a l t o r i , c o m p l e t e n e s s of the v a r i e t y of B o r e l s u b g r o u p s , e t c . d e v o t e d to the c l a s s i f i c a t i o n and elements:

c h a r a c t e r i z a t i o n of v a r i o u s

s e m i s i m p l e , unipotent,

regular, subregular, etc.

d e t a i l e d o u t l i n e the r e a d e r m a y c o n s u l t the t a b l e of c o n t e n t s . an a l g e b r a i c a l l y c l o s e d f i e l d .

The second part is s u c h c l a s s e s of For a more A l l of t h i s i s o v e r

I had p l a n n e d to i n c l u d e two t a l k s on r a t i o n a l i t y

q u e s t i o n s , but t h i s a i m was not r e a l i z e d .

B e c a u s e of t i m e l i m i t a t i o n s t h e r e

had to b

In the f i r s t p a r t the m o s t s e r i o u s of

gaps in the a c t u a l d e v e l o p m e n t .

t h e s e i s the o m i s s i o n of a l a r g e p a r t of the p r o o f of the e x i s t e n c e of a q u o t i e n t of a g r o u p by a c l o s e d s u b g r o u p .

A l s o the p r i n c i p a l s t r u c t u r a l and c o n j u g a c y

r e s u l t s about c o n n e c t e d s o l v a b l e g r o u p s a r e u s e d without p r o o f , but t h i s i s n o t s o s e r i o u s s i n c e the L i e - K o l c h i n t h e o r e m is p r o v e d and f r o m t h e r e on the p r o o f s , by i n d u c t i o n , f o l l o w f a i r l y c l a s s i c a l l i n e s .

In the s e c o n d p a r t the B r u h a t l e m m a

f o r r e d u c t i v e g r o u p s i s u s e d without p r o o f (but a f a i r l y c o m p l e t e p r o o f i s i n d i c a t e d f o r the c l a s s i c a l g r o u p s ) a s a r e v a r i o u s p r o p e r t i e s of r o o t s y s t e m s and r e f l e c t i o n g r o u p s (for which a c o m p r e h e n s i v e t r e a t m e n t m a y be found in B o u r b a k i ' s book).

Modulo a f e w o t h e r p o i n t s l e f t to be c h e c k e d by t h e r e a d e r

I have attempted a coherent development.

IV

It i s a p l e a s u r e to thank m y c o l l e a g u e s a t the T a r a I n s t i t u t e , youn~ and old, f o r t h e i r h o s p i t a l i t y and f r i e n d s h i p to m y wife and m e d u r i n g o u r v i s i t and f o r t h e i r s t i m u l a t i n g i n f l u e n c e on m y t a l k s .

It i s a s p e c i a l p l e a s u r e to be

a b l e to thank h e r e S h r i V i n a y V. D e o d h a r who in a d d i t i o n h a s w r i t t e n up these notes.

Robert Steinberg U n i v e r s i t y of C a l i f o r n i a

TABLE

Chapter

OF CONTENTS

Affine algebraic varieties, affine algebraic groups and their orbits

1.1

Affine algebraic varieties

1.2

Morphisms

1.3

Closed subvarietie s

1.4

Principal open affine subsets

1.5

A basic l e m m a

1.6

Product of varieties

1.7

Notion of affine algebraic groups

.............

5

1.8

Comorphisms

.............

5

1.9

Linear algebraic groups

.....................

7

i . I0

Zariski-topology on varieties

I. I I

Noetherian spaces

1.12

Irreducible c o m p o n e n t s of an algebraic group

1.13

Hilbert's second t h e o r e m

Chapter

II

...................

1

......................

2

........................

2

of varieties

................

2

...........................

3

........................

4

in algebraic groups

.................

9

...........................

i0

....

.....................

First Part: Jordan decompositions, dia~onalizable groups

12 14

unipotent and

2.1

Definitions and preliminary results

............

2.2

Jordan decomposition for an e n d o m o r p h i s m

2.3

Jordan decomposition for an e n d o m o r p h i s m (~tinued) .................................

26

2.4

Jordan decomposition for group-elements

29

2.5

Kolchin's ~ h e o r e m

2.6

Diagonalizable groups

2.7

Rigidity t h e o r e m

.....

......

22 24

...........................

33

.......................

36

...........................

43

VI

Chapter II

Second Part: Quotients and solvable groups

2.8

Solvable groups

..............................

2.9

Varieties

in general

2.10

Complete

varieties

2.11

Quotients

....................................

Appendix to

2.11

Borel subgroups

2.13

Density and closure

2.14

Bruhat lemma

Chapter

III

........................... and projective

2.12

46

t

°

,

.

~

.

.

~

.

°

.

.

.

.

°

°

°

°

.

°

°

~

.

.

.

°

°

.

.

varieties

.

°

i

47

~

o

e

O

0

O

l

O

.

.

o

6

*

~

l

.

.

49 54

o

.

......

O

O

O

,

.

O

.

.

.

.

.

.

°

.

~

.

.

.

°

°

.

.

°

.

°

6

.

.

,

58

o

,

61

°

................................

Reductive and sere,simple and subregular elements

algebraic

65 72

groups,

regular

3.1

Definitions and examples

3.2

Main theorem

3.3

Some representation

3.4

Representation

3.5

Regular

elements

............................

93

3.6

Unipotent classes

............................

100

3.7

Regular

elements

3.8

Regular groups

elements in simply connected, ......................................

3.9

Variety

of B o r e l s u b g r o u p s

3.10

Subregular

......................

on s e r n i s i r n p l e g r o u p s theory

...................

theory (continued)

(continued)

elements

............

..............

..................

76 77 79 83

110

sernisirnple

....................

.........................

116 128 140

A p p e n d i x on t h e c o n n e c t i o n w i t h K l e i n , a n singularities ................................

156

References

159

..................................

Chapter I Affine algebraic varieties,

affine algebraic groups

and t h e i r o r b i t s

Throughout this chapter,

1.1.

k w i l l d e n o t e an a l g e b r a i c a l l y

Affine algebraic varieties.

c o p i e s of k.

Classically,

Let

a subset

k n d e n o t e t h e c a r t e s i a n p r o d u c t of n V of k n i s c a l l e d an a l g e b r a i c s e t if it i s

t h e s e t of z e r o s of a s e t of p o l y n o m i a l s in k IX 1 . . . . . x 2 + y2 = 1, a l i n e in s p a c e ,

closed field.

Xn].

kn itself, the circle

e t c . a r e e x a m p l e s of s u c h s e t s .

But this notion is unsatisfactory

s i n c e it is not i n t r i n s i c .

H e n c e we d e f i n e an

( a b s t r a c t ) a f f i n e a l g e b r a i c v a r i e t y in t h e f o l l o w i n g way: It i s a p a i r ( V , A ) , w h e r e w i t h v a l u e s in k.

V i s a s e t and

A is a k-algebra

of f u n c t i o n s on V

This pair satisfies the following properties:

(1) A i s f i n i t e l y g e n e r a t e d a s k - a l g e b r a . (2) A s e p a r a t e s

p o i n t s of V i . e .

given

x ~ y ~V, there exists

f~A

such that

f(x) ~ f(y). (3) E v e r y

x cv

k-algebra homomorphism

i.e.

Remarks.

t h a t p o i n t (to b e d e n o t e d a s

Examples

(I)

; k i s t h e e v a l u a t i o n at a p o i n t

¢(f) -- f(x) V f , A .

B y (2), t h e p o i n t

respondence

~ : A

with the

x EV ex).

k-algebra

i s u n i q u e l y d e t e r m i n e d b y t h e e v a l u a t i o n at T h u s , t h e p o i n t s of V a r e in o n e - o n e c o r -

homomorphisms

of A i n t o

k.

of a f f i n e a l g e b r a i c v a r i e t i e s :

(kn, k Ix I ..... Xn] ). (It is called the affine space of dimension n).

-2-

(2) V ~ k n, a n a l g e b r a i c s e t in e a r l i e r s e n s e , A = k IX1, . . . , X n l / V • (3) Let A be a f i n i t e l y g e n e r a t e d k - a l g e b r a without n i l p o t e n t e l e m e n t s . t h e r e e x i s t s an i n t e g e r n ) 0 Let V = {(a 1 . . . . .

and an exact s e q u e n c e : 0 - - - ~ I - - ~ k [ X l , . . . , X n ] - - ~ A - - p 0 .

an)¢kn/g(al .....

a l g e b r a i c variety.

Then

an) = 0 ~ g ¢ I t

" T h e n (V,A) is a n a f f i n e

(This is a consequence of H i l b e r t ' s Nullstellensatz: see

c o r o l l a r y to lemma 1 of 1.13).

In fact, as we shall prove l a t e r , any affine

a l g e b r a i c v a r i e t y is obtained in this way.

1.2.

M o r p h i s m s of affine a l g e b r a i c v a r i e t i e s .

algebraic varieties.

Let

(U,A), (V, B) be affine

Then a m o r p h i s m f : ( U , A ) - - - ~ ( V , B )

i s a m a p f:U

> V

such that the a s s o c i a t e d m a p f* defined by the c o m p o s i t i o n with f, t a k e s

B

into A. f* is c a l l e d the c o m o r p h i s m a s s o c i a t e d to f.

Remarks.

(1) F o r

u£U,

the point f(u) E V i s given by: e

f(u)

=e

u

o

Thus

f is c o m p l e t e l y d e t e r m i n e d b y f*. (2) If f: (U, A)-----) (V,B) and g : (V,B)-----~ (W, C) a r e m o r p h i s m s of affine a l g e b r a i c v a r i e t i e s then so i s

i. 3.

gof:

(U,A)

v(W,C)

and

( g o f ) * = f * o g *.

Subvarieties of affine algebraic v a r i e t i e s . Let (V,A) be an affine algebraic

v a r i e t y and V ' C V .

If ( V ' , A / v , )

i s an affine a l g e b r a i c v a r i e t y i n i t s own r i g h t ,

then it i s c a l l e d a s u b v a r i e t y of (V,A).

It can be e a s i l y s e e n that

( V ' , A / v , ) is

a s u b v a r i e t y if and only if V' is the s e t of z e r o s of a s e t of e l e m e n t s i n A.

(The

n o v i c e should check this. )

1.4.

Principal open affine subsets.

and f £ A .

Then

Vf = ~ x ~ V l f ( x ) = % I

Let

(V, A) be an affine algebraic variety

ex(f)# 0 ~ is calleda principal open subset j

-3

-

of V. It can be seen that (Vf,Af) is an affine algebraic variety. Here Af= A~].

i . 5.

A basic lemma.

Here,

we p r o v e an i m p o r t a n t l e m m a w h i c h will be u s e d

q u i t e o f t e n l a t e r on.

Lemma.

Let

( U , A ) , (V, B)

be a morphism. then

f(U)

Proof.

Let

f*: B

is an algebraic

Let

0--~I

be affine algebraic varieties J,A

be the associated

s u b v a r i e t y of V a n d f

~B

,*A----~0

(e o f ~ ' ) ( g ) = O.

such that

quotients to

--e v : A - - - ~ k

algebraic variety,

let

v cV

such that

e v = e v o f,~

hence there exists

comorphism.

Then$or

as

B

separates

V.

is onto,

is an i s o m o r p h i s m .

gel, ef(u)(g) =

e v ( g ) = 0 V g ~ I. Now

u ~ U such that

(U,A) g

= e V

v = f(u)

If f*

> (V, B)

be exact.

Let ucU.

Conversely,

f : (U , A ) -

f : U - - ~ f(U)

Claim: f(U)=(v~Vlev(g) =0 Vggl~. U

and

This proves the claim.

Hence

Clearly,

ev

is an affine o f* = e .

V

Hence

U

f(U)

is an algebraic

subvariety of V and B/f(U)---~-~ ~. Hence (f(U), B) is a subvariety of (V, B). Clearly, there exists g~ : A --~ -B such that g'~of~ and f~' o g~ are respective I identities. The morphism g defined by g~ is such that g o f and f o g are

respective identities. Hence the lemma.

Proposition I. Every abstract affine algebraic variety is isomorphic to a sub-

variety of the affine algebraic variety (kn, k [Xl~...,Xnl ) for suitable n. Proof. Let (V,A) be an affine algebraic variety. A is finitely generated say

by fl ..... fn" Define ~ : V

~k n, given by: ~(v) = (fl(v)..... fn(V)). It can be

easily seen that the corresponding map Clearly ~* maps

~* is given by ~*(X i) = fi i~i$ n.

k [X1,...,Xn] onto A. Hence by the above lemma, ~(V) is

-4

a s u b v a r i e t y of k n and ~ : V - - ~

-

~(V) is an i s o m o r p h i s m .

Hence the

proposition.

1.6.

P r o d u c t s of affine a l g e b r a i c v a r i e t i e s .

algebraic varieties. UxV.

Then e l e m e n t s of A ~ B

Let

(U,A), (V, B) be affine

can be t r e a t e d a s f u n c t i o n s on

E x p l i c i t e l y , (a@b)(x,y) = a(x). b ( y ) , x e U, y ~ V .

T h e n (UY, V, A@B)

can be s e e n to be an affine a l g e b r a i c v a r i e t y , c a l l e d the p r o d u c t of iV, B).

H e r e , only p r o p e r t y (3) is to be v a r i f i e d .

k-algebra homomorphism. defined b y : exists

NOW,

~ : A @B----~k be a

T h i s gives r i s e to ~1 : A ~

~l(a) = ~ ( a @ l )

x~U,y~V

Let

suchthat

and ~2(b) = ~(1Ob); a £ A , ~1 = e x '

(U,A) and

k, ~2 : ]3 ~ b e B.

k

Hence t h e r e

~2 = ey.

( a ~ b ) ( x , y ) = a(x). b(y) = ex(a), ey(b) = ~l(a) . ~2(b) = ~ ( a ® b ) .

Thus

~ = e ( x , y ).

A g a i n the m a p s A c - - - ~ A ~ B , UxV

give r i s e to m o r p h i s m s

U~V ~ I

U;

b V which, in fact, a r e the p r o j e c t i o n s .

(UxV, A ~ B ) variety

Bc-~A~B

h a s the following u n i v e r s a l p r o p e r t y :

(W,C) and m o r p h i s m s

p l : (W,C)----~ (U,A) and p2 : (W, C)-----~ (V, B),

t h e r e e x i s t s a u n i q u e m o r p h i s m p : (W,C) 7f1 o p = P l ' ~:2 o p = P2'

Given an affine a l g e b r a i c

~ (U x V , A ~ B) such that

T h i s p r o p e r t y follows i m m e d i a t e l y f r o m a c o r r e s -

ponding u n i v e r s a l p r o p e r t y in t e n s o r p r o d u c t s of c o m m u t a t i v e a l g e b r a s o r e l s e can be v e r i f i e d d i r e c t l y .

As an e x e r c i s e the n o v i c e m a y wish to p r o v e the

i m p o r t a n t fact that each of the m o r p h i s m s

Pl" P2 above i s open (maps open

sets onto o p e n sets , in the Zariski topology defined in l.lO).

- 5

-

N o t i o n of a f f i n e a l g e b r a i c g r o u p s , An a f f i n e a l g e b r a i c g r o u p is a p a i r

1.7. (G,A)

such that

(1)

(G,A)

(2)

G is a group

(3)

The group operations are morphisms i.e.

i s an a f f i n e a l g e b r a i c v a r i e t y

m : GxG

~

G, m ( x , y ) = x . y

and

i: G

v G, i(x) = x

-1

are morphisms.

Examples (1)

Let

of a f f i n e a l g e b r a i c g r o u p s : V b e an n - d i m e n s i o n a l

(GL(V), k [2711 ,...,Tnn]D)

vector space over

where

(2)

Then

D is the determinant

k IT11 , ...,Tnn3:Dis the ring obtained from D -I

k.

of

(Tij)

and

k [TII ) ...,Tnn] by adjoining

(This will be discussed in I. 9).

SL(V)

is an algebraic subvariety of GL(V)

and is an affine algebraic

group in its own right. (3)

The group of diagonal m a t r i c e s in GL(V) as subvariety of GL(V) is an affine algebraic group.

(4)

The invertible elements of any finite dimensional associative k - a l g e b r a .

(The groups in (i), (2) and (3) a r e called l i n e a r algebraic groups. )

A l i n e a r algebraic group is an affine subvariety of GL(V), for some finite dimensional vector space V, which is a subgroup also.

i. 8.

Comorphisms in affine algebraic groups. Let (G, A) be an affine

algebraic group. ations.

Let m : Gx G - - ~ G and i : G----~G be the group o p e r -

These give r i s e to comorphisms m* : A

Consider the m o r p h i s m

: ~x : G -

- G

~ A ~ A and i*: A

~ A.

given by ~ x(y) = v x (= re(y, x)) for

-6

a fixed x ~ G .

-

This gives rise to a comorphism ~ y : A - - - ~ A .

This in fact is

a k-algebra automorphism of A, since {~x is an automorphism of G as a

A l s o , ~xy = ~y Oex , hence ~ *xy = ~*x ° ~y* .

variety.

Thus,

(~* : G - - - - > A u t o -

m o r p h i s m of A, _.2) a n d

F

let

r

F 1 C

) =F ItJ

irreducible.

F i is a closed F, F

is a c l o s e d

(F f ~ F 2) U . . . Hence

U(F

,~Fr).

F 1 ~ F C_F i f o r

-

some

i~2.

12

-

T h i s c o n t r a d i c t s t h e i r r e d u n d a n c y of V = F l U . . . U F r .

F 1 i s a m a x i m a l c l o s e d i r r e d u c i b l e s u b s e t of V. has the above property.

A g a i n , if F

then F -- (F ~ F 1 ) V . . . l ] ( F implies

F = F.. 1

f~ F r)

is a m a x i m a l closed i r r e d u c i b l e subset,

gives F ~ F. for some 1

i, w h i c h in t u r n

(V,A)

H e n c e the t h e o r e m ,

Corollary.

Let

is true for

V, e n d o w e d with the Z a r i s k i - t o p o l o g y .

Proof.

also

H e n c e a l l the m a x i m a l c l o s e d i r r e d u c i b l e s u b s e t s of V o c c u r

e x a c t l y once in F I ~ . . . U F r ,

Zariski)

Similarly Fi(i~2)

Hence

be an affine a l g e b r a i c v a r i e t y .

T h e n the a b o v e t h e o r e m

(In f a c t , we p r o v e : (V,

is n o e t h e r i a n ) .

A s a c o n s e q u e n c e of H i l b e r t ' s b a s i s t h e o r e m , we h a v e : E v e r y f i n i t e l y

generated k - a l g e b r a is noetherian.

H e n c e A is n o e t h e r i a n i . e . e v e r y i d e a l

of A is f i n i t e l y g e n e r a t e d (which is e q u i v a l e n t to s a y i n g : A s a t i s f i e s the m a x i m u m c o n d i t i o n on i d e a l s ) . With e v e r y c l o s e d s u b s e t

by: A

U of V, we a s s o c i a t e an i d e a l I(U) of A d e f i n e d

I(U) = {f ~ A/f(x) = 0 V x

vanishing on U.

EU~i.e.

Since U is closed, it follows that U

set of zeros of I(U). Thus, U ~ U', U I(U) ~ I(U'). Hence the m a x i m u m minimum

I(U) is the idealofall elements of is precisely the

and U' closed)imply that

condition on ideals of A

condition on the closed sets of V.

Thus

implies the

(V, Zariski) is noetherian.

Hence the previous theorem is applicable.

I. 12.

Irreducible components of an affine alogebraic group.

an affine algebraic group,

Let (G,A) be

Since G has a group s t r u c t u r e on it and the group

operations a r e m o r p h i s m s , the i r r e d u c i b l e components of G have a special

-13

nature which is described

Proposition

1.

Let

in t h e f o l l o w i n g p r o p o s i t i o n :

(G,A)

be an affine algebraic group.

c o m p o n e n t s of G a r e d i s j o i n t . containing i n d e x in

e, t h e n

G.

G°

Further,

-

If G °

is the irreducible

is a (closed)normal the irreducible

Then the irreducible c o m p o n e n t of G

s u b g r o u p of G havir~g a f i n i t e

c o m p o n e n t s of G a r e p r e c i s e l y t h e

c o s e t s of G° .

Proof.

L e t , if p o s s i b l e ,

two components intersect.

x b e l o n g s to two d i s t i n c t i r r e d u c i b l e G

as a variety permutes

components.

the irreducible

s l a t i o n b y a n e l e m e n t of G

component than

V.

V.

Since an a u t o m o r p h i s m

components.

This clearly contradicts the irredundancy

G are disjoint.

Let

components.

x 6G ° ; then

xG °

of

it f o l l o w s that e v e r y e l e m e n t Take an irreducible

E a c h of i t s e l e m e n t s b e l o n g s t o a n i r r e d u c i b l e

a u n i o n of i t s i r r e d u c i b l e

such that

components and since the left-tran-

is an automorphism,

of G b e l o n g s to two d i s t i n c t i r r e d u c i b l e

Hence 3 x ~G

component other

of t h e e x p r e s s i o n

Hence the irreducible

of G

as

c o m p o n e n t s of

is a l s o a c o m p o n e n t and it c o n t a i n s

x.

Hence by disjointness, xG ° = G ° . H e n c e G ° . G ° d G ° . F o r a s i m i l a r a r g u m e n t , -1 G° = G° , and for yEG arbitrary, y G ° y -1 = G ° . H e n c e G ° i s a n o r m a l s u b g r o u p o f G. Conversly,

let

x-lF

or

= G°

Remark

1.

G°

Clearly, F

its cosets are also irreducible

be an irreducible

F = xG ° .

c o m p o n e n t of G.

c o m p o n e n t s o f G.

Choose

x ~F,

then

Hence the result.

is the smallest

c l o s e d s u b g r o u p of G h a v i n g f i n i t e i n d e x in

(Any c l o s e d s u b g r o u p of f i n i t e i n d e x i s o p e n a l s o ) .

Remark

2.

(exercise).

If S i s a c l o s e d s u b s e m i g r o u p ,

t h e n it i s , in f a c t , a s u b g r o u p

G.

-

Remark

3.

e

Remark

4.

-

F o r an a l g e b r a i c g r o u p , the i r r e d u c i b l e

components are the same. identity

14

and call

G

We call

G°

components and connected

t h e ( c o n n e c t e d ) c o m p o n e n t of t h e

c o n n e c t e d if G = G ° .

A s an e x a m p l e , w e c o n s i d e r

c o n s i s t s of t w o c o m p o n e n t s .

The groups

G = On .

Here

G ° = SO n

G L n , S L n , SP2 n , D i a g ,

so that

G

Superdiag ....

on t h e o t h e r h a n d a r e a l l c o n n e c t e d .

1.13.

Hilbert's second theorem.

a finite one.

The assumption that

Until now k

k

could have been any field, even

is algebraically

closed will now be brought

into play.

Notation. algebra

Henceforth, A

we d e n o t e an a f f i n e a l g e b r a i c v a r i e t y

of f u n c t i o n s on V

times also written

k IV] .

(V,A)

is not mentioned unless required,

Similarly,

by a v a r i e t y

V

by V.

The

and is some-

we m e a n an affine

algebraic variety.

We

start with a definition. Let V

in V

be a variety and U C V.

Then

U

is epals

if (I) U

is irreducible

(2) U

contains a dense open subset of U'.

The main proposition is :

Proposition I. Let

U,V

be varieties } ~ : U ~

V

U' C U be an 6pats. T h e n o((U') is an ~pais in V.

be a m o r p h i s m .

Let

-15-

Proof.

l~' i s a v a r i e t y in i t s own r i g h t and

a~/U' i s a m o r p h i s m .

Hence,

without l o s s of g e n e r a l i t y , one m a y a s s u m e t h a t U' is d e n s e in U. U ' c o n t a i n s an open d e n s e s e t which i s p r i n c i p a l . any open s e t i s e m p t y o r d e n s e ) . U'

itself may

H e n c e , without l o s s of g e n e r a l i t y ,

0J

forsome

fgk[U].

Again, ~(U)

own r i g h t , and h e n c e one m a y a s s u m e that f o l l o w s that

,

$

(Since U' = U i s i r r e d u c i b l e ,

be a s s u m e d to b e a p r i n c i p a l open d e n s e s e t .

U' = { x ~ U / f ( x ) ~

: k IV]

are integral domains.

,

Further,

~(U)

k [U] i s i n j e c t i v e .

Let

i s a v a r i e t y in i t s

i s in f a c t d e n s e in V.

A l s o , both of k [V]

(U and V a r e both i r r e d u c i b l e ) .

It

and k [U]

We now s t a t e a l e m m a

which w i l l b e p r o v e d l a t e r .

L e m m a 1.

L e t A and B be i n t e g r a l d o m a i n s , A ~_ B, and A f i n i t e l y

generated over

B.

L e t f ~ 0, f ( A .

fer any algebraically with

Then t h e r e e x i s t s

c l o s e d f i e l d F and a h o m o m o r p h i s m

o~1.

It can be seen that X is nilpotent iff all the eigenvalues of X are equal to zero.

Definition.

An e n d o m o r p h i s m

X on V is s a i d to be u n i p o t e n t if X - I

is

nilpotent.

It can be seen that X

From

is unipotent iff all the eigenvalues of X

are equal to I.

the above, it follows immediately that the restriction of a semisimple

(respectively nilpotent, unipotent) e n d o m o r p h i s m

to an invariant subspace is

-23

-

again semisimple (respectively nilpotent, unipotent) and similarly for quotients.

It c a n a l s o be s e e n t h a t an e n d o m o r p h i s m w h i c h i s s e m i s i m p l e and n i l p o t e n t m u s t be i d e n t i c a l l y z e r o .

Lemma. space

Any c o m m u t i n g s e t of e n d o m o r p h i s m s of a f i n i t e d i m e n s i o n a l v e c t o r

V can be put s i m u l t a n e o u s l y in an u p p e r - t r i a n g u l a r f o r m in s u c h a way

that the s e m i s i m p l e e l e m e n t s a r e d i a g o n a l .

Proof.

Let ~

be a c o m m u t i n g s e t of e n d o m o r p h i s m s .

the d i m e n s i o n of V.

If d i m V ~ 1, e v e r y e n d o m o r p h i s m i s s c a l a r and the

lemma is trivially true. C a s e (i) ~ V =

Z ~(k

We u s e induction on

A s s u m e the l e m m a f o r s p a c e s

W with d i m W < d i m V.

contains a non-scalar semi-simple endomorphism

Va:, w h e r e

follows mat for

Vc
End Vi/vi.1 are unipotent°

H e n c e ~i(G) = I d e n t i t y . Vi/vi.1

is

S is

Consider a composition

G-module V1 ~i~ r.

is the corresponding representation.

on ~i(G).

Hence

But

V with r e s p e c t to G, i . e .

H e n c e the a c t i o n of G on V i / v i . 1

i s o n l y one t r a c e - v a l u e

Since

~ V o = (0) f o r

is a s i m p l e

~ is i n j e c t i v e .

Let E l e m e n t s of G

is a l s o u n i p o t e n t a n d t h e r e

H e n c e by l e m m a 2 a b o v e , l ~ i ( G ) ~ ~ i -

H e n c e e v e r y s u b s p a c e of V i / V i . l

s i m p l e , it f o l l o w s t h a t

Vi/vi.1

is G-invariant.

i s of d i m e n s i o n 1.

This

proves the required result.

Remarks.

(1) T h e a b o v e t h e o r e m h o l d s f o r a r b i t r a r y f i e l d s (not n e c e s s a r i l y

a l g e b r a i c a l l y c l o s e d ) (check t h i s ) . (2) A r g u i n g a s i n L e m m a 2 one s e e s t h a t a s u b g r o u p of GLn(k} with j u s t c o n j u g a c y c l a s s e s is f i n i t e , of o r d e r at m o s t

r n2.

r

By m o d i f y i n g the p r o o f

s o m e w h a t , one c a n p r o v e t h a t if c h a r k = 0, S i s a s u b g r o u p of A u t V w i t h o u t u n i p o t e n t e l e m e n t s a n d with r

t r a c e s o n l y , t h e n IS~ ~ r n 2 , n = d i m e n s i o n of V.

One c a n t h e n e a s i l y d e d u c e t h a t o v e r f i e l d s of c h a r a c t e r i s t i c

0, e v e r y t o r s i o n

s u b g r o u p of A u t V with the e l e m e n t s of b o u n d e d o r d e r s i s f i n i t e a n d t h a t e v e r y t o r s i o n s u b g r o u p of GLn(Z~) (i. e. m a t r i c e s with i n t e g r a l c o e f f i c i e n t s a n d h a v i n g i n v e r s e s with i n t e g r a l c o e f f i c i e n t s a l s o )

i s f i n i t e , of o r d e r ~(2n+1) n 2 .

-

35

-

T h e s e r e s u l t s a l l go b a c k to B u r n s i d e . (3) K o l c h i n ' s t h e o r e m i n c i d e n t l y p r o v e s that e v e r y u n i p o t e n t group i s n i l p o t e n t s i n c e the group of u p p e r t r i a n g u l a r u n i p o t e n t m a t r i c e s is so (check this).

T h i s r e s u l t has an i n t e r e s t i n g c o n s e q u e n c e :

Proposition.

L e t G be a u n i p o t e n t a l g e b r a i c g r o u p ( i . e . an a l g e b r a i c group

c o n s i s t i n g of u n i p o t e n t e l e m e n t s ) .

Let G act on an affine v a r i e t y V.

Then

every orbit is closed.

Proof.

Let ~ : G x V - - ~ V be the a c t i o n .

W r i t e o GL(W(f))

Hence by Since k[V]= ~..W(f),

f~k[V]

C o n s i d e r a n o r b i t O of V

Consider O.

T h e n by

c o r o l l a r i e s I and 2 to p r o p o s i t i o n 1 of 1 . 2 , it follows that O is open in O and - O is a u n i o n of o r b i t s (of s m a l l e r d i m e n s i o n s ) .

Since ~)- O

is a p r o p e r

-

c l o s e d s u b s e t of O , t h e r e e x i s t s

36

-

f ~ k [~)] such that f ~ 0 on ~)-'~) and

f ~ 0 on O. Since O - 0 perties.

(*) is a union of orbits, it follows that xSf(x ~ G) has similar pro-

N o w {x ~/w(f), x ~ G ~

theorem, it has a c o m m o n

is a unipotent group.

Hence by Kolchin's

eigenvector fo" Since fo ~ 0 and every element

of W(f) is zero on O - O, it follows that f also has the above mentioned o property ($). Now, x$(fo) = fo ~ x E G . being fixed.

Hence fo i n c o n s t a n t and c o n t a i n s

i s c l o s e d in O.

But f

is already

o

O.

Hence

~ on O.

fo(x'l.v) = fo(V) V x

B u t t h e n the s e t {Xlfo(X) =• t

Hence it is the whole of (~.

Corollary.

Thus

f

o

-- A on

0 on ~ ) - O and O - O i s n o n - e m p t y by a s s u m p t i o n .

H e n c e fo = 0, a c o n t r a d i c t i o n to the f a c t : fo ~ 0. O is closed.

~G,v

H e n c e ~ ) - O i s e m p t y i , e.

T h i s p r o v e s the p r o p o s i t i o n .

E v e r y c o n j u g a c y c l a s s of a u n i p o t e n t a l g e b r a i c g r o u p i s c l o s e d .

T h e m o s t i m p o r t a n t e x a m p l e of a u n i p o t e n t a l g e b r a i c g r o u p , i n c i d e n t a l l y , i s the a d d i t i v e g r o u p G a d e f i n e d b y Ga(k) = (k, k IX] ) with a d d i t i o n the g r o u p operation. t 4--~[:

T h i s g r o u p m a y be s e e n to be u n i p o t e n t e i t h e r f r o m the i s o m o r p h i s m :]

(t~k)

t e r m s of the b a s i s

2.6.

orelse

d i r e c t l y f r o m the f o r m of the r i g h t t r a n s l a t i o n s in

1,X,X 2....

of k I X ] .

Diagonalizable Groups.

Definition. A n (affine) algebraic group is said to be diagonalizable if it is commutative and consists of semisimple elements.

-

37

-

The m o s t i m p o r t a n t e x a m p l e i s the m u l t i p l i c a t i v e group Gin(k) of k, e q u a l to

(GL(k), k I X , X - I ~ ) .

P r o p o s i t i o n 1. F o r a n a l g e b r a i c g r o u p G, the following s t a t e m e n t s a r e equivalent: (a)

G is d i a g o n a l i z a b l e .

(b)

G i s i s o m o r p h i c to a c l o s e d s u b g r o u p of s o m e

D n ( i . e . of the group of

d i a g o n a l m a t r i c e s in GL n) o r , e q u i v a l e n t l y , of s o m e (c) k [G~

n GL 1 .

i s s p a n n e d , as a v e c t o r s p a c e , by the c h a r a c t e r s .

(A c h a r a c t e r is

a m o r p h i s m of G into GL 1.)

Proof.

(a)

~

(b).

T h i s i s obvious f r o m the p r o p o s i t i o n of 1.9, p r o p o s i t i o n 2

of 1.13 and p r o p o s i t i o n 3 of 2 . 4 . F o r a c l o s e d s u b g r o u p of Dn, X m ll l...

(b) ==~ (c}.

m i £ Z , 1 ~ i ~ n. of D n t o / h e

X nmnn

is a character,

(By Xii, we m e a n the c a n o n i c a l f u n c t i o n taking an e l e m e n t

i th (diagonal} entry}.

Obviously,

k [G] c o n s i s t s of p o l y n o m i a l s

which a r e l i n e a r c o m b i n a t i o n s of such c h a r a c t e r s .

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

span k [G 1 . (c) ~

(a).

Let

f be a character

Hence

~y f = f(y).f VY

vector for

~y* ,y ~ G.

semisimple

and any two

Definition.

The

pointwise

by

X(G)

Then

In other words,

Since characters ~y ,

characters

multiplication.

and is denoted

~G.

of G.

group

(or simply

character

group

G

G

Clearly

X(G)

By*

is

is diagonalizable.

form

is called the character X).

EG.

is an eigen-

k [G] , it follows that

Hence

of an algebraic The

every

span

commute.

f(xy) = f(x).f(y) ~x,y

a group under group

is abelian.

of G

and

-

The

character

groups.

We

now

group

for arbitrary

Proposition

algebraic

role in the theory of diagonalizable

clear as we proceed

a proposition

2. For

-

plays an important

This will become

prove

38

for diagonalizable

with the development.

groups.

(This proposition

holds

groups).

a diagonalizable

group

G, X(G)

is finitely generated.

Proof. Since G is diagonalizable, k [G] is spanned by X(G). But then k [G] is finitely generated as k-algebra. which g e n e r a t e s

k [G 7

Hence there exist characters

as k-algebra.

Let

XI,...,Xn

H be the s u b g r o u p g e n e r a t e d by

r1 C l e a r l y e l e m e n t s of H a r e of the f o r m : X 1 . . . X r n with r. E77'.

X 1, . . . . X n.

n

Also, any f 6 k [G]

1

is a l i n e a r c o m b i n a t i o n of e l e m e n t s of H.

r T h e n X = ~_. ~ j . % ~ j )~j@H. We m a y j=l ! a s s u m e that the ~Tj s a r e all d i s t i n c t . The p r o p o s i t i o n now follows f r o m the Claim.

H=X(G).

Let

XEX(G).

following g e n e r a l l e m m a :

Lemma

I. Distinct characters

independent

Proof.

as k-valued

Let, if possible,

functions

~ o + i__~_l~i~i.= = O, w h e r e

and r

on

into

k

are linearly

distinct characters.

i

~ i s a r e d i s t i n c t c h a r a c t e r s of H into k*

is m i n i m a l with this p r o p e r t y ( r ~ l ) .

~o(ho) ~ O(l(ho). Consider

H

H.

there exist relations between

r

Let

of an (abstract) group

Choose h t~ H such that o

-39-

r 0-- ~ o ( h o . h) + ~- ~ i ~ i ( h o . h ) ) V i=l r

hEH

= °(o(ho)" ~ o ( h ) + ~ ~ i " °(i(ho)" °(i(h)" i=l r Also,

0 = ~o(ho) + ~ i ~ i ( h o ). i=l r 0 =~ (~i(ho) - ¢_,dim V, w i t h e q u a l i t y f o r a d e n s e o p e n s e t of v ' s

Then

d i m T(V) v i s f i n i t e ,

in V.

(Such

v's

are

called simple or nonsing~lar. )

This result,

needed later,

also uses Noether's

theorem,

refi~d as follows: A

(~) If A i s a f i n i t e l y g e n e r a t e d i n t e g r a l d o m a i n o v e r a p e r f e c t f i e l d k t h e n there exists a generating set IXl,X2 ..... some

d) i s a l g e b r a i c a l l y

Xnt s u c h t h a t [ X l , X 2 . . . . .

independent over

k and f o r e a c h

i •d,

X d l (for xi i s

-

separable algebraic over say

Fi(x 1.....

k(x 1 . . . . .

refinement as well.

To prove o

] v

t. = 0 f o r a t l J

equations for

with (monic) minimal polynomial, k Ix 1 . . . .

Proposition (and

i • d.

n unknowns,

,xi.1].

3, we m a y a s s u m e

d = dimV).

Let

t : xi

is nonzero,

V to b e a f f i n e .

v : x.~ 1

v. be a p o i n t 1

~-t i i s a t a n g e n t v e c t o r a t

v

S i n c e t h i s i s a h o m o g e n e o u s s y s t e m of

we g e t

n >,,dim T(V) v >r d = d i m V w i t h

e q u a l i t y on t h e r i g h t on t h e s e t of p o i n t s w h e r e s o m e /~Fi~ "$xj l

T h e u s u a l p r o o f of

in M u m f o r d ' s b o o k , t a k e s c a r e of t h i s

It r e a d i l y f o l l o w s ( s e e 2 . 1 1 ) t h a t

J~'Sxj

n-d

Xi_l)

as given, e.g.,

W e a p p l y ($) w i t h A = k [ V 3

iff

-

x i ) , w i t h c o e f f i c i e n t s in

Noether's theorem,

of V.

61

a n o p e n s u b s e t of V.

(n-d) th

o r d e r m i n o r of

As may be checked, the minor

f a r t h e s t to t h e r i g h t w o r k s out to

(it i s l o w e r t r i a n g u l a r ) w h i c h i s

i>dk~Xi J not identically 0 because of the~separability in (*) so that the above open set is nonempty,

2.12.

hence dense, as asserted.

Borel subgroups.

Throughout this section, G will denote a connected,

affine algebraic group.

Definition.

A m a x i m a l c o n n e c t e d s o l v a b l e s u b g r o u p of G i s c a l l e d a B o r e l

subgroup.

Remarks.

(1) B o r e l s u b g r o u p s e x i s t f o r d i m e n s i o n r e a s o n s .

(2) A B o r e l s u b g r o u p i s a l w a y s c l o s e d ,

s i n c e t h e c l o s u r e of a c o n n e c t e d

s o l v a b l e s u b g r o u p i s a g a i n a c o n n e c t e d s o l v a b l e s u b g r o u p of G.

Example.

Let

G = GL(V)

all upper triangular

for some vector space

e l e m e n t s of G L ( V ) .

Then

V.

Let

B be t h e s e t of

B i s a B o r e l s u b g r o u p of G.

-

62

-

T h e b a s i c r e s u l t is a s f o l l o w s :

T h e o r e m 1.

T h e B o r e l s u b g r o u p s of G a r e c o n j u g a t e to e a c h o t h e r .

one of t h e m , then G / B

Proof.

Let

is complete.

B be a B o r e l s u b g r o u p of m a x i m u m d i m e n s i o n .

t h e o r e m 1 of 2 . 1 1 , we h a v e : a r e p r e s e n t a t i o n W 1 E IP(V) w h o s e s t a b i l i z e r is G/B ~

If B i s

~: G

F r o m the

> GL(V) and a point

B and s u c h that the r e s u l t i n g o r b i t m a p

G . W 1, xB ,,---~x.W 1 is an i s o m o r p h i s m .

Note that

K e r ~ C_ B,

h e n c e is a l s o s o l v a b l e .

Using Borel's fixed point t h e o r e m repeatedly, one has a flag w : 0 =

W ° OWl ..... C

Wn=VE~(V

) suchthat

is now clear that such a g necessarily

e(g)(Wi)CW.1

G w = B and h e n c e a m a p

It

belongs to B (i.e. B = ~g ~G/~(g)(Wi)

C__Wi V 0 -~ i ~ n)). In o t h e r w o r d s , one h a s a m a p s u c h that

V°~i~n'VgEB"

G/B ~Gw.

G -T-~(V),'~(g)

= g. w

Since this map dominates

the e a r l i e r one (the map: e a c h f l a g g o e s to i t s v e r t e x : w , , - - ~ W l , ) and v i c e v e r s a s i n c e the e a r l i e r one was an i s o m o r p h i s m , we s e e that t h i s m a p i s a l s o an i s o m o r p h i s m .

Let

@ be the o r b i t Gw.

C l a i m : F o r any o t h e r o r b i t Let

w'E~'.

Then

Gw, f i x e s the f l a g w ' .

in u p p e r t r i a n g u l a r f o r m ) . s o l v a b l e and h e n c e so is G.

8' (for the a c t i o n of G on ' ~ ( V ) ) , d i m @'~ d i m 8.

Since Gw ° I"

H e n c e ~(Gw,)

k e r ~ is solwable, it f o l l o w s t h a t Gw,

Thus

v

(all the fibres of the rnorphism

dimension)

orbits.

G ~

6'

and similarly, d i m @ = d i m G - d i m B.

proving the claim.

is

G vfrl ° is a c o n n e c t e d s o l v a b l e s u b g r o u p of

H e n c e by m a x i m a l i t y of d i m B, d i m G w' ° ~ d i m B.

d i m G w°

is s o l v a b l e (being

This proves that 8

Also, d i m 8' = d i m G

, g ~

g.w

Hence

is of m i n i m u m

I

are of the s a m e

d i m 8' ~ d i m ~ ,

dimension a m o n g the

H e n c e by corollary 3 to proposition 1 of 1.13, @ is closed.

Thus

@

-

63

-

i s a c l o s e d , c o m p l e t e (in f a c t , p r o j e c t i v e ) v a r i e t y . proves that G/B

But then G / B ~ 9 .

This

is complete projective whenever B has maximal dimension

a m o n g the d i m e n s i o n s of the B o r e l s u b g r o u p s .

Now, l e t B' be a n y o t h e r B o r e l s u b g r o u p . variety G/B

in a n a t u r a l way; ( i . e . b ' .

(gB) = b ' g B , b ' E

b y B o r e l ' s f i x e d p o i n t t h e o r e m , ~ xB E G / B This shows that x-lB'x is x'lB'x.

C B.

Hence x ' l B ' x

But then B'

= B.

L e t B ' a c t on the c o m p l e t e B', g £G).

Then

such t h a t b ' x B = xB V b ' ~ B ' . i s a B o r e l s u b g r o u p and h e n c e , s o

This proves that all Borel subgroups are

c o n j u g a t e and in p a r t i c u l a r , have the s a m e d i m e n s i o n .

This proves that G/B

is complete projective for all Borel subgroups.

C o r o l l a r y 1.

If P i s a c l o s e d s u b g r o u p of G, then the f o l l o w i n g c o n d i t i o n s

are equivalent:

(a) G/p (b) P

Proof.

is complete. contains a Borel

(a) ~

translations. theorem)

(b).

subgroup.

L e t B be a B o r e l s u b g r o u p .

Then b y B . F . P . T . ,

B has a fixed point xP.

L e t B a c t on G / p

by left

( a b b r e v i a t i o n f o r B o r e l ' s fixed p o i n t T h i s c l e a r l y g i v e s (b) b e c a u s e

P then

c o n t a i n s the B o r e l s u b g r o u p x - l B x . (b) ~

(a) .

Let

P _~ B, a B o r e l s u b g r o u p .

Then the map 7:

G---~G/p

i s c o n s t a n t on c o s e t s of B and h e n c e g i v e s r i s e to a ( s u r j e c t i v e ) m o r p h i s m : : G/B

r~ G / p .

This shows that G/p

is complete, because

G/B

is.

Thus: B o r e l s u b g r o u p s a r e the ' s m a l l e s t ' a m o n g t h e s e t of c l o s e d s u b g r o u p s P with G / p

complete,

i . e . a m o n g the " p a r a b o l i c s u b g r o u p s " .

-

C o r o l l a r y 2.

64

-

T h e m a x i m a l t o r i of G a r e a l l c o n j u g a t e to e a c h o t h e r and so a r e

the m a x i m a l , c o n n e c t e d u n i p o t e n t s u b g r o u p s .

Proof.

Let

T,T'

be two m a x i m a l t o r i .

a Borel subgroup. each other.

B.

B and B '

a r e c o n j u g a t e to

T and T ~ a r e c o n t a i n e d in s o m e

It now f o l l o w s f r o m the t h e o r e m 2 of 2 , 8 , t h a t T and T '

a r e c o n j u g a t e in B.

A g a i n , if U is m a x i m a l c o n n e c t e d u n i p o t e n t s u b g r o u p ,

U ~ B for some Borel subgroup

in 2 . 3 ) .

T is c o n n e c t e d s o l v a b l e , T C B,

S i m i l a r l y , T ~C B ~. But then

H e n c e we m a y a s s u m e that

Borel subgroup

then

Since

B

(B i s n i l p o t e n t by K o t c h i n ' s T h e o r e m

Now, by t h e o r e m 2 of 2 . 8 , U C Bu"

T h i s s h o w s that

U = B u.

Now

!

the c o n j u g a c y of s u c h

U s f o l l o w s f r o m the c o n j u g a c y of B o r e l s u b g r o u p s .

C o r o l l a r y 3. (a) If 04 is an a u t o m o r p h i s m ( e n d o m o r p h i s m ) of G, i d e n t i t y on B, then (b)

04 i s the i d e n t i t y of G. ZG(B) C ZG(G).

Proof. Then

(a) C o n s i d e r the m o r p h i s m

~ : G - - ~ G g i v e n by ~(x) = o< ( x ) . x -1

~ is c o n s t a n t on the c o s e t s of B.

quotient morphism, Now G / B of 2 . 1 0 , ~

H e n c e by the u n i v e r s a l p r o p e r t y of the

~ f a c t o r s to ~ : G / B - - - ~ G g i v e n by ~(gB) = ~(g) = ~ ( g ) . g - 1

is c o m p l e t e and i r r e d u c i b l e w h i l e G is a f f i n e .

H e n c e by p r o p o s i t i o n 1

is c o n s t a n t and c l e a r l y this c o n s t a n t = e, the i d e n t i f y e l e m e n t .

This

p r o v e s that ~ = I d e n t i f y on G.

(b) T h i s f o l l o w s by a p p l y i n g (a) to the i n n e r a u t o m o r p h i s m s by e l e m e n t s of

ZG(B). R e m a r k . In the s a m e way, one can p r o v e that if G a c t s on an affine v a r i e t y , then any point f i x e d by B i s f i x e d by G.

-

65

-

C o r o l l a r y 4. If B, a B o r e l s u b g r o u p , is n i l p o t e n t , then so is G.

In fact, we

p r o v e G = B.

Proof. G/B -~

We p r o c e e d by i n d u c t i o n on d i m B. G.

If d i m B = 0, then B = l e ~

Now G i s affine, c o n n e c t e d and G / B

G = ~ e t = B.

is complete.

Hence

So l e t d i m B >_,1. Since B is n i l p o t e n t , t h e r e e x i s t s a c l o s e d

s u b g r o u p C C_ ZB(B ) s u c h that d i m C ~,,1.

By c o r o l l a r y 3 above, C C ZG(G)-

Hence G / C i s an affine (connected) a l g e b r a i e group.

( T h e o r e m 1 of 2.11).

C l a i m . B / C , which i s a s u b g r o u p of

G/C, i s

Consider: G ~

, the c a n o n i c a l m o r p h i s m .

G/C~

G/C/B/c !

i n fact a B o r e l s u b g r o u p . T h e n ~ is

c o n s t a n t on c o s e t s of B and h e n c e f a c t o r s t h r o u g h G / B .

T h u s , we have

G]B -~G/c/B]C; ~ issurjeetive. T h u s

iscompletesince

G/B

is so.

B/C

G/c/B/c

i s a l r e a d y c o n n e c t e d and s o l v a b l e .

follows that B/C i s a B o r e l s u b g r o u p of G / C . B/C

and

is a g a i n n i l p o t e n t .

Hence by c o r o l l a r y 1, it

Now d i m B / C < dim B and

Hence by i n d u c t i o n h y p o t h e s i s , G / C = B / C .

This

shows G = B and c o m p l e t e s the p r o o f of the c o r o l l a r y .

2.13.

Density and Closure.

D e f i n i t i o n . A s u b g r o u p C of G is c a l l e d a C a r t a n s u b g r o u p if C = Z(T) ° for some m a x i m a l torus

T in G.

(We l a t e r p r o v e that

Z(T) i s c o n n e c t e d ,

so that C = Z(T)).

All C a r t a n s u b g r o u p s a r e conjugate s i n c e a l l m a x i m a l t o r i a r e .

-

Proposition

1.

If T , C

66

-

are as above, then

T

is the unique maximal

t o r u s of

C.

Proof.

Clearly,

corollary

T

is maximal

2 of 2.12,

in

all maximal

C.

Also,

tori in

C

T

is normal

in

are conjugate.

C

and by

Hence

the pro-

position follows.

Corollary

Proof. B

1.

Let

of C

theorem

All Cartan subgroups

C = Z(T) °, T

such that 2 of 2 . 8 ,

are nilpotent.

a maximal

C_~ B 2 T . B = T.B u .

Now Also,

a direct product decomposition.

torus in T T

isa

G.

maximaltorus

is normal

It follows that B

B u are so). Hence by corollary 4 of 2.12, C = B

We

now

Lemma

prove

an important

1 (Density).

Choose a Borel subgroup

in

B.

in

B.

Hence by

Hence the above is

is nilpotent (since T and C

and

is nilpotent.

lemma.

T h e u n i o n of t h e C a r t a n

subgroups

of G

contains a dense

o p e n s u b s e t of G .

Proof.

Fix a Cartan

j u g a c y of C a f t a n

subgroup

subgroups,

C = Z(T) °, T

a maximal

one has to prove that

K =

torus.

By the con-

U g C g-1 g~G

contains

a d e n s e o p e n s u b s e t of G.

Let

¢ So=~(x,y)~GXG/x'Iyx

EC~

7

irreducible, being the image of G X C @(x,y) = (x, xyx-l). since

. Clearly S O is closed. under the m a p

@ : GxG

S O isalso ~

GxG,

Again, S o is m a d e up of cosets of C)~II ~ in G X G,

(x,y)~S o implies that (xc, y) C S o ~ c ~C.

B y using a theorem on

compatibility of quotients and products (Borel's book, page 179), one has

-

G3CG/cxII~-~G/cXG.

67

-

Hence 7 : G x G ~

G/C•

G, given by ~ (g,g')=

(gC, g'), is just a quotient map. ~ is open and h e n c e the i m a g e of a c l o s e d s u b s e t c o n s i s t i n g of c o m p l e t e c o s e t s of C x [11 is c l o s e d . isclosed.

T h u s S = ~ (S o)

It is i r r e d u c i b l e as well since S O is. I S = ~(xC, y ) / x - l y x ~ C ~ J .

Now c o n s i d e r Pl : G / c X G - - - ~ G / c

and P2 : G / c X G

Pl(S) = G / C and the f i b r e of xC is x c x ' l .

---~G.

Clearly,

It now follows that the f i b r e s of

Pl have the s a m e dimension. Hence,

dim C = d i m e n s i o n of a fibre = dim S - dim Pl(S) = dim S - d i m G / c

.

Hence dim S = dim C + dim G / C = dim G.

Again, c o n s i d e r P2:S - - ~ P2(S).

is an etpais in S, hence P2(S) is an e~pais, i . e . of P2(S).

Note that P2(S) = K(=

S

P2(S) contains an open subset

U g c g ' l ) . Hence the l e m m a is p r o v e d if gEG

we p r o v e P2(S) = G. Claim:

d i m P2(S) = d i m G,

This c l e a r l y shows dim P2(S) = dim G and hence

P2(S) = G, both being i r r e d u c i b l e and of the s a m e d i m e n s i o n . W e observe that C has the following property:

(,)

suchthat xC/x'ltx C

There exists

isfi°ite.

F o r : By the c o r o l l a r y to the p r o p o s i t i o n in 2.7 t h e r e e x i s t s t { T C_C, such that ZG(t) = ZG(T).

If C' is any C a f t a n subgroup containing t, then

t CC' : Z G ( T ' ) ° ~

T'C_ ZG(t ) = ZG(T ).

by p r o p o s i t i o n 1 above. Thus:

Hence W ' C Zo(W)° : C = >

Hence C' = C.

C is the unique C a r t a n s u b g r o u p containing t.

x-ltx~C

~

texCx'l

T = T'

~--~ xCx -1 = C ~

is the unique m a x i m a l t o r u s of C.

Also,

XENG(C) ~

Now, X £ N G ( T ) , since

W

NG(T)° = ZG(T) ° = C (by c o r o l l a r y 2

-

of the t h e o r e m of 2 . 7 ) .

68

-

It now follows that the n u m b e r of d i s t i n c t c o s e t s

with x "1 tx ~ C ~ o r d e r of N(T)/N(T) o which is f i n i t e .

xC

Hence C s a t i s f i e s

the p r o p e r t y (*).

C o m i n g b a c k to the c l a i m , we s e e that f o r the m o r p h i s m P2/s: S o v e r t EC

), P2(S) ( P 2 : G / C w G is f i n i t e .

~- G ; S = {(xC, y ) / x ' l y x ~ C ~ ) , the f i b r e

Hence d i m S = d i m P2(S) = d i m G.

This p r o v e s the

c l a i m and h e n c e the l e m m a .

(Note that we have u s e d L e m m a 2 of 1.13 which gives: (*~) F o r a d o m h l a n t m o r p h i s m f : U - - - ~ V , d i m U = d i m V it s o m e f i b r e is f i n i t e , n o n - e m p t y . we have not p r o v e d this l e m m a in t h e s e n o t e s .

But

We have, h o w e v e r , p r o v e d

P r o p o s i t i o n 3, Appendix to 2 . 1 1 , which y i e l d s (**) with s o m e r e p l a c e d by most.

T h i s is enough f o r o u r proof h e r e s i n c e f r o m P r o p o s i t i o n (b) of 2 . 7 ,

it r e a d i l y follows that m o s t f i b r e s of P2/s_ above a r e f i n i t e .

A similar remark

a p p l i e s to o u r l a t e r a p p l i c a t i o n s of L e m m a 2 of 1 . 1 3 . ) Remark.

Let D be a n y s u b g r o u p of G s a t i s f y i n g the p r o p e r t y (*).

f r o m the proof of the above l e m m a , it c l e a r l y follows that

Then

~) g Dg "1 c o n g~G

r a i n s a d e n s e open s u b s e t of G.

L e m m a 2. ( C l o s u r e ) .

Let G act on a v a r i e t y V.

Let H be a c l o s e d s u b -

g r o u p of G and U C V be a c l o s e d s u b s e t of V, i n v a r i a n t u n d e r the a c t i o n of H.

Assume

G / H to be c o m p l e t e .

Proof.

Let S = f ( x H , v ) / x ' l v ~ u ~ ,

it follows that S is w e l l defined.

Then G.U

S

~G/H~,V.

is c l o s e d .

Since h(U)C_ U V h ~ H ,

Since U is c l o s e d , S i s c l o s e d in G / H X V.

Hence P2(S) is c l o s e d in V, s i n c e G / H is c o m p l e t e . which p r o v e s this l e m m a .

But then P2(S)= G . U ,

-

Theorem

1.

element

x

69-

(a) T h e u n i o n of t h e B o r e l s u b g r o u p s

of G i s a l l of G

of G i s c o n t a i n e d i n a B o r e l s u b g r o u p

i.e.

every

(i. e. i n a c o n n e c t e d s o l v a b l e

subgroup). (h}

Every semisimple

element

is contained in a connected

Proof.

is contained in a torus.

unipotent group.

(a) Since every Cartan subgroup is contained in a Borel subgroup, it

follows from the Density lemma that

G.

Every unipotent element

U B' all Bore1 B'

contains a dense open subset of

Now let G act on itself by conjugation. Choosea Borel subgroup B.

Take

V--G, H= B, U - - B in the closure lemma. T h e n G.B = ~ gBg "I = U B' g~ G all Borel B' is closed. This clearly proves (a).

(b) S i n c e e v e r y e l e m e n t and solvable,

(b)

is contained in a Borel subgroup,

immediately

follows from theorem

which is connected

2 of 2 . 8 .

Remark. F o r an arbitrary algebraic group G (connected), it is not true that each of its elements is contained in a connected abelian subgroup.

Consider one

G to be the group of upper triangular 2 x 2 matrices of determinant

(k of c h a r .

0).

Let

x =

.

Then it can be easily checked that

-1 the only connected

subgroup containing

x

is

G

itself and

G is not abelian.

W e s t a t e h e r e a n e x t e n s i o n of t h e a b o v e t h e o r e m .

Theorem

2.

Every surjective

Borel subgroup invariant.

endomorphism

In p a r t i c u l a r ,

F o r t h e p r o o f of t h i s a n d r e l a t e d m a t t e r s ,

of a n a f f i n e g r o u p k e e p s s o m e

every automorphism

see A. M .S. Memoir

does so.

No. 80.

(In

particular, for an inner automorphism ix, ix(B) = B for some B. Hence x ~B

-70

-

(since every Borel subgroup is its own normalizer)).

Corollary

0.

If B i s a B o r e l s u b g r o u p of G, a c o n n e c t e d g r o u p ,

then

Z(B) = Z(G).

Proof.

We have

Z(B) C Z(G)

x E B', some Borel,

Corollary

Proof.

Let

B = Bu.T

by Theorem

1. ( F u r t h e r

be such that

closure).

Gv ~ a maximal

Gv ~

by Theorem

Let

torus.

T, a maximal

(by t h e o r e m

2, h e n c e

1, C o r o l l a r y

3(b).

x E B since

B'

If x {~Z(G), t h e n is conjugate to

G act on an affine variety Then

torus.

G.v,

V.

Let

B.

v~V

t h e o r b i t of v , i s c l o s e d .

Let

T ~ B, a Borel subgroup.

2 of 2 . 8 ) ; S = B . v = B u . T . v

= Bu.v,

since

T.v

Then

= v.

Thus

S is closed, being the orbit under th~ action of a unipotent group (by the proposition of 2.5). Hence by the closure lemma, as S is invariant under B, G. v= G. S is closed. Corollary 2.

Let G

be an affine group.

(a) A n y semisimple conjugacy class,

or m o r e generally (b) any conjugacy class meeting a Cartan subgroup is closed.

Proof.

By theorem

1 above,

every semisimple

and hence in a Cartan subgroup. Now t o p r o v e (b), l e t torus.

Let

1 above,

Corollary

3.

Let

T h u s (b) c l e a r l y

a Caftan subgroup.

G act on G by conjugation.

corollary

Proof.

x ~C,

t h e o r b i t of x, i . e .

a Borel subgroup, it follows that

x

and

be arbitrary. B'

a c t i n g on

implies Let

Then

(a).

C = ZG(T)

ZG(S)

, T

a maximal

Hence by

is closed.

is connected.

Fix a Borel subgroup G/B

o

G x = ZG(X)_~ T.

the conjugacy class,

If S i s a t o r u s in G , t h e n

x EZG(S)

element is contained is a torus

B.

Since

x (B',

h a s a f i x e d p o i n t (by B . F . P . T .

has a fixed point in G/B.

Let

),

W b e t h e s e t of a l l f i x e d

-

p o i n t s of

x

Then

complete also.

71

-

W i s a n o n - e m p t y c l o s e d s u b s e t of G / B .

A g a i n , x E ZG(S)

and hence

h a s a f i x e d p o i n t in W (by B . F . P . T . ) . that

S. g B - - g B

and

Borel subgroup

x. gB = gB.

W invariant.

In o t h e r w o r d s ,

Hence both

g B g "1 = B ' , s a y .

theorem 2 of 2.8.

S keeps

Now

S and

x @ZB,(S)

Thus

W is

Hence

~ gB~G/B

S

such

x b e l o n g to t h e s a m e w h i c h i s c o n n e c t e d by

T h u s ZB,(S) C ZG(S)° and hence x ~ ZG(S)°. This shows

that ZG(S) = ZG(S)°. This proves the corollary.

Remarks.

(1)

T h e f i r s t p a r t of t h e a r g u m e n t

shows that any connected

solvable group and any element in its centralizer can be put in a Borel subgroup. (2) A C a r t a n s u b g r o u p (3)

C = ZG(T) °

i s , in f a c t , = Z G ( T ) .

T h e a b o v e c o r o l l a r y i s n o t t r u e in c a s e

sistingofsemisimple

[0 :]

can be seen that form:

.a.1

C o r o l l a r y 4.

Let

S is a c o m m u t a t i v e

Consider G = PSL2(~), x = [:

elements.

ZG(X) -- s e t of d i a g o n a l m a t r i c e s with

a ~ C .

Thus

t EG be semisimple.

Then

elements,

i.e.

Proof.

u E ZG(t)

be an u n i p o t e n t e l e m e n t .

It f o l l o w s t h a t B.

t

t,u

I

C B.

G as well.

U s e t of m a t r i c e s

ZG(t)/ZG(t)o

e v e r y u n i p o t e n t e l e m e n t in

i s t h e J o r d a n d e c o m p o s i t i o n of x. t,u ~ B also.

(Let

Let

T

x = t .u

!

1 of 2, 8.

T h i s p r o v e s the c o r o l l a r y .

t = t', u = u').

Hence

It

of t h e

c o n s i s t s of is in

x = t.u.

o ZG(t ) .

Clearly,

this

B containing

x.

be t h e J o r d a n d e c o m p o s i t i o n in !

c o n n e c t e d by t h e o r e m

ZG(t)

Choose a Borel subgroup

T h e n b y p r o p o s i t i o n 3 of 2 . 4 , x = t . u H e n c e by u n i q u e n e s s ,

0~.

ZG(X) is not c o n n e c t e d .

semisimple

Let

group con-

I

is the decomposition

Now u ~ Z B ( t ) , w h i c h is

ZB(t) C ZG(t) ° , s o t h a t

u E ZG(t)°.

in

-72 -

2.14

Bruhat Lemma.

(a) Let G be a c o n n e c t e d a l g e b r a i c group.

a m a x i m a l t o r u s in G.

Let

Let T be

B be a B o r e l s u b g r o u p of G s u c h that B D Z(T)~_T.

T h e n the c a n o n i c a l m a p i :

Z(T~X~kN(T)~T)_, • ,--,

>

B'~G/B is a b i j e c t i o n .

( i (Z(T) . n. Z(T)) = B . n . B , n 6 N ( W ) ) .

T h i s l e m m a i s s o m e t i m e s e x p r e s s e d in a d i f f e r e n t f o r m , viz. (a') Any two B o r e l s u b g r o u p s of a c o n n e c t e d a l g e b r a i c group have a m a x i m a l t o r u s in c o m m o n .

It can be s e e n that (a)0,

~R I

B = T.U,

maximal,

This subgroup contains an

wo( of N(T) w h i c h a c t s on V a s t h e r e f l e c t i o n r e l a t i v e to ~ ( i . e .

. x = x -(~,~)

Let

P S L 2.

Let

B-=

T.U-

R.

W i s g e n e r a t e d by

Let

wE,~t~R

(Note t h a t

.

U b e the g r o u p g e n e r a t e d b y

U" b e the g r o u p g e n e r a t e d b y I X ~ , , < 0 , T normalises

U, U ' . }

ER ~ .

Then

U is a

c o n n e c t e d , u n i p o t e n t s u b g r o u p of G, B is a B o r e l s u b g r o u p a n d the

c a n o n i c a l m o r p h i s m : -[]-- X o ( - - ~

U is a n i s o m o r p h i s m

of v a r i e t i e s .

p r o d u c t is t a k e n in a n y o r d e r . ) T h e c a n o n i c a l m o r p h i s m s : U-x B -

generate

-Uq B are also isomorphisms

s u b s e t of G a n d is c a l l e d t h e Big C e l l .

of v a r i e t i e s .

T XU

U-B

(The

~ B and

i s a d e n s e open

(Clearly, analogous statements are

t r u e in c a s e of U ' } .

We o m i t t h e p r o o f s (which a r e q u i t e l o n g a n d m a y be found in § 13-14 of B o r e l ' s book}, b u t s h a l l u s e t h e s e r e s u l t s in w h a t f o l l o w s .

A s a n e x a m p l e , the r e a d e r m a y w i s h to v e r i f y t h e s e f a c t s f o r the g r o u p G = SL n . Take

T a s the d i a g o n a l g r o u p .

¢ < ( i , j ) ( d i a g (t 1 . . . . .

For each

-1 tn}} = t i . t j

is a r o o t .

corresponding morphism. ) A root W(i,j)

i,j, i ~ j, ~(i,j} : T---~k (xc ~ ' .

Since V ~

i s one d i m e n -

t

sional,

V~, = D = D .

T h u s D i s the u n i q u e l i n e kept i n v a r i a n t by B.

f o r a n y o( s i m p l e , wof (1) i s a l s o a weight o f / l ' . ~ t

~

Again,

= ~ - (A,,~*)~.

-

Hence

()k,~*)~

(b) L e t

0 and

~

-

is dominant.

V 1, V 2 b e t w o i r r e d u c i b l e

weight

85

A . Let V = V I ~ V

2.

representations

with the same highest

Choose non-zero vectors

v.1 6 V.z (i = 1,2)

corresponding to the dominant character 4.

Let v = V l + V 2 E V.

the G-subspace generated by v. W e have, W

=

= ~U[

Let W B.v>

be

=

= k.v + lower weight spaces, since v = Vl+V 2 corresponds to the weight A . W/~ V 2 is a G - s u b m o d u l e of the irreducible m o d u l e v 2 (v 2 ~ W

by above).

It f o l l o w s t h a t

is i n j e c t i v e .

Since

also.

W is isomorphic

Hence

Hence

(c)

V 1 and

W / ~ V 2 = {0 t .

Pl'V = v 1 f 0 and to V 1.

V 2 and does not contain

V 1 is i r r e d u c i b l e , Similarly

is surjective

"Lectures

to

V 2.

on C h e v a l l e y

It i s s o v i t a l f o r o u r f u r t h e r d e v e l o p m e n t t h a t we s h a l l i n d i c a t e

In A = k [ G ] , l e t A ~ be t h e s p a c e of f u n c t i o n s

f(b-x) = ~(b-).f(x)

forall

on B - = U . T

is n o n - z e r o .

Pl

> V1

V2 are isomorphic.

group" (p.210).

character

Pl:W

W is isomorphic

T h e p r o o f of t h i s p a r t m a y b e found in t h e a u t h o r ' s

a proof:

Hence

b'EB-, as

T

x~G.

()kEX(T)

normalizes

submodule.

(Vl

canbe

e x t e n d e d to a

U . ) S u p p o s e we know t h a t

G a c t s on A ~k v i a r i g h t t r a n s l a t i o n s ,

V A b e an i r r e d u c i b l e

f w h i c h s a t i s f y (*):

locally finitely.

is finite dimensional).

A)~

Let

By (a), t h e r e T

~ .

exists a highest weight vector f corresponding to s o m e highest weight the big cell U U B

= U-.T.U, w e have : f(U'.tu) = A (t). f(u) by (*). In particular,

f(U'.t) = ~(t).f(1). f ( u - ) = ~ ( t ) . f(1). has

~

Also, since f is a highest weight vector, f(u-.t) = ~(t). Now f(1) f 0, s i n c e o t h e r w i s e

as its highest weight.

that the function polynomial,

The proof that

AA

f = 0.

Hence ~ = A'

is not zero,

and

V

or equivalently,

f d e f i n e d on t h e b i g c e l l by f ( u " tu) = ~ (t) e x i s t s on G a s a

requires

mentioned book.

On

further argument,

w h i c h m a y be f o u n d in t h e a b o v e

We s e e , i n c i d e n t l y , t h a t t h e i r r e d u c i b l e

representations

of

-

G are all induced representations,

86

-

i n d u c e d f r o m one d i m e n s i o n a l r e p r e s e n t a -

t i o n s of B.

This p r o v e s the t h e o r e m c o m p l e t e l y .

H e n c e f o r t h , we w r i t e

X(T) m u l t i p l i c a t i v e l y and r e s e r v e the a d d i t i o n s i g n f o r

functions (in k [G] o r k IT] }. T h e W e y l g r o u p W d e f i n e s an e q u i v a l e n c e r e l a t i o n a m o n g the c h a r a c t e r s on T(~-,,w(~),

~EX(T) ,w 6W).

It can be e a s i l y s e e n t h a t e a c h e q u i v a l e n c e c l a s s

c o n t a i n s e x a c t l y one d o m i n a n t c h a r a c t e r .

F o r the c l a s s [ ~ ] ,

define Symm[~]

to be the s u m (as f u n c t i o n s on T) of a l l c h a r a c t e r s b e l o n g i n g to it. representation

G--~GL(V),

C l e a r l y X~ E k [ G ] . characters,

V~

we define X~: G ~

Consider

X ~ on T.

For a

k by X ~(g) = T r a c e (~(g)).

For each class [~]of

(equivalent)

h a s a c o n s t a n t d i m e n s i o n , ~ ~ [~t]. It f o l l o w s t h a t on T, X e v

i s j u s t a s u m of S y m m ['~] s. weight.

L e t & b e i r r e d u c i b l e with ~

S i n c e a n y w e i g h t of ~

l o w e r than A

a s the

highest

o t h e r than A i t s e l f , i s of o r d e r s t r i c t l y

and ~ h a s m u l t i p l i c i t y 1, it f o l l o w s f r o m the a b o v e t h a t X % on

T is given by X~

= Symm

['~] +

~__

Symm

[~]

~

(*)

~t d o m i n a n t We denote X ~a

by j u s t X A and S y m m [ ~ 3 by j u s t S y m m ~ .

F r o m (*), it

immediately follows that Symm ~ = X

+~ - C w . l ( x ) .

X E X(T) X X characters are linearly independent, hence C X = Cw_I(x) ~ X, ~ w .

X.

Now

This means

that the elements of an equivalence class (of characters under action of W) occur with the same coefficient. Hence f = ~ C x. S y m m X X

and the S y m m s

span

k[T]W.

F u r t h e r , let ~- a N . S y m m X = 0. Now, c h a r a c t e r s o c c u r i n g in X S y m m X a r e d i s t i n c t f r o m t h o s e o c c u r i n g in S y m m X'(X ~ X ' ) . Since c h a r a c t e r s a r e l i n e a r l y i n d e p e n d e n t , it follows that a X = 0 ~ X.

{ S y m m ~} is a basis of k i T ] W.

The r e l a t i o n s ( * ) a n d ( * * ) t h e n i m p l y t h a t

is simply connected, the fundamental weights f ~ i t

(b) If G cters.

T h i s p r o v e s that

Let X i denote X A . .

Now, for any ~

are in fact chara-

dominant, ~ =-~T~n.i with i

l

n.>10. 1

X

Since

= Symm ~ +

~__

Symm

dominant

~ +

~-

w.~ +

w6 W

ni ,~A can be e a s i l y s e e n that X ) ~ - - [ r E i

~

Symm

~,

~. d o m i n a n t

is a s u m of X ' ~ s

with ~ ~ l .

Since

t h e r e e x i s t only f i n i t e l y m a n y c h a r a c t e r s (dominant) which a r e l e s s than ~ , it follows, by r e p e a t e d a p p l i c a t i o n of the above a r g u m e n t , that X A is a p o l y !

n o m i a l in X i s. = 0.

We w r i t e

A g a i n , l e t p be a p o l y n o m i a l in n v a r i a b l e s with P ( X 1 , . . ~Xn) r1 p in the f o r m p = a . X 1 . . . .

the t e r m of h i g h e s t o r d e r .

Xr n

+Pl'

(We u s e a l a x i c o g r a p h i c o r d e r ) .

m e n t s i m i l a r to the one above, it follows that a = 0. p is i d e n t i c a l l y z e r o .

~

k[r]W

isomorphism.

Xrl... 1

Xrn n

is

T h e n by a n a r g u -

T h u s it can be p r o v e d that

Thus X 1 , . . . , X n g e n e r a t e f r e e l y k [ T ] W a s k - a l g e b r a .

We now p r o v e s i m i l a r s t a t e m e n t s f o r

C[G]

where

C [G] .

C o n s i d e r the r e s t r i c t i o n map :

, which is well defined. W e claim that it is in fact, an

Since X • ,

for ~ a dominant character, is in C [ G ] , sur-

jectivity is obvious from the above results for k IT] w.

Further, let f~C [G]

-

s u c h that f / T = 0. gEG.

-

Now for a n y x s e m i s i m p l e C G, g x g - l £ T f o r s o m e

Hence f(x) = f(gxg -1) = 0 (f is a c l a s s - f u n c t i o n ) .

the s e t of s e m i s i m p l e e l e m e n t s , i . e . T

89

-- Z(T)) and

U C is d e n s e C cartan

that f = 0 on G.

Hence

in

Thus f is z e r o on

f = 0 on

U gTg -1 = ~,) C {since g EG C cartan G (Density l e m m a of 2.13). It now follows

~ is i n j e c t i v e .

Now the s t a t e m e n t s for

C [G] are

obvious f r o m those for k [ T ] w .

T h i s p r o v e s the t h e o r e m c o m p l e t e l y .

Note.

Taking G

=

SL n and c h o o s i n g c o - o r d i n a t e s p r o p e r l y on T = the group

of d i a g o n a l m a t r i c e s , we see that the above t h e o r e m is j u s t the f u n d a m e n t a l t h e o r e m on s y m m e t r i c p o l y n o m i a l s .

In fact, the above m e t h o d of proof is

c o m p l e t e l y p a r a l l e l to one of the s t a n d a r d p r o o f s of that t h e o r e m .

T h i s t h e o r e m has s o m e i n t e r e s t i n g c o r o l l a r i e s :

C o r o l l a r y 1.

If f C C [ G ]

and x E G , then f(x) = f(Xs).

P r o o f . F o r a n y r e p r e s e n t a t i o n ~ , we m a y w r i t e ~ (x) i n s u p e r d i a g o n a l f o r m with ~ (Xs) a s i t s d i a g o n a l (by the l e m m a of 2 . 1 ) .

T h u s X ~ ( x ; = X~(Xs).

Now

the c o r o l l a r y (1) follows f r o m (a) of the t h e o r e m .

C o r o l l a r y 2.

(a) The s e m i s i m p l e c l a s s e s a r e in o n e - o n e c o r r e s p o n d e n c e with

e l e m e n t s of T / W ( i . e . s e t of o r b i t s of T u n d e r the a c t i o n of W. ) (b) If G is s i m p l y c o n n e c t e d , then T / W /~r

is i s o m o r p h i c to the a f f i n e - s p a c e

u n d e r the map:

¢ : T/w

~/A r ; ~(t-) = (Xl(t) . . . . .

Xr(t)),t-~ T/W •

w

Proof.

(a) C o n s i d e r the m a p ~

: T / W - ~ (Conjugacy c l a s s e s of s e m i s i m p l e

e l e m e n t s ) given by : ~ (t-) = I t ] . Let t , t ' E W gTg.1

=[t'3.

suchthat It]

90

Clearly, ~ i.e.

~ g~G

is well defined and s u r j e c t i v e . suchthat

gtg "1 = t ' .

, o and T a r e c o n t a i n e d in ZG(t ) and a r e m a x i m a l t o r i t h e r e .

hEZG(t') ht,h -1 = t ~.

such that h g T g ' l h -1 = T. In o t h e r w o r d s ,

t ~- t

Thus hgffN(T).

Now, Hence

A l s o , h g t g - l h -1 =

u n d e r the a c t i o n of W.

T h i s p r o v e s the

i n j e c t i v i t y of ~ and h e n c e (a).

(b) C o n s i d e r the m a p

We p r o v e : (1) (2)

~ ." T - - ~ / ~ r, given by:

O*(k [ ~ r 2 )

~(t) : (Xl(t), . . . , X r ( t ) ) .

= k IT3 w.

F i b r e s of ~ a r e j u s t the o r b i t s u n d e r W.

F r o m the t h e o r e m above, (1) is c l e a r . T o p r o v e (2), we o b s e r v e the following fact: If x , y a r e two e l e m e n t s of T which lie in d i f f e r e n t o r b i t s , then t h e r e e x i s t s a f u n c t i o n f E k I T ] W such that f(x) = 0, f(y) f 0. s e t s of T,

For:

Hence the c o r r e s p o n d i n g i d e a l s

so t h e i r s u m i s k i T ] of x and

The o r b i t s of x and y a r e f i n i t e , h e n c e c l o s e d s u b -

. Write

1 = i+j, i ~ I ,

I and J have no c o m m o n z e r o and j gJ.

Then i is

0 on the o r b i t

1 on the o r b i t of y.

the r e q u i r e d p r o p e r t i e s . b e l o n g to the s a m e o r b i t .

Let f = ~ w.i. Then f clearly satisfies wEW T h i s p r o v e s that w h e n e v e r $(t) = O(t'), t and t ' The c o n v e r s e is c l e a r l y t r u e .

F u r t h e r , k I T ] is

integral over kiWI w. (gek IT] satisfies

- ~ - {X-w.g) : 0, which is a m o n i c wEW p o l y n o m i a l in X with c o e f f i c i e n t s in k I T ] W). T h u s a h o m o m o r p h i s m of k IT] w

into k c a n be lifted to a h o m o m o r p h i s m of k I T ] into k.

it i s now e a s y to see that corollary.

~ is onto.

Thus

U s i n g (1),

(b) is p r o v e d and h e n c e so is the

-

91

-

Corollary 3. Let x, y be semisimple elements in G.

Then the following

statements are equivalent: (i) x and y are conjugate. (2) X f(x) = X~(y) for every irreducible representation ~. (3) ~ (x) is conjugate to ~(y) in GL(V~) f o r e v e r y i r r e d u c i b l e r e p r e s e n tation ~. If G is s i m p l y connected, then (2) and (3) a r e r e p l a c e d by :

(2') Xi(x)=Xi(y) V l.'isr. (3')

~Ai(x) is conjugate to

Proof. (i) --~ ( 3 ) ~ (2) ~

(1).

~Ai(y) in GL(V{3 ~ i ) V 1,< i .,Jr.

(2) is clear.

C l e a r l y , one m a y a s s u m e that x, y G T.

Since Xe(x) = X{)(y) f o r

e v e r y i r r e d u c i b l e r e p r e s e n t a t i o n ~ , the t h e o r e m shows that f(x) = f(y)VfEk[T] W. As seen in the p r o o f of c o r o l l a r y (2), k I T 3 w

s e p a r a t e s o r b i t s of W.

follows that x and y belong to the s a m e o r b i t of W i . e .

It now

x and y a r e c o n -

jugate. In c a s e of G being s i m p l y connected, ( 1 ) < ~

(2')~

(3') is p r o v e d in e x a c t l y

the s a m e way. Remark.

It is, h o w e v e r , not known whether a s i m i l a r r e s u l t holds f o r o t h e r

e l e m e n t s of G.

Corollary 4. x C O

is unipotent iff XA(x) = X A (i) V

dominant character

i.e. the variety of unipotent elements is closed and is defined by equations {XA(x) = XA(1); )% a dominant c h a r a c t e r ~ . )

A s i m i l a r r e s u l t follows, in c a s e of s i m p l y connected g r o u p s , with X ~ s

-

replaced by

Proof.

iff

x

X~x)

92

-

X' s.

Ai

is unipotent if

xs = 1

= X~l) Vdominant

iff

characters

For a semisimple group

Corolla~g 5.

X~(Xs)

G,

~.

= X ~ l ) Vdominant

characters

(We use corollary (1) and (3)

above).

a conjugacy class is closed iff it is

semisimple.

Proof.~___.

This is already proved in corollary 2 to theorem 1 of 2.13.

proof is given as follows:

Fix

a faithful representation of

polynomial of

x0

and

x0

G.

(semisimple) in the conjugacy class. Consider

Let

S = (x C Glx

X~(x) = X~(x0)

contains the conjugacy class of

Another

x 0.

satisfies the minimal

for all dominant ~ .

Conversely, if

x 6 S

S

then

is closed and

x

is semi-

simple since its minimal polynomial has distinct roots, hence is conjugate to

x0

by corollary 3.

~.

This implication follows from a general lemma:

Lemma.

The closure of every conjugacy class of

element, its semisimple part.

(G

G

contains, along with each

reductive).

Granting this fact, our result follows.

For:

a closed conjugacy class will con-

tain the semisimple part of one of its elements and hence will be semisimple

itself.

Proof of lemma.

Let

S

be a conJugacy class)

x G S.

We can assume, after

- 93

conjugation, that -~-

x

x~B=T.U

(Co ~ d i m f i b r e

G 2 be t h e c l o s u r e

P l ( G 2 ) = G.

(since

it f o l l o w s t h a t

x E G , t h e f i b r e of P l

t semisimple Now f o r a n y

it f o l l o w s t h a t

d e f i n e d by f ' ( x , y ) = f(xy) - f ( y . x )

f•k[G]isarbitrary,

Let

e l e m e n t s a r e d e n s e in

whenever

f'E k [GgG]

~:GxTxT

at

In o t h e r

Further,

choose

(Such t e x i s t s ) .

x mzdimG 2- dimG

= d i m f i b r e at

-

95

T ~_ f i b r e at t.

-

t ~dimT

= r, since

T h i s p r o v e s (2).

Remark.

By the a b o v e m e t h o d of p r o o f , one c a n show:

c o n t a i n s a n a b e l i a n s u b g r o u p of d i m e n s i o n ~ r . Define

Si, Gi(i = 3, 4 . . . )

ZG(X)

The proof runs as follows:

as above. (e.g. define

b e l o n g to the s a m e t o r u s } ) .

For any x gG,

S3 =I(x,y,z}EG~G~G/x,y,z

T h e n (1) the c o m p o n e n t s of a n y e l e m e n t of S i,

h e n c e a l s o of G i , c o m m u t e w i t h e a c h o t h e r , a n d (2) the m a p

f i : G i + l - - - - - ~ G i,

fi(xl .....

xi+ 1) = (x 1 . . . . .

~ Si+l'gi(xl

= (x 1 . . . . .

xi,1), then fi°gi

x i) is s u r j e c t i v e .

( F o r : if gi: S i

= 1 on Si, h e n c e on G1 s o t h a t

f i ( G i + l ) ) . It f o l l o w s t h a t the m a p

P l °'f2 o . . . o fi : G i + l

finite subsets

(x,y 1.....

(Yl . . . . .

a subset such that noetherian).

yi ) with

ZG(y 1.....

Let

z6G

(x,y 1.....

yi}.

y i ) is m i n i m a l .

suchthat

fi+l

at

i.e.

z E c e n t r e of Z G ( y I . . . . .

(x,y 1.

. . . .

y i ).

Also, z 6ZG(X).

one gets:

:~ G is onto.

Consider

Gi+ 2 i . e .

let

G is

zEfibreof

yi } = ZG(y 1. . . . .

y i , z).

In o t h e r w o r d s , z r a n g e s

T h i s s u b g r o u p , b e i n g a f i b r e of f i + l ' h a s

by our earlier argument,

As an immediate corollary,

G i = fi(giGi) ~

(This is possible since

yi, z)6

xi)

Choose an i and such

By c h o i c e of y ' s , Z G ( y 1 . . . . .

o v e r a n a b e l i a n s u b g r o u p of ZG(X). dimension ~ r

yi)~Gi+l.

. . . .

whence our assertion.

If x i s r e g u l a r ,

then

ZG(X) ° i s a b e l i a n .

H o w e v e r it is n o t k n o w n w h e t h e r t h e c o n v e r s e i s t r u e o r n o t .

P r o p o s i t i o n 2. F o r

G = SL n o r

GLn,

(a) A s e m i s i m p l e eleme,,is r e g u l a r iff a l l of i t s e i g e n v a l u e s a r e d i s t i n c t f r o m each other,

(b) A u n i p o t e n t e l e m e n t i s r e g u l a r iff it i s a ' s i n g l e b l o c k ~ i n the J o r d a n - H o l d e r form.

(c) T h e f o l l o w i n g a r e e q u i v a l e n t :

-

96

-

(1) x is r e g u l a r . (2) T h e m i n i m a l p o l y n o m i a l of x is of d e g r e e n ( i . e . the m i n i m a l p o l y n o m i a l = characteristic polynomial). (3) Z(x) is a b e l i a n . (4) k n is c y c l i c a s

k IX]-module.

T h e proof of t h i s p r o p o s i t i o n is s t r a i g h t f o r w a r d and so i s o m i t t e d .

We now c h a r a c t e r i z e 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 .

P r o p o s i t i o n 3.

F o r a s e m i s i m p l e t £ G , the following s t a t e m e n t s a r e e q u i v a l e n t :

(a) t is r e g u l a r .

(b) ZG(t)° is a t o r u s , n e c e s s a r i l y m a x i m a l . (c) t b e l o n g s to a unique m a x i m a l t o r u s . (d)

ZG(t) c o n s i s t s of s e m i s i m p t e e l e m e n t s .

(e)

0< (t) ~ 1 for e v e r y root ¢< r e l a t i v e to e v e r y , o r to s o m e , m a x i m a l t o r u s

c o n t a i n i n g t.

Proof. B = T.U

We choose a t o r u s

T and a B o r e l s u b g r o u p B such that t E T , and

is the d e c o m p o s i t i o n as given in 3 . 2 .

(a) ~-> (b).

Since T C ZG(t)° and d i m T = r = d i m ZG(t) °, it follows that

Z G ( t ) ° : T.

(b) ~-~

(c).

Let tET',

a (maximal) torus.

a maximal torus.

Hence T' = ZG(t) o .

T h e n T C ZG(t) °, which i t s e l f is

T h u s t b e l o n g s to a unique m a x i m a l

t o r u s v i z . ZG(t)°.

(c) ~

(b).

c o n t a i n s t.

T C ZG(t) °.

F o r a n y g g ZG(t) ° , gtg "1 = t, h e n c e gTg -1

Hence by u n i q u e n e s s ,

gTg -1 = T.

Thus

T is n o r m a l in ZG(t) °

- 97 -

O

which is connected. Hence T is central in ZG(T) ° so that Z G ( t ) C

ZG(T ) =T.

This proves (b}. (b) ~

(d).

By c o r o l l a r y 4 to t h e o r e m 1 of 2.13, all the unipotent e l e m e n t s

in ZG(t) belong to ZG(t) °. ZG(t)° being a t o r u s .

Hence

But then e l e m e n t s of ZG(t) ° a r e s e m i s i m p l e , ZG(t) does not contain any unipotent e l e m e n t .

If x EZG(t), then Xs, XuC ZG(t) as well. is s e m i s i m p l e . (d) ~

(e).

F o r a r o o t s y s t e m R of G with r e s p e c t to T, let ~ ( t ) -- 1 for Since t . x ( ( c ) . t -1 = x

X o< C ZG(t ).

This c l e a r l y c o n t r a d i c t s (d).

(b).

x

This p r o v e s (d).

some e(ER.

(e) ~

Hence x u -- 1 and x = x s i . e .

We f i r s t prove:

(aO

w-l(d)~.O

Hence

rut "1 =o41T->0

x°((°( (t)' co() C U w a g a i n .

w'l(0()< 0 Thus

x = ( t u t ' l ) . n w. ( t ' . t - l . t b t ' l ) .

Thus

v~ d i m V V v @V.

v

(see the appendix to 2.11).

holds f o r all e l e m e n t s .

Hence T(C)g = T(Z) . g The equality holds f o r a l m o s t

Since C is h o m o g e n e o u s , the equality

Hence dim C = dim T(C)g ~ g @C. A l s o , dim T(Z)g>~

Thus dim C >~ dim Z, which gives dim C = dim Z.

a dense open s u b s e t of Z.

If C

I

Thus

C contains

w e r e any o t h e r c l a s s C Z, then by a s i m i l a r

a r g u m e n t as above, C' ~ dense open s u b s e t of Z ~

C' and C i n t e r s e c t ,

-

g i v i n g C' - C.

Thus

104

-

C is the unique c l a s s d_. Z and h e n c e

C ~ Z.

This

p r o v e s the t h e o r e m .

C l e a r l y , c o n d i t i o n (*) p l a y s an i m p o r t a n t r o l e in the p r o o f of the a b o v e t h e o r e m . H e n c e we t r y to find out g r o u p s f o r which (*) h o l d s .

P r o p o s i t i o n 1. L e t G be a g r o u p . a group G' isogenous

Then in the following c a s e s , t h e r e e x i s t s

to G and a f a i t h f u l r e p r e s e n t a t i o n

G ' e - - - ~ G L n of it

T

such that (*) h o l d s f o r

~,

the L i e a l g e b r a of G .

dition (•*) h o l d s : (**) T h e t r a c e f o r m of (a)

In f a c t , the s t r o n g e r c o n -

~1 n i s n o n - d e g e n e r a t e on ~ ' .

c h a r k - 0, G a n y s i m p l e g r o u p .

(b) G = G L n (c)

c h a r k ~ 2, G a n y s i m p l e g r o u p of t y p e Bn, Cn, D n.

(d)

c h a r k ~ 2, 3, G any s i m p l e g r o u p of t y p e G2,F4,E6, E 7.

(e)

chark

Proof.

~ 2,3,5,

G any s i m p l e g r o u p of t y p e E 8.

(a) c h a r k = 0.

a l g e b r a of G. Also, ~'

Then

C h o o s e G' = Ad(G) c G L ( ~ ) ,

G' i s i s o g e n o u s to G.

is just the Lie algebra

ad ~ ,

w h e r e ~ i s the L i e

(The c e n t r e of G i s d i s c r e t e ) .

h e n c e the t r a c e f o r m of ~ 1 n on ~ '

i s j u s t the K i l l i n g f o r m of ~ and it i s n o n - d e g e n e r a t e s i n c e ~ i s s i m p l e .

(b)

G = GLn;

the statement i s c l e a r .

(c) C h o o s e the n a t u r a l r e p r e s e n t a t i o n of G a s a c l a s s i c a l g r o u p (SOn o r Sp2~)

Consider SOn . W e claim that its Lie algebra 7 = { X E ~inIX =- xt}. For: O n

is given by functions (~ xij .Xkj- ;ik)i,kE k[Xll,...~n9 [I]. J Hence the Lie algebra ~ consists of derivations at I which vanish on these functions.

- i05

Hence

iff T( ~ x i j . J

T~°~

-

Xkj - ~ i k ) = 0 ~ i , k

(T(xij) .Ski + g i j " T(~kj)) : 0 V i , k

iff X

J iff T(Xik ) + T(Xki ) : 0 7 i , k . i.e.

The Lie a l g e b r a of On

is the set of all skew s y m m e t r i c m a t r i c e s . Since

SOn is the identity component of On, its Lie a l g e b r a is a l s o the s a m e . Let Tf~ = I X ¢ ~ l n / X = X t ~

x¢~}l n

( x + x t) =

2

(x-x +

F u r t h e r , for A C T ,

t) with x + x

2

'

2

t

xt ~TI~,

Thenany

£~.

X-2

BETTY, tr(AB) ffi t r ((AB) t) ffi tr(-BA) ffi - t r (AB). This

shows that tr(AB) = 0. d e g e n e r a t e on

, the s p a c e s o f s y m m e t r i c m a t r i c e s .

Hence ~ i n = 7 ( ~ T { "

Since the t r a c e f o r m is non-

7 In and 7 ,-n~ a r e orthogonal, (**) follows. 0

,ell .

Consider SP2 n

+1

-1 -1

W e claim that its Lie a l g e b r a

~

" -1

2n ( is given by: ~ = / X ~ 1 2 n / X M + MXt = 07 •

(This c l a i m is easily proved in the s a m e way as in the case of SOn. ) that X £ ~ i f f

XM is s y m m e t r i c .

Let Tt~ =

It i m m e d i a t e l y follows that ~ 12n = ~ +

I

We note

X 6 ~ 12n/XM is s k e w - s y m m e t r i c ~ .

Again, for A E ~ , B ~ T ~ ,

A =

MAtM; B = -MBtM.

Also, tr(AB) = tr(MAt. M. (-MBtM)) = tr(M. AtBtM) as M 2 = -I ~-%~AtBt) as can easily be verified = -tr (BA)t = -tr(BA) = -tr(AB).

Hence tr(AB) = 0 since char k ~ 2. Hence

~ 12n = ~e~¢l and the result

follows. F o r (d) and (e), we again choose the adjoint r e p r e s e n t a t i o n .

We o b s e r v e that

the Lie a l g e b r a of a simple group p o s s e s s e s a special b a s i s , called ~ h e v a l l e y

-

basis'.

106

-

We calculate the discriminant of the Killing form with respect to this.

This is a number (integer) which is divisible only by 2, 3 for th~ groups in (d) and by 2, 3, 5 for the group in (e).

Hence (d) and (e) hold if we put the suitable

restrictions on char k.

Definition.

Given a r o o t s y s t e m (of a r e d u c t i v e g r o u p G), a p r i m e

p is said

to be 'good' with r e s p e c t to it if p satisfies: (1) Root s y s t e m s i m p l e , and of type: An : p a r b i t r a r y Bn, Cn, D n : p ~ 2 G 2 , F 4 , E6, E 7 : p ~ 2 , 3 , E 8 : p ~ 2,3,5. (2) Root s y s t e m is no__~tsimple.

Let R = R 1U . . . IJ R k be the simple componentsj !

then p is good with r e s p e c t to each of I~ s (as defined in (I)). 1

Remark.

The p r o p e r t y

T h e o r e m 2.

'p good' is inherited by i n t e g r a l l y c l o s e d s u b s y s t e m s .

If G is r e d u c t i v e and c h a r G (-- c h a r k) is good (with r e s p e c t to

the r o o t s y s t e m of G), then the n u m b e r of unipoteut c o n j u g a c y c l a s s e s is finite. Proof• Let G = GL n (or SLn).

w h e r e A is of the type:

Ii

E v e r y unipotent e l e m e n t is of the f o r m : g . A . g - l ,

0 . . .0~

w h e r e each A. is of the f o r m :

A2. . •0 L

1o ::I 0

1

i

•

•

....

•~.

•

I

. .

Ak ]

0. . . .

l~i~k.

(The J o r d a n n o r m a l f o r m ) .

Let r l , . . . , r

k

-

be the ' b l o c k - s i z e s ' .

Then

r 1.

107

-

r k c o m p l e t e l y d e t e r m i n e the c o n j u g a c y

. . . .

c l a s s (of u n i p o t e n t e l e m e n t s ) to which A b e l o n g s .

In o t h e r w o r d s , the n u m b e r

of d i s t i n c t c o n j u g a c y c l a s s e s = the n u m b e r of c o l l e c t i o n s of i n t e g e r s (r 1 . . . . . such that

r i ~ 0 and

~- r i = n (= p(n), the n u m b e r of p a r t i t i o n s of n into n

non-negative integers}. p e c t i v e of c h a r

r n)

H e n c e the t h e o r e m is t r u e f o r G L n o r

SL n ( i r r e s -

G).

L e t G be any a r b i t r a r y s e m i s i m p l e g r o u p .

T h e n t h e r e e x i s t s an i s o g e n y

!

f : G ~

G, w h e r e G '

is s i m p l y c o n n e c t e d .

Now, the n u m b e r of u n i p o t e n t

,

!

c o n j u g a c y c l a s s e s of G = t h a t of G . In o t h e r w o r d s , we m a y a s s u m e

H e n c e we m a y p r o v e t h e t h e o r e m f o r G .

G to be s i m p l y c o n n e c t e d .

G, b e i n g s e m i -

s i m p l e and s i m p l y c o n n e c t e d , i s a f i n i t e d i r e c t p r o d u c t of s i m p l e g r o u p s H e n c e the t h e o r e m n e e d b e p r o v e d only f o r s i m p l e g r o u p s a b o v e m a y b e a s s u m e d to be d i f f e r e n t f r o m f o l l o w s that c h a r

G i is a l s o good.

c l o s e d s u b s y s t e m of that of G).

G L n i ' ( ~ t : Gi classes.

) GLni

Since GLni

f o l l o w s that

G i , which by the

Since c h a r

G is good, it

(The r o o t s y s t e m of G i is an i n t e g r a l l y

Now the p r o o f of the p r o p o s i t i o n 1 s h o w s that

the c{,ndition (*) is s a t i s f i e d f o r G i. s u i t a b l e i s o g e n o u s group).

A r.

G i.

(In f a c t , (**) i s s a t i s f i e d , by taking a

H e n c e by R i c h a r d s o n ' s t h e o r e m , a n y c l a s s of

is a f a i t h f u l r e p r e s e n t a t i o n ) m e e t s

G i in f i n i t e l y m a n y

i t s e l f h a s f i n i t e l y m a n y u n i p o t e n t c o n j u g a c y c l a s s e s , it

G also has this p r o p e r t y .

T h i s p r o v e s the t h e o r e m .

Remark.

It i s not known w h e t h e r the h y p o t h e s i s on c h a r G i s n e c e s s a r y o r not.

C o r o l l a r y 1.

In a r e d u c t i v e g r o u p

G, with c h a r

G good, the n u m b e r of c o n -

j u g a c y c l a s s e s of c e n t r a l i z e r s of e l e m e n t s of G is f i n i t e .

-

Proof.

Let

108

-

T be a m a x i m a l t o r u s of G.

T h e n the n u m b e r of c e n t r a l i z e r s ,

in G, of e l e m e n t s of T i s fiDite (by c o r o l l a r y to p r o p o s i t i o n of 2 . 7 ) .

Since

any s e m i s i m p l e e l e m e n t is c o n j u g a t e to an e l e m e n t in T, it f o l l o w s that the n u m b e r of c o n j u g a c y c l a s s e s of c e n t r a l i z e r s of s e m i s i m p l e e l e m e n t s i s f i n i t e . Let

x 6G, x = s.u

be the J o r d a n d e c o m p o s i t i o n .

Then

ZG(X) = Z G ( S ) / ~ ZG(U).

CUp to c o n j u g a c y , t h e r e a r e f i n i t e l y m a n y p o s s i b i l i t i e s f o r

ZG(S). ) A g a i n ,

u ~ Z G ( S ) ° (by c o r o l l a r y 4 to t h e o r e m 1 of 2 . 1 3 ) . ZG(S)° i s r e d u c t i v e (by p r o p o s i t i o n 4 of 3.5) and c h a r (ZG(S)° } c a n b e s e e n to be good.

H e n c e up to c o n -

j u g a c y , u has f i n i t e l y m a n y p o s s i b i l i t i e s in ZG(s) ° .

ZG{X) i t s e l f

h a s f i n i t e l y m a n y p o s s i b i l i t i e s in G.

Remark.

Hence

T h i s p r o v e s the c o r o l l a r y .

By u s i n g a s i m i l a r m e t h o d , one can p r o v e the following:

c o n n e c t e d , r e d u c t i v e g r o u p with c h a r G - 0 o r s u f f i c i e n t l y l a r g e . on an a f f i n e v a r i e t y V. p o n e n t s of G v (v ~ V )

Let G act

T h e n the n u m b e r of c o n j u g a c y c l a s s e s of L e v i c o m is f i n i t e .

(If we w r i t e

the u n i p o t e n t r a d i c a l and M r e d u c t i v e , then

T h e o r e m 3.

L e t G be a

G v = M . U , s e m i d i r e c t , with U M is c a l l e d a L e v i c o m p o n e n t . )

L e t G be a r e d u c t i v e g r o u p with c h a r

G good (or with the

property: G has finitely many unipotent conjugacy classes). s e t of a l l u n i p o t e n t e l e m e n t s in G.

L e t V b e the

Then,

(a) V is a c l o s e d , i r r e d u c i b l e s u b v a r i e t y of G and it has c o d i m e n s i o n

r

in

G (r = r a n k of G).

(b) V c o n t a i n s a unique c l a s s of r e g u l a r e l e m e n t s .

(Thus, in p a r t i c u l a r ,

r e g u l a r u n i p o t e n t e l e m e n t s e x i s t in c a s e c h a r G i s good).

T h i s c l a s s i s open

d e n s e in V and i t s c o m p l e m e n t has c o d i m e n s i o n / > 2 in V.

109

-

Proof.

Take a faithful representation

-

in G L n of G.

in G L n f o r m a c l o s e d s e t (A i s u n i p o t e n t It f o l l o w s t h a t group).

V i s a l s o c l o s e d in G.

Fix a Borel

Define

S C G/B~G

subgroup

: G ~U--~G/B×G

( T h i s , of c o u r s e ,

Pl

is t r u e if G i s a n y

B in G. EUt

, U = B u.

It i s c l e a r t h a t

S

A l s o , S i s t h e i m a g e of G x U u n d e r t h e m o r p h i s m

g i v e n by : ~ ( g , x ) -- (gB, g x g ' l ) .

Consider the projection Pl(S) = G/B.

iff ( A - I ) n = 0, a p o l y n o m i a l c o n d i t i o n ) .

by : S = { (gB, x ) / g ' l x g

i s w e l l d e f i n e d and c l o s e d .

Now t h e u n i p o t e n t m a t r i c e s

Hence

S is irreducible.

of G / B X G o n t o t h e f i r s t f a c t o r .

A l s o , the f i b r e s of Pl

Clearly

a r e c o n j u g a t e s of U, h e n c e a r e of t h e

same dimension. Hence

dim S = dim

S = dim G/B + dim U

= dim G - dim B + dim U = dimG

- r (since dimB

Consider the projection

P2

- dimU

o n t o the s e c o n d f a c t o r .

we s h o w t h a t s o m e f i b r e of P2 i s f i n i t e . d i m V. that

Hence

= r).

V has codimension

r

c l e a r l y p r o v e t h a t t h e f i b r e of P2

in G.

over

P2(S) = V.

Now

T h i s p r o v e s t h a t d i m S = d i m P2(S) = Choose

We s h o w : g ' l x g

c~ ~ 0 for all simple roots.

Clearly

x = ~ x

6U~

gEB.

(c~)EU

such

This will

x is f i n i t e , in f a c t c o n s i s t s of o n l y one

e l e m e n t v i z . (B, x). A s s e e n in p r o p o s i t i o n 4 of 3 . 5 , g - 1 = U . n w . b " u ~ U w.

One c a n a s s u m e

Nowwehave:

n xn

n

g-1 xg~U

.x.n "IE'~-X w

w

o( >0

, , also.

i.e.

Hence

i.e.

W

-1 W

E U.

Now

w(~) > 0 whenever

ctK ~ 0.

In p a r t i c u l a r ,

This clearly means

w = Id.

(w t a k e s t h e

wloc;

w(~) > 0 f o r a l l s i m p l e r o o t s fundamental chamber T h i s 0 r o v e s (a).

Unw.x.n-lu-lEu w

b = 1.

~.

into itself).

(The a r g u m e n t

Hence

n

W

E T,

above proves,

so that

g £B.

incidently, that

gBg " 1 ; B @ g ~ B

(for r e d u c t i v e g r o u p s ) ) .

(b) S i n c e

G has only finitely many unipotent conjugacy classes,

V is a finite

-

u n i o n of c o n j u g a c y c l a s s e s .

Since

class in V also has dimension Also,

closure,

its closure

it follows that

Remarks.

dim G - r.

(corollary (b).

Hence this class

Now a n y o t h e r c l a s s i n V

4 to proposition

The statement

is the existance

unipotent elements

1 of 1 . 1 3 ) a n d h e n c e

about codlin is proved later.

(a) a n d (b) h o l d in a r b i t r a r y

(All we r e q u i r e

C o is

Since any class is open in its

C o i s o p e n ( d e n s e ) in V .

(1) T h e c o n c l u s i o n s

(2) T h e s e t of i r r e g u l a r

dim V = dim G - r, it follows that some

equal to

This proves

we shall use them.

of V i s e q u a l t o t h e d i m e n s i o n

i s t h e w h o l e of V .

i s of s t r i c t l y l o w e r d i m e n s i o n cannot be regular.

-

Hence dimension

of a t l e a s t o n e of t h e c l a s s e s .

regular.

110

characteristics

of r e g u l a r

and

unipotent elements).

is closed in G (and has codimension

>/2).

3.7

Regular

elements

We now characterize

Theorem

1.

Let

(continued).

the regular unipotent elements.

G be a reductive

group,

B = T. U, a Borel subgroup containing G.

T.

T

a maximal

Let

x

torus and

be a unipotent element in

Then the following statements are equivalent.

(a) x

is regular.

(b) x b e l o n g s t o a u n i q u e B o r e l s u b g r o u p . (c)

x belongs to finitely many Borel subgroups.

(d) If x E U ,

x =~-~ ~>0

x¢0

~

° ).

It f o l l o w s t h a t x ~ Y B ( y E U ) , a("

-

o

O

which a r e i n f i n i t e l y m a n y in n u m b e r .

( F o r d ~ 0, X_~b(d) ~ B . )

This contra-

d i c t s o u r a s s u m p t i o n in (c) and h e n c e p r o v e s (d).

(d) ~

(a).

C l e a r l y , the e l e m e n t s s a t i s f y i n g the c o n d i t i o n (d) a r e d e n s e in V.

By t h e o r e m 3, r e g u l a r u n i p o t e n t e l e m e n t s a r e a l s o d e n s e in V. such t h a t x ° i s r e g u l a r and s a t i s f i e s (d).

Hence ~ XoE U

O u r c l a i m i s t h a t x and x ° a r e

c o n j u g a t e (in B) and t h i s p r o v e s the i m p l i c a t i o n .

F o r the p r o o f we d e v e l o p e h e r e s o m e m a c h i n e r y which w i l l b e u s e f u l in l a t e r discussions also.

(1) C o m m u t a t i o n F o r m u l a e : For positive roots ~,/3, p o l y n o m i a l in t , u .

(Xc0 T h i s p o l y n o m i a l is i d e n t i c a l l y z e r o if r ~ io( + j ~ ( i , j

is a

integers~> 1) and P i c ( + j / ~ (t,u) = c ( i , j ) t i . u J, c ( i , j ) E k. T h i s can be e a s i l y p r o v e d in the s a m e m a n n e r a s the p r o o f of p r o p o s i t i o n 2 of 3 . 3 .

(2) For

a simple root o~i, let U i =-[[¢~>0

X

Let U' = e("

./~ U i. Then the l.

H e n c e d i m P i - d i m U i = r + 2.

Now, P. 2 B and h e n c e G / p i i s c o m p l e t e . 1

C o n s i d e r the s e t S i ~ _ G / P i × G,

givenby: S i = {(gPi. x)/g'ixgEUil (whichis welldefined). Then Si is c l o s e d and i r r e d u c i b l e (being i m a g e of the m o r p h i s m ~(g,x) = (gPi" g x g ' l ) ) '

GXU i

~ > G / P i ;< G,

By p r o j e c t i n g onto the f i r s t f a c t o r ,

d i m S i = d i m G / P i + d i m U i (by an a r g u m e n t s i m i l a r to one in t h e o r e m 3) = d i m G - (r+2). By t a k i n g p r o j e c t i o n onto the s e c o n d f a c t o r , P2(Si) = Vi = U g u i g gt~G d i m V . ~ d i m S. = d i m G - (r+2). 1 1 Now, b y the p r o p o s i t i o n a b o v e , for some

i.

-1

and

an u n i p o t e n t e l e m e n t x i s i r r e g u l a r iff x EV.1

Hence, from above,

d i m (V - r e g u l a r unl. e l e m e n t s ) = sup d i m V i /A r

given by

1

p(g) = (X 1 (g) .

. . . .

Xr(g)).

Theorem

Let

F

I.

T h e n we h a v e :

b e a n y f i b r e of p.

(a) F

is a closed, irreducible

(b) F

i s a u n i o n of c o n j u g a c y c l a s s e s ,

cteristics.

s u b v a r i e t y (of G)

which has codimension

r

in G.

f i n i t e in n u m b e r in c a s e of g o o d c h a r a -

-

(c) F

117

-

c o n t a i n s a u n i q u e c l a s s of r e g u l a r e l e m e n t s .

d e n s e in F (d) F

and i t s c o m p l e m e n t h a s c o d i m e n s i o n

c o n t a i n s a u n i q u e c l a s s of s e m i s i m p l e

in F , a n d i s c h a r a c t e r i z e d

Proof.

Since there always exists

r-tuple

(el

contains a semisimple

this class is unique. corresponds

to

S.

class

S.

in F , i s in t h e c l o s u r e of a n y of

F

such that

X.(t} = C. ~ i , f o r a p r e g i v e n 1

1

is always non-empty.

c l o s e d and i s a u n i o n of c o n j u g a c y c l a s s e s F

This class is closed,

b y a n y of t h e s e p r o p e r t i e s .

t ET

C r ), it f o l l o w s t h a t

. . . . .

>~ 2 (in F ).

elements.

has the minimal dimension among the classes the classes

T h i s c l a s s i s o p e n and

(since

Clearly

is

c X s are class functions}. i

B e c a u s e of c o r o l l a r y

3 to t h e o r e m

B y p r o p o s i t i o n 4 of 3 . 6 , p i c k t h e r e g u l a r c l a s s Since

F

Xi(x} = Xi(Xs), it f o l l o w s t h a t

C q F

Also,

2 of 3 . 4 , C which

a s w e l l and i s

unique. We claim that contain

y.

C i s d e n s e in F .

Let

yEF

H ' , o p e n in G, be s u c h t h a t

N o w , Yu £ Y -1 s " H'/% V, w h e r e reductive group

So l e t

O

ZG(Ys) ,

u n i p o t e n t e l e m e n t s in

Now b y t h e o r e m

Xi(x) = Xi(Ys) ~{ i, it f o l l o w s t h a t

Z

o

x~G.

Hence

H, o p e n in F ,

Consider

x EC

ZG(Ys )°. of t h e

3 of 3 . 6 , t h e c l a s s of a l l r e g u l a r

ZG(Ys )° i s d e n s e in V.

Since

is

H ' / ~ F = H.

and

V i s the s e t of a l l u n i p o t e n t e l e m e n t s

ZG(Ys )°, w h i c h i s a l s o in y - 1 H'(% V. s

G(Ys} , s o

be a r b i t r a r y

H e n c e ~ u, r e g u l a r u n i p o t e r t

Hence x ~F.

x = Since

Ys" u x

£H'

and

x u = u.

= u i s r e g u l a r in

a n d a l s o in H /% F = H.

This proves the

claim.

Now

C is irreducible

g----~gxog

-1

, x o E C fixed}.

of C a n d o t h e r c l a s s e s i s o p e n in F s i o n -- r .

( b e i n g i m a g e of G u n d e r t h e c o n j u g a t i o n m a p

F

F

itself is irreducible.

of s t r i c t l y l o w e r d i m e n s i o n .

as well, since

Hence

Hence

C = F.

But then

also has codimension

r.

Hence

Now F

is union

d i m F = d i m C. C

C i s r e g u l a r and h a s c o d i m e n T h e f a c t t h a t c o m p l e m e n t of C

-

has

codlin ~2,

3.6.

118

-

i s d e r i v e d e a s i l y f r o m a s i m i l a r s t a t e m e n t in t h e o r e m 3 of

This proves (a), (b) and (c).

(d) F o r a n y c l a s s

S 1 ~ F , we p r o v e the e q u i v a l e n c e of the f o l l o w i n g s t a t e -

ments: (i) S~ is the (unique) (ii)

S

1

semisimple class

S.

is closed.

(iii) S 1 h a s m i n i m a l d i m e n s i o n a m o n g c l a s s e s in F . (iv) S 1 b e l o n g s to the c l o s u r e of any of the c l a s s e s of F .

(iv) ~

(iii) ~

(ii).

s i t i o n 1 of 1 . 1 3 .

(ii) ~ ' ( i )

(i) ~

T h e s e follow i m m e d i a t e l y f r o m c o r o l l a r y 4 to p r o p o -

(Note that F i s c l o s e d . )

i s a l r e a d y p r o v e d in c o r o l l a r y 5 to t h e o r e m 2 of 3 . 4 .

(iv) f o l l o w s f r o m L e m m a in 3.4 and u n i q u e n e s s of S.

in F , t a k e x E K ,

then X s E S

also, XsEClK

(Take a n y c l a s s E

~SficlK).

T h i s p r o v e s the t h e o r e m c o m p l e t e l y .

T h e o r e m 2.

T h e r e g u l a r c l a s s e s have a n a t u r a l s t r u c t u r e of a v a r i e t y , i s o -

morphic to/~r

u n d e r the m a p p : G r e ~ - - ~ r

where

Gr e g

i s the open

v a r i e t y of G of r e g u l a r e l e m e n t s .

T h e p o i n t s to b e p r o v e d are: (1)

p i s a m o r p h i s m and i t s f i b r e s a r e j u s t the ( r e g u l a r ) c l a s s e s .

(2)

p ~ ( k [ ~ r l ) = k ~Greg~ Int G

(3)

If f ~ k [ / ~ r j , x E G r e g , then f i s d e f i n e d at p(x) iff p*(f) is d e f i n e d

a t x.

-

119

-

T h e p r o o f of t h e s e p o i n t s i s s t r a i g h t f o r w a r d and i s o m m i t t e d f r o m h e r e . (e. g. x,y

r e g u l a r and p(x) = p(y)~v_--~P(Xs) = p ( y s ) ~ = ~ x s c o n j u g a t e to y s ~

x

c o n j u g a t e to y}.

We now g i v e a f i n a l i m p o r t a n t c h a r a c t e r i z a t i o n of r e e-ular e l e m e n t s (in c a s e of s i m p l y c o n n e c t e d g r o u p , of c o u r s e } .

T h e a r e m 3.

Let x ~G, p : G--~/~r

surjective, i.e.

as before.

Then

x i s r e g u l a r iff (dp) x i s

iff (dXi) x (1 • i ~r} a r e l i n e a r l y i n d e p e n d e n t .

We p o s t p o n e the p r o o f of t h i s t h e o r e m f o r a while and g i v e a d e v e l o p m e n t which w i l l e v e n t u a l l y p r o v e i t and at the s a m e t i m e w i l l p r o d u c e a c r o s s - s e c t i o n to the c o l l e c t i o n of r e g u l a r c l a s s e s .

Cross-Sections.

L e t G , T , ~ 1 ' . . . . g r " X1 . . . . .

Xr be as before.

Pick

n i 6 N(T}, a r e p r e s e n t a t i v e f o r w i = r e f l e c t i o n r e l a t i n g to ~ i " Consider X_, .n.X , n_.X . n r = C. ~'I 1 ~2" z" g r

We s h a l l show t h a t C i s a c r o s s -

s e c t i o n of the r e g u l a r c l a s s e s . We h a v e , n l X ¢ ~ 2 = X W ~ ( v f 2 ) . n 1 ,• n l n 2 X ~ 3 P r o c e e d i n g in t h i s way, we get, C = X ~ I ' X ~ 2 ' ' "

= Xwlw2(~3). nln 2 etc. X~r "nl .... nr

where

~ i = (Wl . . . . w i . 1 ) ' ( ~ i) i 1 , < i ~ r . !

Now the following f a c t s a b o u t t h e s e (1)

~1'" "'~r

P i s can be v e r i f i e d e a s i l y :

a r e j u s t the p o s i t i v e r o o t s which a r e m a d e n e g a t i v e by

w_ 1 = w -r I " " " Wl-1 "

(This f o l l o w s e a s i l y f r o m the fact: w i p e r r ~ u t e s a l l the

p o s i t i v e r o o t s o t h e r than a~.. ) 1

(2) /~1 . . . . .

~r

are linearly independent

a c o n s e q u e n c e of (1), s ~ i + t ~ j

(for ~ i = Hi+ e a r l i e r ~ . s ) . H e n c e , a s 3 i s not a r o o t f o r i ~ j, s and t i n t e g e r s ~ 1 .

-

(3)

120

-

A s a c o n s e q u e n c e of the c o m m u t a t i o n f o r m u l a e in p r o p o s i t i o n 2 of 3 . 6 , it

follows that Thus

X/~ i and X/~j

C may be written

commute elementwise.

U w . n w with w = w 1 . . .

of an a f f i n e r - d i m e n s i o n a l

w r , the t r a n s l a t i o n b y n w

space.

W e now a i m to p r o v e : r

Theorem

4.

Let

C =-[]- X

n. be a s a b o v e .

i=l ~ i 1 (a) C is a closed subset of G and isomorphic, as a variety, to /~r the co-ordinates

c o m i n g f r o m t h o s e of X '

(b) C is a c r o s s - s e c t i o n

s.

0( i

of the c o l l e c t i o n of r e g u l a r c l a s s e s .

H e r e (a) f o l l o w s f r o m the a b o v e d i s c u s s i o n . T o p r o v e (b) we r e q u i r e :

Theorem

5. p : C

is an i s o m o r p h i s m

Before proving this theorem,

of v a r i e t i e s .

we c o n s i d e r the e x a m p l e

r o o t s a r e ~o((i, i+l)~,

and the c o r r e s p o n d i n g

G = S L n. T h e s i m p l e

unipotent groups are

1 1 0 V i , V(kl~ ' ' k r ) E V ~ Consider and n ~

(* * )

.v

r

(k I ..... kr)

with

Trv(Y) = A~ T r ~ 76A. y . n ~ : VA~-~V

is the injection.

~--~+~(kj-mj)~j. J

, where

From

71"A:V----~ V A

($~) above, y.v

is the projection

contributes to the

t r a c e only if kj " mj ~ j . Hence: (i) The contribution is zero if m. < 0 for some j i . e . if ~ is not a dominant 3

weight:

for the right side of

(2) If ~ is dominant ~ A the contribution,

(~-~) has no term back in

V k.

and mj = 0 for some j, then t. does not occur in

In other words, only those t[s m a y o c c u r w h e r e J

But then in that case ~ i >~j"

m. ~ 0. J

Hence the contribution is a polynomial in e a r l i e r

t'. S. (3)

If

~ =Ai'

then

m@ = O V j ~ i)

Hence the contribution is A

ci

independent of the

observe that each nonzero since also

xq i %v_ . ~i : ~ i _ ~ i ,

cj's, by

yj, j ~ i, wj~i = ~i

niV~i_~ i = V~i_Q i

slnce

V,

i l

We claim that

ci ~ O.

acts as a nonzero scalar on and

thus Z %[

~i+~(iis not a weight on

(**).

c.t. with

dim

V~i = 1).

:

neither is

n i ~ <X~i,X_~i>.

~

X

wi(~i + ~ i ) = ~ i This gives

<scalar by

i c.i = 0

Hence

But also

For this we

V - 2~).

(**),

gives

-% =X)% i<since Hence

-

123

-

a c o n t r a d i c t i o n s i n c e ni(V~i } = V~ i" ~i"

T h i s c l e a r l y p r o v e s the r e s u l t .

!

Remark.

By c h o o s i n g n j s p r o p e r l y , it is possible to p r o v e :

Xi(Y) -- t i + p o l y n o m i a l / 7A in e a r l i e r

t's. J

P r o o f of t h e o r e m 5. F r o m the above i e m m a , it follows that each ti c a n w r i t t e n in the f o r m : Xi(Y) + p o l y n o m i a l in e a r l i e r

I

Xj(y) s.

In view of the p a r t (a) of

of t h e o r e m 4, it follows i m m e d i a t e l y that p:C . . . ~ r

is an i s o m o r p h i s m of

varieties.

We now p r o v e t h e o r e m 3 and p a r t (b) of t h e o r e m 4 s i m u l t a n e o u s l y b y p r o v i n g :

T h e o r e m 6.

L e t x C G, C c G a s above.

T h e n the following c o n d i t i o n s a r e

equivalent: (1) x is r e g u l a r (2) (dp)

is surjective, X

(3) x is c o n j u g a t e to s o m e e l e m e n t in C.

(1) ~ v ~ (2) i s the t h e o r e m 3 and by (3), C c o n s i s t s of r e g u l a r e l e m e n t s , one f r o m each c l a s s ( T h e o r e m 1 and 5).

Proof.

(3) ~

(2).

C l e a r l y , one m a y a s s u m e x E C .

(dp) x : (TC) x

~ iT/A

(TG} x.

(dp}

Hence

r) p(x)

is s u r j e c t i v e .

Now by t h e o r e m 5,

A l s o , (TC) x is a s u b s p a c e of

is surjective. X

(2) ~

(1).

We prove:

x i s i r r e g u l a r = ~ dX 1 . . . . .

dX r

are linearly inde-

p e n d e n t at x ( i . e . e q u i v a l e n t l y (dp) x is not s u r j e c t i v e ) .

Step I. We need p r o v e the above s t a t e m e n t for s e m i s i m p l e e l e m e n t s only.

-

124

-

F o r : By l e m m a 2 of 3 . 7 , the s e t of s e m i s i m p l e , i r r e g u l a r e l e m e n t s is d e n s e in the s e t of a l l i r r e g u l a r e l e m e n t s . e l e m e n t s at which dX 1 . . . . .

dX r

a r e l i n e a r l y d e p e n d e n t is c l o s e d .

a b a s i s to the t a n g e n t s p a c e a t e. to the t a n g e n t s p a c e at y. of the b a s i s e l e m e n t s .

Hence o u r c l a i m is c l e a r if the s e t of i r r e g u l a r Now choose

T h e n the left t r a n s l a t i o n by y g i v e s a b a s i s

C o n s i d e r the dual b a s e . dX i is a l i n e a r c o m b i n a t i o n

Now f r o m the choice of the b a s i s , it is c l e a r that dX i

is a v e c t o r field on G ( i , e . the c o e f f i c i e n t s p o l y n o m i a l f u n c t i o n s ) . m a t r i x f o r m e d by t h e s e c o e f f i c i e n t s .

T h e n dX 1. . . . .

dX r

C o n s i d e r the

are linearly dependent

at a point y iff a l l the r - m i n o r s of this m a t r i x v a n i s h at that point.

This is

c l e a r l y a p o l y n o m i a l c o n d i t i o n and the set of points at which it is s a t i s f i e d is a c l o s e d set.

T h i s p r o v e s I.

H e n c e we m a y a s s u m e that x is s e m i s i m p l e .

We m a y f u r t h e r a s s u m e that x

b e l o n g s to T .

Step If. F o r a class function X E k [ G ] , the

tangent space to T

at x.

(dX)x = 0 iff d X x / [ = 0, w h e r e

Consider the big cell U .T.U.

Clearly, the

tangent space to it at x is the s a m e as the tangent space to G .

w e m a y consider of roots.

r

U .T. U

instead of G.

Consider the m a p

~ :K ---~U'.T. U

=~4-,,~nx (~,~)~0 i 1

} ~ 0i.e.

= 0 (det

wi

-- 0.

Thus

w.(~-') = ~-'. i =

-1}.

1

#

and

It, n o w f o l l o w s

Hence terms

in s k e w -

f -- s k e w ~ , p r o v i n g the

h o l d s in c a s e of c h a r k ~ 2.

But the lemma considered

m a y b e v i e w e d a s a f o r m a l i d e n t i t y to b e p r o v e d in t h e g r o u p a l g e b r a of X ( T ) over

~

f r o m t h e h y p o t h e s i s (*).

H e n c e i t c o n t i n u e s to h o l d e v e n if c h a r k = 2).

-

127

-

We now u s e an i d e n t i t y due to Weyl viz. f = skew ~ = ~ .'~-(1- ~0 ( p r o p o s i t i o n 3 of 3.5). Hence f(x) -- 0 so that - ~ d X i = 0.

It now follows i m m e d i a t e l y that

1

(dX1) x . . . . , (dXi) x . . . . , (dXr) x a r e l i n e a r l y d e p e n d e n t . i m p l i c a t i o n (2) ~

(1) ~

(3).

(1).

Let x ~ G

be r e g u l a r .

(This is p o s s i b l e b y t h e o r e m 5. ) ~egular

((3) ~

T h i s p r o v e s the

(2) ~

the s a m e f i b r e of p.

(1)).

Pick

y~C

such that Xi(Y) = Xi(x) ~ i.

A s a l r e a d y p r o v e d , a n y e l e m e n t of C is

T h u s , x and y a r e both r e g u l a r and b e l o n g to

Hence by t h e o r e m 1 above,

x and y a r e c o n j u g a t e .

T h i s p r o v e s the i m p l i c a t i o n and the d e v e l o p m e n t in t h e o r e m s 1 to 5.

Problem

(Open)

Can we find a n o r m a l f o r m f o r n o n - r e g u l a r e l e m e n t s i n

a r b i t r a r y s i m p l e g r o u p s c o r r e s p o n d i n g to the one c o n s i s t i n g of s e v e r a l J o r d a n b l o c k s in SL n ?

T h e o r e m 7.

Let F

be a f i b r e of the map p:G

~r

.

(See t h e o r e m 1 a b o v e . )

(a) T h e r e g u l a r e l e m e n t s of F a r e j u s t the s i m p l e o n e s .

(An e l e m e n t in F

is s i m p l e o r n o n - s i n g u l a r if the d i m e n s i o n of the t a n g e n t s p a c e to F a t that point e q u a l s the d i m e n s i o n of F . (b) F

Such e l e m e n t s f o r m a d e n s e open set in F . )

i s n o n - s i n g u l a r in c o d i m e n s i o n 1.

(c) The i d e a l of F

in k [GJ ( i . e . the ideal of f u n c t i o n s in k [G]

on F ) i s g e n e r a t e d by I X i - Cit

if F -- p ' l ( c 1 . . . . Cr). l~ 0. ucgB.

Hence ~ u'~U

Write u'-- -N-x Ic ). ~>0 w(~)0 w~)>0

suchthat

u =gu'.

133

-

nwu' = - ~

Hence

) . 77-

(c'

~>o

-

x

Xw(~) w(~) /~>o

w(~) 0.

Wk_ l . w k ( ~ k ) < 0 , it follows that C ~ k

normalizes

u'j.

V = [

Let

of Vj

V. w i t h 1

Vi cyclic for

x and

n i = d i m V..1 If y E ZG(X), t h e n

Bij

T h e s p a c e of s u c h h o m o m o r p h i s m s generator

Here all the conjectures

are a w

( E x e r c i s e : do t h i s . ) We m a y v e r i f y C o n j e c t u r e 3 d i r e c t l y

with

V i.

S L n.

u ( u n i p o t e n t ) in n o r m a l f o r m we c a n e a s i l y p r o d u c e

xEG,

a q u o t i e n t of V.

G = GL n or

y has the

an x - m o d u l e h o m o m o r p h i s m

has dimension

rain (ni, nj),

m a y go to a n y e l e m e n t of V i w h i l e if i ~ j

Vj a

of Vj i n t o

(If i > j , t h e n we

-

136

-

apply this to the dual. ) Hence d i m ZG(X) = .~'-. m i n ( n i , nj) = n+2 .~_, m i n ( n i , nj) (one l e s s f o r SLn).

T h u s d i m C(x) = n 2- n - 2 . ~ • m i n ( n i , nj), an e v e n n u m b e r . 1 $ If t h e r e is a s i n g l e block, then in SLn, d i m ZG(X) = n - 1 -- r so that x is r e g u l a r as a s s e r t e d e a r l i e r .

(See p r o p o s i t i o n 2 of 3 . 5 . ) If t h e r e a r e two b l o c k s

of size n - I and i, then dim ZG(X) -- r+2 and x is in the c l a s s of " s u b r e g u l a r e l e m e n t s " which we shall study p r e s e n t l y .

Henceforth, we a s s u m e G to be a simple algebraic group.

T h e o r e m 2 ( R i c h a r d s o n ) . Let

P be a p a r a b o l i c s u b g r o u p of G.

Let U p be

the u n i p o t e n t r a d i c a l of P. (a) T h e r e e x i s t s in U p a d e n s e open s u b s e t of e l e m e n t s , each of which is c o n t a i n e d in a finite n u m b e r of c o n j u g a t e s of U p (in G).

(b)

GUp =

LJ gUpg -1 is a c l o s e d , i r r e d u c i b l e s u b s e t of d i m e n s i o n = d i m G gEG

dim P / U p . (e) If G has a f i n i t e n u m b e r of u n i p o t e n t c l a s s e s ( e . g . in e a s e of good c h a r a cteristics), then G U p c o n t a i n s a u n i q u e d e n s e e t a s s as its own. under

In this case, C /% U p

is dense in U p

C of the s a m e d i m e n s i o n

and f o r m s a single class

P.

We o b s e r v e that t h e s e facts have a l r e a d y b e e n p r o v e d f o r P = B.

(See t h e o r e m 3

of 3 . 6 , )

Proof. We m a y a s s u m e that P c o n t a i n s B, a fixed B o r e l s u b g r o u p .

We u s e

h e r e s o m e of the s t a n d a r d facts about such p a r a b o l i c group.= T h e s e f a c t s a r e : P has the following d e c o m p o s i t i o n :

P = ?rip. Up, w h e r e

group and the p r o d u c t is s e m i - d i r e c t . fied with a s u b s y s t e m R p

Mp

is a r e d u c t i v e

The root s y s t e m of M p

can be i d e n t i -

of R g e n e r a t e d by s i m p l e r o o t s , M p i s g e n e r a t e d

-137 -

!

by T and those Xo(s such that o ( E R p .

t

U p is g e n e r a t e d by those Xc<s such

that ¢<ER + - R +p" We give the proof in s e v e r a l steps: (1) Let W p be the Weyl group a s s o c i a t e d to R p (i.e. the group g e n e r a t e d by wE , o<E

p).

Let W

be the s u b s e t of W defined by :

weW/w(Rp)> 0

Then: (a) If w E w P , w ' g W p , then l(ww') = l(w) + l(w'). ( H e r e

.

l(w) denotes

the m i n i m a l length of an e x p r e s s i o n of w as a p r o d u c t of s i m p l e reflections~ (b) W = W P . W P

with u n i q u e n e s s of e x p r e s s i o n .

This is a s t a n d a r d l e m m a and

we a s s u m e it.

(2)

G = U Uw. n w . P . w•W P

For:

By the r e f o r m u l a t i o n of B r u h a t l e m m a , G =

it is sufficient to p r o v e that f o r WoE W, Uwo. nwo. w E wP.

~) "~/ Uwo. nwo. B.

Thus,

U w. n w. P f o r a suitable

By (1) above, w ° = w . w ' with w E w P , w'C Wp.

Since l ( w . w ' ) =

l(w) + l(w'), it can be e a s i l y checked that Uwo = Uw.w, = Uw. nWUw' . Hence Uwo. nwo. B = Uw. (3) F o r For:

w~W

P

nw

. U w , . n w.nw, .B = Uw. nw. Uw, . n w , . B Cn w U .wP.. -

, d i m (Up/% WUp)~< dim U p - l(w).

U p t% WUp is g e n e r a t e d by those Xc¢ s such that (1) o(C R +-

(2) w(0 0, R w ~ R +-

p,

R +

w h e r e R w = fo¢>0/w(,C) < 0 ] so that l(w) =]Rw~ Hence ( R + - R p ) - K w _ ~ R w. + (We note that R + - R p - K w contains /~ such that w ( ~ ) ~ R p , so that equality m a y or m a y not hold above). dim Up

Hence dim (Up C%TM UID) = I K w [ ~ I R + - R p I -IRw[ =

- l(w).

(4) L e t w E W P

be fixed.

given by : fw(X,y) = xyx "I

C o n s i d e r the m o r p h i s m

fw: U

w

X (U

to

~

w

Up)----> U p

Then U p contains a dense open set U'P, w such

-

that f ' l ( z )

is finite V z ~ U '

W

Proof.

138

-

. P,w

Case (i).

fw is dominant.

By l e m m a 2 of 1.13, U p contains a dense

' w such that dim f w ' l ( z ) : dim (Uw×(U P (~ Wup)) - dim Up, open s u b s e t Up, !

!

V z ~ U p , w. f-l(z) W

By (3), it follows that dimf-l(z)w : 0 ~ z EUp, w.

In o t h e r w o r d s ,

is finite.

Case (ii). f is not dominant. In this case, let Up,_ = U - Image of fw' w w p which is dense open in Up. Also, f-1 (z) is empty V z ~ U' This w P, w" p r o v e s (4).

We a r e now in a position to p r o v e the t h e o r e m . !

!

(a) Let U'P : w[~ £ w P U p , w' a finite i n t e r s e c t i o n (Up, w is as in (4)). Then P . By choice of U'p, w , Z ~ U p ~ U'p is dense open in Up, Let z E U p', w E W X.W

Up

for finitely m a n y

x EUw.

Since G =

U _ ] m u w.w.p, it follows w E w--

that z belongs to a finite n u m b e r of conjugates of U p (in G,) (b) Consider S C G / p x G ,

definedby: S :

well-defined since P normalizes subset of G / P K G.

Up.

lf(gP, x)/g'lxg6UpI. " S

is

Also, S is a closed irreducible

(S is the image of G X U p

under the m o r p h i s m

(x,y) " - * " (xP, xyx-1). ) By p r o j e c t i n g onto the f i r s t f a c t o r , d i m S = dim G / p + dim U p (fibres a r e conjugates of Up). By p r o j e c t i n g onto the second f a c t o r , d i m S -- dim GUp, since the f i b r e above any z 6 U'P is finite. Hence dim G U p -- dim G / p +

dim U p = dim G - dim P / U p .

A l s o , G U p is c l o s e d since S is c l o s e d and G / p

(b).

is complete. This p r o v e s

-

139

-

(c) If G has f i n i t e l y m a n y u n i p o t e n t c l a s s e s , then G U p is m a d e up of finitely many classes.

Hence it c o n t a i n s a c l a s s

C such that d i m C = d i m G U p .

C l e a r l y C is d e n s e open in G u p a n d h e n c e is u n i q u e . Consider x£Cf%Up. d i m Cp(x) = dim P - d i m Zp(X) ~> d i m P - d i m ZG(X) = dim P - d i m G +

dim C

= dim P - dim G + dim G - dim P / U p = dim Up. A l s o , Cp{X) C _ U p ~ C

CUp.

It follows that C t3 U

P

d i m Cp(X) = d i m C ~ U p

Hence d i m Cp(X) = d i m C F % U p V X 6 C ~ U p .

is a s i n g l e c l a s s u n d e r = dim Up.

Thus

P.

A l s o , by above,

C /% U p is d e n s e in U p .

This

proves theorem completely.

C o r o l l a r y 1.

If x EC (C is the c l a s s m e n t i o n e d in (c) above), then

d i m ZG(X) = r + 2 . d i m 03 . X _

Proof.

P i c k w ' ~ W p such that w ' ( R $

ufiWu For:

) =

P

Rp

!

(such w

exists).

Then

=Up. U i%W'u is g e n e r a t e d by t h o s e

X I

I

o( s such that o~>0 and w (~() > 0. + , + +P C _ R and w (Rp) C R"P = R p .

+ Since w ' E W p , w'(R + - Rp) = R + -R + + Hence R - R p is p r e c i s e l y the set d e s c r i b e d above.

A l s o , C I% U p

is d e n s e in U p .

= r+ 2.dim ~

Hence U/%w U = U p .

Hence by c o r o l l a r y to p r o p o s i t i o n 2, d i m ZG(X)

o X

Alternate proof. PB ~ ~

S i n c e x E U p and P n o r m a l i s e s

V P ~P.

Hence

U p , it follows that

dim ~ x >~ dim P / B •

X

Now, dim P / B = dim P - dim B.

Hence

r+2.dim ~5 >~ r+2.(dim P - dim B) X

= dim P - dim Up.

(This can easily be checked).

-

Hence

r + 2 . d i m (~x>~dim ZG(X ).

Hence

d i m ZG(X) = r + 2 . d i m ~5 • Th's eventually proves

C o r o l l a r y 2.

If x , C

c o n j u g a t e s of U p

Proof.

But d i m ZG(X) >,, r + 2 . d i m ~ x

is always true.

P.

c o n t a i n i n g x.

Hence ~ p E P as

d i m ~5x = d i m P / B .

a r e a s in c o r o l l a r y 1, then ZG(X) i s t r a n s i t i v e on the

Let x gUp/'lgUp.

gPup-- g u p

3.10.

-

X

Remark.

under

140

(The l a t t e r s e t i s f i n i t e . )

Then

g-1 x, x E U p t % C , which is a s i n g l e c l a s s

s u c h that

P normalizes

P'8:~x-- x .

Up.

Thus

gp e Z G ( X ) .

Also,

T h i s p r o v e s the c o r o l l a r y .

Subregular Elements.

A f t e r t h e s e g e n e r a l i t i e s , we s h a l l now c o n s i d e r the (unipotent} e l e m e n t s which d i m ZG(X) = r+2 (r = r a n k G). elements.

x for

Such e l e m e n t s a r e c a l l e d s u b r e g u l a r

In what f o l l o w s , we s h a l l p r o v e the e x i s t e n c e of s u b r e ~ u l a r (uni-

potent) e l e m e n t s , show that t h e y f o r m a d e n s e c l a s s in the v a r i e t y of a l l i r r e g u l a r u n i p o t e n t e l e m e n t s and then d i s c u s s s o m e of t h e i r c h a r a c t e r i z a t i o n s . We s t a r t with the f o l l o w i n g i m p o r t a n t l e m m a s .

Lemma

I.

Let

maximal torus

(1)

G

be simple

and

B a fixed

subgroup

of

G.

Fix

a

T and a s i m p l e r o o t ~ .

F o r any s i m p l e ~ ~ o< s u c h that

(2) F o r a n y s i m p l e ~ ~

(*)~

(~,o() #

0 and a n y x c U ~ , ~ y B g P ~ / B

(*).

suoh that y-lxy

satisfying

Borel

, define

V

"/5

=

f xC U / ~ p r e c i s e l y

n !~,

yBI s¢ lB

-

C ~

where n'~.~

=I(1 ~,~)

141

-

if this integer is 0 or different from char G. otherwise.

Then Vo(,~ is open and dense in U o . ) supportin

{e4,~}.

Uo< . (Note that U~is a semidirect

Let R ,/5 be the system of roots having

We may assume U~ = 'X ~ r > 0 r'

rCR~ mayboma0eaboo,

Also

Similar assumptions

_

i

arootsyotemwi,h

-

{~,~

142

-

as b a s i s .

~ a s e (i).

Rc~

' Here, no< ~

is of type A 2.

I, U~ = X/~.

=

X

+

.

By commutation formulae, -i xo4(t)-l,x ~ (a).x/$+~0Xr(Cr)" Then

r¢~ n~o.bl

-1

-1 "nc