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, <;$(x)- ~ : is a g r o u p h o m o m o r p h i s m .

~x i s c a l l e d the r i g h t t r a n s l a t i o n of A by x £ G .

Similarly %x' the l e f t

t r a n s l a t i o n of A b y x can b e d e f i n e d .

Lemma.

L e t B be the s u b s p a c e of A s p a n n e d

f gA).

Then

Proof.

If f = 0, t h e r e i s nothing to p r o v e .

by i~'x (f)' x~ G~(for a

fixed

B is finite dimensional,

So l e t f ~ 0.

n

Let m*f = ~ gicghDgi, hiEA i=l

such that n is m i n i m a l .

Claim. { il is a basis of B. g- I ~ i ~ n Firstly,

giI s a r e l i n e a r l y i n d e p e n d e n t ( f r o m m i n i r n a l i t y of n).

reason,

h'ls a r e a l s o l i n e a r l y i n d e p e n d e n t .

G such that

(hi(xj))(i '

,

n

J)

Hence t h e r e e x i s t ~xjl 1 ~- j ~n in

has a non-zero determinant.

( f y f)(x) = f ( x . y ) = ~ gi(x) . hi(Y). i=l n ~ H e n c e ~y* f = i~= l hi(Y)" gi

F o r the s a m e

A g a i n , f o r a n y y ~G,

I.

n

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

~xj f = ~ hi(xj).g i i=l

1 ~ j _~n.

Hence f o r e a c h i, gi i s a l i n e a r c o m b i n a t i o n of ( ~ j f ) l _<_j~n" Thus~

g i ~ B V l~-i~n.

From I above,

B is s p a n n e d by ~ g i l 1_~ i&n

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

Remarks.

In the a b o v e l e m m a , one can t a k e the l e f t - t r a n s l a t i o n s

)%G

x eG

l

o r the l e f t and r i g h t t r a n s l a t i o n s

fo° o )&G*

i n s t e a d of the

-7-

right-translations ~ G "

1. 9,

For

Linear algebraic groups. v£V

Let V be an n - d i m e n s i o n a l v e c t o r - s p a c e / k .

and aS£ V ~, c o n s i d e r the m a t r i x c o e f f i c i e n t m

V, a S

: GL(V)

~ k

defined by : mv, a , ( T ) = a*(T(v)).

Clearly,

v,

mv+v,, a S = mv, a, + m v , , a , ~

!

a,b

m y , a S +b* = m y , a S + m y , b S J

L e t A' b e t h e k - a l g e b r a g e n e r a t e d b y be the f u n c t i o n : GL(V) well-defined).

V!

~V ~

~ VS

{mv, as/VEV,

a*~V*J.

Let D ~ A '

* k given by D(T) = d e t e r m i n a n t of T (which is

Let A be the r i n g o b t a i n e d f r o m A ' by a d j o i n i n g D " l .

(GL(V), A) is an affine a l g e b r a i c group.

Then

Any s u b v a r i e t y of (GL(V), A) which

i s a l s o a s u b g r o u p is c a l l e d a l i n e a r a l g e b r a i c group.

Lemma.

Let

(G,A) be an affine a l g e b r a i c group,

subspace invariant under ~G.

Define

~: G

B ~A

be a f i n i t e d i m e n s i o n a l S

) GL(B) by ~<(x) = ~ x / B . Then

c< is a m o r p h i s m of affine a l g e b r a i c g r o u p s and the m a t r i x c o e f f i c i e n t s on G (pulled b a c k v i a ~ ) g e n e r a t e the s a m e l i n e a r s p a c e a s

Proof.

Let v t B ,

a~

BS.

Then m v,a* : G L ( B ) ~

B .

k is g i v e n by:

mv, aS (T) = a*(T(v)). n

Let

mSv = ~ gi ~ hi (n m i n i m a l ) . i=l

T h e n as p r o v e d e a r l i e r ,

gi 1 ~ i ~ n - _

a b a s i s of the l i n e a r s p a c e s p a n n e d by the r i g h t - t r a n s l a t e s of v and ~hil 1 ~_ i _~n i s a b a s i s of the l i n e a r s p a c e s p a n n e d by the l e f t - t r a n s l a t e s of v.

-8-

,

Now

#

(mv, a , o ~)(y) = a (Q((y) (f)) = a * ( ~ y f) n

n

= a*( ~[~ hi(Y), gi ) = ~__a (gi). hi(Y) i 1 n

or

Again,

m y , a* o 0 ( =

1 (D oo0(y)

1 o0( =Doo(oi. D

_

1 det. ( ~ ( y ) )

#

I

7" a (gi). h i i=l

I.

= det. ~ ( y - 1 ) = det. ( ~ o i ) ( y ) . G is a morphism.

and i : G - - - ~

Hence

Hence f r o m I a b o v e ,

I__ o ~ £A. D Thus:

(i) ~

is a morphism.

Next, ~' s are linearly independent. shown to be in the subspace

Hence choosing a

W s p a n n e d by the m a t r i x c o e f f i c i e n t s .

space generated by left-translates of v space spannedby

properly,

G_W.

From

I, clearly,

W

h i can be T h u s the C linear

{ ~ k G B ~ . Hencethe lemma.

We now p r o v e a f u n d a m e n t a l p r o p o s i t i o n : Proposition.

A n y ( a b s t r a c t ) affine a l g e b r a i c g r o u p i s i s o m o r p h i c to a l i n e a r

algebraic group.

Proof.

Let

(G, A) be an affine algebraic group.

g e n e r a t e d , s a y by fl = 1 ..... f k 6 A .

Then A i s f i n i t e l y

L e t B be the l i n e a r s p a c e s p a n n e d by:

l~i_-n~. Then by an earlier lemma, under o( : G

~G"

B

is finite dimensional and obviously invariant

Hence the previous l e m m a applied to B

• GL(B) such that the linear space W

as the one spanned by matrix coefficients of G. is a k-subalgebra which contains Hence

A = Image ~*.

W

gives a m o r p h i s m

spanned by ~ G B is the s a m e But then the image of

which, in turn, contains ~fiI 1 ~ i~ n"

The proposition now follows from the l e m m a in'1,5.

-9-

A final r e m a r k .

J u s t as in the c a s e of a b s t r a c t a l g e b r a i c v a r i e t i e s , so in the

c a s e of a l g e b r a i c g r o u p s

(G,A) e v e r y t h i n g can be r e t r a c t e d into A.

p o i n t s of G a r e the k - a l g e b r a h o m o m o r p h i s m s x : G - - ~ in G is r e f l e c t e d by the c o m u l t i p l i c a t i o n m earlier);

x and y a r e m u l t i p l i e d by:

: A-

k.

• A ~A

The

The m u l t i p l i c a t i o n (f " ~ ' g i

(xy)(f) = ~gi(x).hi(Y).

~ hi

A thus b e c o m e s

a c o a l g e b r a a s well a s a n a l g e b r a with c o m p a ~ t i b l e c o m u l t i p l i c a t i o n and m u l t i p l i c a t i o n (Hopf a l g e b r a ) and o t h e r c o n d i t i o n s to r e f l e c t the group l a w s .

1.10.

Z a r i s k i - t o p o l o g y on v a r i e t i e s .

variety.

Let"~={FC_V/F

Let

(V,A) be an affine a l g e b r a i c

is the s e t of z e r o s of a s e t of f u n c t i ~ n s in A ~ i . e .

E l e m e n t s of "~ a r e j u s t the a l g e b r a i c s u b s e t s of V.

Now the following s t a t e -

m e n t s can be e a s i l y v e r i f i e d :

(1)

~, v ~

(2)

Let F~ g'~ V ~ ( A . Then

(3)

t~ F~

Let F~, be the set of zeros of

f~,l~ ~ E ~

i s the s e t of z e r o s of { f ; ~ p t ~ , / ~ , ~ , . ~ .

Let F 1 , F 2 £ t ' ~ .

Let F i be the s e t of z e r o s of

Then F l U F 2 is the set of z e r o s of

{

fl 1

{f:l~,Ai;

i = 1,2,

]

2 ;~1 E /~1 ~2 E A 2

H e n c e ~ d e f i n e s a topology on V. of "~ ).

(The c l o s e d s e t s a r e j u s t the e l e m e n t s

T h i s topology is c a l l e d the Z a r i s k i - t o p o l o g y on V.

It can be v e r i f i e d

that a m o r p h i s m of v a r i e t i e s i s c o n t i n u o u s with r e s p e c t to this topology.

We s t a t e h e r e an i m p o r t a n t r e s u l t , which follows f r o m the ' H i l b e r t b a s i s theorem'.

The r e s u l t is: 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 is n o e t h e r i a n , i . e .

s a t i s f i e s the m a x i m a l c o n d i t i o n on i d e a l s .

-

1. t 1.

Let (1)

10

-

W e n o w d e f i n e t h e n o t i o n s of n o e t h e r i a n s p a c e a n d i r r e d u c i b l e

(X,"J) b e a t o p o l o g i c a l s p a c e .

space.

We have the following definitions:

(X, ~ ) is s a i d to be n o e t h e r i a n if it s a t i s f i e s t h e m i n i m a l c o n d i t i o n on

closed sets i.e. element.

e v e r y n o n - e m p t y f a m i l y of c l o s e d s e t s c o n t a i n s a m i n i m a l

(It f o l l o w s i m m e d i a t e l y

t h a t a s p a c e is n o e t h e r i a n iff a n y d e c r e a s i n g

s e q u e n c e of c l o s e d s e t s t e r m i n a t e s terminates

iff a n y i n c r e a s i n g

s e q u e n c e of o p e n s e t s

iff a n o n - e m p t y f a m i l y of o p e n s e t s h a s a m a x i m a l e l e m e n t ) .

(2)

(X, ~ )

(3)

A

is called irreducible

if it i s n o t a u n i o n of t w o p r o p e r c l o s e d s u b s e t s .

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

topology.

if it i s i r r e d u c i b l e

(It c a n be e a s i l y s e e n t h a t a s u b s e t

iff f o r e v e r y

U,V

U t% V t% A fi ~.

open in X, s u c h t h a t

T h u s in a n i r r e d u c i b l e

A

UnA space,

in t h e s u b s p a c e

of a s p a c e

X is irreducible

~ ~, V t%A fi @, we h a v e every non-empty open set is

dense).

Lemma.

Let

(V,A)

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

Then

(V,A)

is irredu-

c i b l e i f f A i s an i n t e g r a l d o m a i n .

Proof. S=

Let

(V,A)

{x~V[f(x)

be i r r e d u c i b l e

= 01

and

T=

and

f . g = 0, f o r

{xEV/g(x)

= 0J.

Then

in V and

V = S%J T .

Since V is irreducible,

or

Conversly,

l e t A b e an i n t e g r a l d o m a i n .

g = 0.

open sets Further,

Vf a n d

Vg; f ~ 0, g ~ 0.

any o p e n s e t

V is i r r e d u c i b l e .

Consider

S and

V = S orV

T

= T.

are closed Hence f = 0

Consider the principal

f . g ~ 0, it f o l l o w s t h a t

S c o n t a i n s a p r i n c i p a l open s e t

Now it follows immediately Hence

Since

f,g EA.

Vf, f ~ A

V f C ~ V g ~ #. and

f ~ 0.

that every two non-empty open sets intersect.

-11

Remarks.

(a)

A m o r p h i c i m a g e of a n i r r e d u c i b l e

(b) If ( U , A ) (UxV,

-

a n d (V, B) a r e i r r e d u c i b l e

variety is irreducible.

algebraic varieties,

then so is

A@B).

(Exercise:

P r o v e t h i s u s i n g (1), (2), (3) in t h e d e f i n i t i o n of a v a r i e t y ) .

We now prove a basic theorem for noetherian spaces.

Theorem.

E v e r y n o e t h e r i a n s p a c e c a n be e x p r e s s e d

closed irreducible

subsets.

If s u c h an e x p r e s s i o n

subset may be omitted, then the irreducible determined

as the maximal irreducible

called irreducible

Proof.

Let

such that

c i b l e s u b s e t s of F ] .

F 1 and

and

(Such s u b s e t s a r e

w h i c h a r e c l o s e d and

a s a u n i o n of f i n i t e l y m a n y c l o s e d i r r e d u -

If p o s s i b l e ,

Now F

l e t ~ ~ ~.

So l e t

F ~ "~ w h i c h i s m i n i m a l

itself cannot be irreducible.

F 2 are expressible

s u b s e t s ; but t h e n s o i s = ~

if no

subsets involved are uniquely

Let ~ = IF ~V

F 1 I F 2 a r e p r o p e r c l o s e d s u b s e t s of V. that both

i.e.

c o m p o n e n t s of V).

c a n n o t be e x p r e s s e d

(V i s n o e t h e r i a n ) .

is irredundant,

s u b s e t s of V.

V be a noetherian space.

F

a s a f i n i t e u n i o n of

T h e m i n i m a l i t y of F

F = F 1 U F2; in ~

shows

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

F , w h i c h is a c o n t r a d i c t i o n s i n c e

V, in p a r t i c u l a r ,

Hence

is expressed

F ~ ~.

Hence

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

cible subsets.

C o n s i d e r an i r r e d u n d a n t irreducible

subset

irreducible

subset.

Then

F =F/1V=

expression

1 $ i -~ r .

(F~F1)U...

Now F F I F i i s c l o s e d

V = FIU...UF r.

If p o s s i b l e ,

U(F ~F

(i >_.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<(g) ~ O, ~ e x t e n d s

toahomomorphism

We p r o v e the p r o p o s i t i o n u s i n g t h i s l e m m a . Since U' = I x ~ U / f ( x ) ~ 0 ~ * ( g ) ~ 0, ~ ( g ) ~ B W = ~y~'q/g(y) ~ 01. Claim.

Consider

i n j e c t i v e onto B. lemma,]~ =e

P

p~U'.

~: A ~ F

e

such that

o4: B ~

F

s u c h t h a t ~ ( f ) ~ 0.

T a k e A = k [ U ] , B = ~ * ( k ~V] ).

i s d e n s e in U, f ~ 0.

Hence b y t h e l e m m a ,

(g t k IV] ) h a v i n g the s a i d p r o p e r t i e s .

q

Also,

o ~

,-1

: B --~k

Let

eq o ~ * - l ( ~ * ( g ) )

So l e t q 5 W .

which i s w e l l - d e f i n e d s i n c e = eq(g) = g(q) ~ 0.

: A - - - ~ k s u c h that ~ / B = eq o ~ *-1

and ~(f) ~ 0.

~

Also, for h~k[V],

Then ,.

But then Hence

h(oc(p)) = e (p)(h) = ep o ~ e ' ( h ) = ~ (~*(h)) =

eq o o~':"l(o<~(h)) = eq(h) = h(q).

is

H e n c e b y the

f o r s o m e p ~ U b y p r o p e r t y (3) of v a r i e t i e s and ~(f) = f(p) ~ 0.

Hence q = ~(p).

3

T h e n W i s open in V , W ~ ~, h e n c e d e n s e in V.

W C_~ ( U ' ) , which c l e a r l y p r o v e s the p r o p o s i t i o n .

g(q) ~ 0.

g ~ 0, g ~ B

-16-

!

Hence

W ¢ ~ (U).

P r o o f of L e m m a 1. The l e m m a is p r o v e d in s e v e r a l steps: Step 1.

Let F b e a field,

R be a s u b r i n g o f F

and x ~ 0 ~ F .

Then e v e r y

h o m o m o r p h i s m of R into an a l g e b r a i c a l l y c l o s e d field K can be extended to

R [xj or R [ x ' l j . Proof.

Let o(: R

~, K be a h o m o m o r p h i s m .

Let

P = k e r n a l of a( .

P i s a p r i m e i d e a l . C o n s i d e r R p , the l o c a l i s a t i o n of R at P. ,-,~ ~ (a) gives r i s e t o a : R p , K d e f i n e d b y : , ~ ( a ~ ) = o ( ( s ) , a ¢ R , s e E - P.

Then

clearly

If

can be extended to R p [ x ] o r R p [x - 1 ] , then c l e a r l y o( is extended to R [ x ] or R [ x - 1 ] .

Also, ~ ( R p )

is a field.

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

we m a y a s s u m e that o((R) i t s e l f if a field, say IT. F o r a p o l y n o m i a l g(X) with c o e f f i c i e n t s in R, let g(X) denote the p o l y n o m i a l ~ R [ X ] obtained via ~ .

Olearly, a P.I.D.

Hence I i s g e n e r a t e d by s o m e

sanidoalin

{ ]w ichis

~o(X).

(i) If ~o(X) i s not a n o n - z e r o c o n s t a n t , then choose ~tCK such that ~o(:k) = 0 (K is a l g e b r a i c a l l y closed). and ~ ( x ) = ~ .

Define ~ : R [ x ] - - ~ K

C l e a r l y , ~ e x t e n d s u n i q u e l y to the whole of

the step.

by ~ = R [x].

~on

R

Hence

(I)

(ii) If ~o(X) is a n o n - z e r o c o n s t a n t , it can be a s s u m e d to be 1. e l e m e n t b of R such that o((b) # 0 is i n v e r t i b l e .

Again, every

Hence t h e r e e x i s t s

go(X) ~ R [ X ~ s u c h that go(X) = 0 and go i s of the f o r m : go(X) = 1 + a l X + . . . + a m X Now c o n s i d e r x °1 i n s t e a d of x. to R [x " l ] o r t h e r e e x i s t s

Either

m

II , a F...~a m E P .

I holds in which case

~

is extended

fo(Y)~ R [Y~ such that

to(Y) = 1 + b l Y + . - . + bn Yn, blr...IbnEP and fo (1) = 0.

III

-

17

-

A g a i n , both go and fo can be a s s u m e d to be of m i n i m a l d e g r e e a m o n g s t the p o l y n o m i a l s s a t i s f y i n g above m e n t i o n e d c o n d i t i o n s (II and III).

A s s u m e , with-

out l o s s of g e n e r a l i t y , n ~ m . b1

bn

Now

0 = 1 +-~-+..

Thus

0 -- x m + b l x m ' l

and

0 = 1 + alx +...+am.lX

m-I

and t h i s c o n t r a d i c t s the minimality

+~ff

.

+ . . . + bn.X m - n

+ a m (I-. b

of m.

x

m-I

-..-b

n

x

m-n

)

T h i s c o n t r a d i c t i o n p r o v e s the step.

Note that if x E R and o<(x) ~ 0, then f r o m III, it follows that ~ e x t e n d s to R [x-l~.

F o r , if not, (III) holds.

Hence

fo(Y) = 1 + b l Y + - + b n y n ,

But then 0 = x n + b I x n'l +...+

b n.

b l , . . . , b n g P, fo (1) = 0. Hence

~(x) n = 0 which

contradicts

,((x) ¢ 0. Step 2.

Let

that

R o D R

xEF,

x~0

Proof. of,<

Let I . On

~" =

~

lemma, now

~

Let

be as in Step I.

and ~

o¢ extends x~R

"~ = ~,

'/R"-

follows

Step 3.

F,R

°

every

R o

R o Then

R o

subring

is a valuation

of F, R' ~ R

by : (R' , ~<')~(R"

chain in

~

element,

say

is a valuation

be as in Step i.

T h e n ~ e x t e n d s to 0<: S - - ~ K .

be amaximal

of F

ring,

such

i.e.

o.

' subring

define an order

has a maximal

F,R

R o.

x-l~R

(R', ~')

Then

that

or

to

Let

is bounded

and

, ~

" ) if R' D R " and

above.

(Ro, ~o)'

is an extension

Clearly,

Hence

by Zorn's

from

Step I, it

ring.

Let

(F _~) S 2 R

be an integral

extension.

-18-

Proof.

Let 0 # x ~ S .

Then

n x

x satisfies:

n-I +

alx

+

. . .+

a2 x+al+-~--+...+

io e°

a n

al~..~an £ R

= 0

an n-H-j1 - =0 X

x £R

Hence

[x-Ij.

Now, let R o be a s in Step 2.

C_R ° a n d h e n c e

then R [ x - 1 3 Hence

x ~ R o or x -1 £ R o.

Then e i t h e r x e R o.

Inany case,

xERo,

If x -1 ~ R o,

sothat

S C_R o.

~ extends to ~ : S - - ~ K.

Step 4.

D e d u c t i o n of the l e m m a f r o m s t e p s 1 to 3.

domains.

A = B [Xl, . , X n ] .

i n d u c t i o n on n.

0 ~ f 6A

be given.

Let A ~ B be i n t e g r a l

We p r o v e the l e m m a by

A s s u m e the l e m m a p r o v e d f o r n = 1.

so that A = B 1 [ X n ] .

T h e n ~ 0 ~ g' ~ B '

L e t B 1 = B[x 1 . . . . x _ ~ ,

with the r e q u i r e d p r o p e r t y .

Now

given this

g' ~ 0, by i n d u c t i o n h y p o t h e s i s , ~ 0 ~ g ~ B with the r e q u i r e d

property.

Solet

n = 1, A = B I x ] ; 0 ~ f ~ A

Case (i). x is t r a n s c e n d e n t a l o v e r

B.

given.

Let f = bo+ blX + . . . + bmX

m

with

b m ~ 0. Choose g = b m (In fact a n y b i ~ 0 will do). If ~ : B - - - ~ K , ~ ) ~ 0 m then ~ ~ ( b i) X i is not i d e n t i c a l I y z e r o . Hence choose ~ 6 K with i=0 m ~i ,~ ~ ~(bi). ¢ 0. Define ~ by •(x) = ~ . T h e n ~ i s the r e q u i r e d e x t e n s i o n . i=0 C a s e (ii), x s a t i s f i e s a p o l y n o m i a l in B [ X ] . C o n s i d e r F = q u o t i e n t field of A.

It follows that

a l g e b r a i c o v e r B. integral over

B [ x J is a l g e b r a i c o v e r

Now choose

g ¢ 0, g ~ B

B (such g e x i s t s ) .

such that ,<(g) ~ 0.

Let

(f) ~ 0.

Hence f-1

is a l s o

such that gx and gf-1 a r e

~ : B-----~K be a h o m o m o r p h i s m

It follows f r o m Step 1 that c< e x t e n d s to B [ g - 1 3 .

f r o m Step 3, ~ e x t e n d s to B [ g - l , gx, g f - l ~ . B Ix "I = A.

B.

Hence ¢< e x t e n d s to A. T h i s p r o v e s the l e m m a .

Now

But then B [ g - 1 gx, g f - 1 3 ~

Further, f-16 B [g-l,gx,

gf-13.

Hence

-19

The lemma

-

mentioned above has s o m e fine applications.

C o r o l l a r y to l e m m a

1. E v e r y p r o p e r i d e a l

I of k [ X I ~ . . . , X n ]

has a zero

in k n, Proof,

I can be assumed

x m o d I, B = k, f = t .

to b e m a x i m a l .

Then

Consider

A .= k I X l r . . , X n ] / I ,

~ g ~ 0 with required properties.

xi =

Consider

1

k

~

Id

k and

Id(g) ~ 0.

T h i s e x t e n d s to A ~ k .

Let ~(~)

= a i . ( a l . . . , a n)

i s c l e a r l y a z e r o of I.

Corollary

1 to t h e P r o p o s i t i o n

a c t i n g on a v a r i e t y

V.

Proof.

Consider

Fix

morphism, Now that

v eV.

1.

Let G

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

G --~-~V g i v e n by ~ ( g ) = g . v .

G itself is an dpais.

Hence

~ ( G } -- G . v = t h e o r b i t t h r o u g h go'V EU C G.v

follows that

of a s m a l l e r

Proof.

for some

-1 g.v E ggo U CG.v.

C o r o l l a r y 2.

The closure dimension.

Since

G.-v

e q u a l to

G.v

go ~ G " Hence

~

is a

~ (G) c o n t a i n s a o p e n s e t in ~ ( G ) . Let

U be a o p e n s e t in G . v

Since

G.v

G.v =

is invariantunder

such G, it

U gU, w h i c h i s o p e n i n G . v. g~G

of a n o r b i t i s a u n i o n of G . v

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

(We s h a l l d e f i n e t h i s t e r m p r e s e n t l y ) .

G. v i s o n e .

Clearly,

W e d e f i n e , f o r an i r r e d u c i b l e

tr. deg. k k IV].

t h e u n i o n of t h e o t h e r o r b i t s i s variety

For a reducible variety,

d i m e n s i o n of a n y i r r e d u c i b l e V irreducible,

v.

Then

i s i n v a r i a n t u n d e r t h e a c t i o n of G, i t f o l l o w s t h a t i t i s a

u n i o n of o r b i t s of w h i c h c l o s e d in G . v .

be a c o n n e c t e d a l g e b r a i c g r o u p

component.

W closed, then

tion, as well as the minimum

V, t h e d i m e n s i o n to b e

we t a k e the m a x i m u m

It c a n b e p r o v e d t h a t if V ~ W;

d i m V ~ d i m W.

(Thus the maximum

c o n d i t i o n , on c l o s e d i r r e d u c i b l e

condi-

sets is satisfied).

-

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

G. v

20

-

i s a u n i o n of G . v

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

dimension.

Corollary

3.

O r b i t s of m i n i m a l

dimension

are closed.

Hence closed orbits

exist.

The proof is obvious from corollary

Remarks.

(1) T h e c o r o l l a r i e s

of c o n n e c t e d n e s s Consider i.e.

on

1, 2, 3 r e m a i n

G

(exercise).

the particular

case when

x(y) = Xy = x y x - 1

2.

t r u e if we r e m o v e

the assumption

G a c t s on i t s e l f b y c o n j u g a t i o n ,

Then the orbits are the conjugacy classes.

Thus we

have:

Corollary

4.

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

group is open in its closure

and its closure

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

of s t r i c t l y

Proposition

Let

of a l g e b r a i c

2.

o~ : G ~

(a)

~ (G) i s c l o s e d i n G ' .

(b)

o<(G °) = ~ ( G ) °

(c)

dimG

G

= dimKer~+dim

be a rnorphism

(a)

Let

are the various each other. minimum cular,

G act on G c o s e t s of o ( ( G )

groups.

!

v i a o( i . e . in G'.

x(g') = ~(x).g.

Hence by corollary

o<(G) = ~ (G). 1 i s c l o s e d .

Then the orbits

Hence the orbits are all isomorphic

Hence the orbits have the same dimension,

dimension.

dimensions.

Im ~. !

Proof.

smaller

to

which is obviously the

3, t h e o r b i t s a r e c l o s e d .

In p a r t i -

-21

(b) C o n s i d e r ~ : G - - - ~ ( G ) .

~(G)

(G°} b e i n g i r r e d u c i b l e ~ ~ (G) ° . having f i n i t e index in ~ ( G ) .

-

i s a n a l g e b r a i c g r o u p in its own r i g h t . But then ~ (G°) is a c l o s e d s u b g r o u p

Hence ~ (G°)_~ ~ (G) °.

(c) T h i s follows at once f r o m a g e n e r a l e l e m e n t a r y fact about m o r p h i s m s , whose p r o o f we s h a l l o m i t , v i z .

L e m m a 2.

Let f : U - - ~

V be a m o r p h i s m of a l g e b r a i c v a r i e t i e s with U

i r r e d u c i b l e and f(U) d e n s e in V (such a m o r p h i s m is c a l l e d a d o m i n a n t morphism).

T h e n d i m f - l ( w ) ~ d i m U - d i m V Vw EV.

The e q u a l i t y holds

for w in s o m e open d e n s e s u b s e t of f(U). (The p a r t d e a l i n g with e q u a l i t y i s p r o v e d in the Appendix to 2.11 below}.

Remark.

Let

A n a l t e r n a t e proof f o r (a) (without u s i n g c o r o l l a r y 3) is as follows:

G ° act on

its closure. x~-S,

xS -I

G'

as before.

But then and

S

S

Then

S = o((G°),

is irreducible,

intersect. n

Hence

hence

being

so is S.

x CS. S c S.

Thus n

orbit of Hence S

= S.

I, is open

in

for any Hence

o<(G °)

is c l o s e d .

Now ~ ( G ) = U ~(hi)" ~ ( G ° ) , w h e r e G = U hi G ° is the c o s e t i=l i=l decomposition. H e n c e ~ (G) is closed also.

Remark.

If k above is not a l g e b r a i c a l l y c l o s e d , then e s s e n t i a l l y a l l of the

r e s u l t s above do not hold, a s the e x a m p l e f : ]R,

~ H~• , f(x) = x 2 ' shows.

Chapter II

F i r s t Part:

2. i .

Jordan decompositions, unipotent and dia~onalizable groups

Definitions and preliminary r e s u l t s . Let V be a finite dimensional

vector space over k.

(k will be algebraically closed unless stated o t h e r -

wise).

D e f i n i t i o n . An e n d o m o r p h i s m

X on V is s a i d to be s e m i s i m p l e if it i s

d i a g o n a l i z a b l e ( i . e . the e i g e n v e c t o r s of X span V).

It can be seen that X is semisimple iff the minimal polynomial of X has

distinct roots iff k IX] as a k-algebra is semisimple.

Definition. An endomorphism X on V is said to be nilpotent if Xn = 0 for some integer n >~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<

Let

is the e i g e n s p a c e of A c o r r e s p o n d i n g to ~ Gk.

dim

S and A c o m m u t e .

A.

< dim V

Also, S(V ) C_ V

VS

H e n c e by i n d u c t i o n , the l e m m a is t r u e f o r

It

5 ; since V~ V ~ @ k .

C l e a r l y , the l e m m a is p r o v e d f o r V in this c a s e .

Case (ii) ~

does not have a non-scalar semisimple element,

if ~

consists

of scalar elements only, then the l e m m a is obvious. If not, choose A ~ that A

is not a scalar-element.

Since k is algebraically closed, A

eigenvalues in k. Let ~ be one such value. Let V ~ eigenspaee; then ~0~ ~ V ~ acts on V / V~

~V.

Also,

S(Vo( )C

V

such

has

be the corresponding V SC$.

Hence S E ~

also. Now, by induction, the l e m m a is true for V ~

and V / V

Clearly, the l e m m a is true for V also. Hence by induction, the l e m m a is true for all vector spaces (finitedimensional).

Corollary.

For a set ~

of semi-simple endomorphisms, the elements of S

commute iff they can be simultaneously diagonalized.

-

2.2.

24

-

Jordan decomposition for an end omor~hism.

Proposition

I.

Let V

be an n-dimensional vector space over an

algebraically closed field k.

Let X t e n d V

T h e n there exist e n d o m o r p h i s m s

S, N ~ E n d

X = S + N ; S is semi-simple, Further, S, N

(the set of e n d o m o r p h i s m s V

with the properties:

N is nilpotent and S and N c o m m u t e ,

are uniquely determined by the above conditions.

uniquely determined e n d o m o r p h i s m s

of V).

These

are polynomials (without constant term)

in X (and are called the semisimple and nilpotent parts of X).

Proof.

Consider the m i n i m a l polynomial

f(T) of X, f(T) = - ~ (T-~) n ~ o<~k follows that oK, involved above, is an eigenvalue of X. Let

It

V Q¢= {v E V / ( X - o()m(v) = 0 for s o m e integer rn~. T h e n it can be proved n no< e a s i l y that: V = V ~ ; V ~ = K e r ( ~ - ~ ) ~ ; and ( T - ~ ) is the m i n i m a l

p o l y n o m i a l of X on V ~

Hence ~

is the only e i g e n v a l u e of X on V a~

Define S on V o¢ to be the s c a l a r - m u l t i p l i c a t i o n by ~ . V.

Clearly, S is s e m i s i m p l e ,

on each V ¢ ~ a n d h e n c e on V. required properties.

T h i s d e f i n e s S on

S and X c o m m u t e and (X-S) i s n i l p o t e n t T h u s X = S + (X-S) = S + N s a t i s f i e s the Hence by the C h i n e s e

Again, { ( T - ~)n~e-I a r e c o p r i m e .

R e m a i n d e r ' l ~ h e o r e m , t h e r e e x i s t s a p o l y n o m i a l p(T) s u c h that p(W) - o ( ( m o d ( T - ~ ) n ¢ ) On

V c<, ( X - ~ ) n¢( = 0. Hence

and p(T) - 0 ( m o d T ) p(X) = ~

on V ~

follows that p(T) does not have constant t e r m

if V ° = ~ 0 1

. Hence

•

p(X) = S.

It clearly

(p(T) - 0 Inod T n for s o m e !

n ~I).

Hence

S, N

are polynomials without constant t e r m in X.

If X = S + N

is another decomposition with the said properties, then S' c o m m u t e s

with X

!

and hence with S and with S and

N.

N (being polynomials in X).

Also, X = S + N = S'+ N'.

Hence

Similarly

N

S'- S = N - N'.

commutes N o w sums,

!

-25

products

of c o m m u t i n g

semisimple

are semisimple

(respectively

S - S = N - N

is semisimple

S = S , N =

Remark.

N I

.

-

(respectively

nilpotent),

e.g.

nilpotent) endomorphisms

by the above lemma.

a n d n i l p o t e n t a n d h e n c e e q u a l to

Hence

0.

Thus

Hence the proposition.

If X

is invertible,

so is

S (having the same eigenvalues).

And

S -1

is a polynomial in X. For.

nor

Let p(T) be the polynomial such that p(T) - o((mod (T- p<)

eigenvalue of X.

Then

p ( X ) = S.

But for X

invertible, c< ~ 0.

); o( an

Hence

p(T)

no< and f(T) ---[[(T - o()

are coprime.

1 = p(T). q(T) + r(T) . f(T). H e n c e S -I = q(X).

Thus

Proposition

2.

Hence

~ q(T), r(T) such that

ld = p(X).q(X);

Any automorphism

x

of a f i n i t e d i m e n s i o n a l

a s a p r o d u c t of a s e m i s i m p l e

phism which commute

with each other.

i.e.

x = s.u ; s

semisimple,

commute. the uniquely determined

(without constant term) parts

Hence

S "I is a polynomial in X.

be uniquely expressed

Further,

since f(X) = 0.

in

X.

Such

s

u and

vector-space

and a unipotent

unipotent; u

automorphisms

s

and

can

automor-

u

are unique. s

and

u

are polynomials

These are called the semisimple

and unipotent

of x .

Proof.

Let

invertible, commute

so is and

proposition position

x = S + N be the decomposition

u

S.

s = S and

is unipotent.

1 since given

x = S + N.

polynomials

Let

From

n = I+S'IN.

Further,

x = s.u,

in

Then

the uniqueness

S = s

the remark

(without constant term)

as in proposition

and

N = s.

to p r o p o s i t i o n X.

1,

x = s.u,

Since s

x

is

and u

follows from that in

(u - I )

give the decom-

1, i t f o l l o w s t h a t

s,u

are

-26

-

Note.

T h e a b o v e n o t i o n s c a n b e d e f i n e d , w i t h a m o d i f i c a t i o n , f o r an a r b i t r a r y

field

K (not n e c e s s a r i l y

in

algebraically

E n d V to be s e m i s i m p l e

semisimple.

ever,

if K is p e r f e c t ,

For.

Let ~'~Gal

if t h e c o r r e s p o n d i n g

~ - N = N.

we d e f i n e an e n d o m o r p h i s m

endomorphism

(~/K)"

= a-N.o'S. Since

K

Let

X = S + N be t h e d e c o m p o s i t i o n of X E E n d V . semisimple,

Also, o- X = X,

is perfect, S £ E n d V

rN

nilpotent and

H e n c e by uniqueness, r S = S and as a- S = S Vo- ~ G a l (K/K). Similarly

N

e x a m p l e brings out this fact. Let

Let

be transcendental over

V ; k p.

Let

Consider the endomorphism

Then the semisimple

T h e following

k o be any field of characteristic

k o.

(This

~ E n d V.

This is not true in the case of a field which is not perfect.

T

How-

t h e n t h e s e p a r t s do b e l o n g to E n d V.

follows immediately f r o m Galois theory).

Let

on V ~)K ~ i s

and n i l p o t e n t (or u n i p o t e n t ) p a r t s a r e in E n d V.

~" X = r S + a- N, with a- S

r S . rN

e.g.

T h e a b o v e t w o p r o p o s i t i o n s w i l l go t h r o u g h , b u t we c a n n o t a s s e r t

whether the semisimple

Then

closed),

k = ko(T ). T h e n

k

on V g i v e n b y : X =

p ~ 0.

is not perfect,

0

1 0

. 1

a n d n i l p o t e n t p a r t s do n o t b e l o n g to E n d V, a s t h e reade~r ~

may check.

2.3.

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

partially extend this idea

(continued).

of d e c o m p o s i t i o n in the c a s e of i n f i n i t e d i m e n s i o n a l

spaces.

Let

V

We now

be a vector space (not necessarily finite dimensional).

-

Definition.

X E End V

27

-

is called locally finite if V =

~ V ~ where each V A AEA. is a finite dimensional subspace invariant under X; or, in other words, if e a c h v ~ V i s c o n t a i n e d in a f i n i t e d i m e n s i o n a l s u b s p a c e i n v a r i a n t u n d e r

X.

We know that f o r an i n f i n i t e d i m e n s i o n a l s p a c e , an e n d o m o r p h i s m m a y be i n j e c t i v e without b e i n g s u r j e c t i v e .

Lernma.

Let

H o w e v e r , we h a v e :

X C E n d V be l o c a l l y f i n i t e .

T h e n the f o l l o w i n g s t a t e m e n t s

are equivalent. (1)

X is an a u t o m o r p h i s m

(2)

A l l the e i g e n v a l u e s of X a r e n o n - z e r o ( i . e . X is i n j e c t i v e ) .

Proof.

(1) ~

(2) i s c l e a r .

(2) :::~(]). Let X X(V;~) _C V A

(i. e. X is i n j e c t i v e and s u r j e c t i v e ) .

for each ~

be injective. Consider

with each V ~

injective and hence surjective.

Note.

Thus

X

finite dimensional.

V'

S, N E End

is

Y of X p r e s e r v e s the

Then t h e r e e x i s t e n d o m o r -

V, s u c h that (1) X = S + N ( 2 )

is a f i n i t e d i m e n s i o n a l s u b s p a c e i n v a r i a n t u n d e r

N(V') ~ V' and X/V, = S/V, + N/V,

X/v A

X and h e n c e is l o c a l l y f i n i t e .

P r o p o s i t i o n 1. L e t X E E n d V be l o c a l l y f i n i t e . phisms

Now

itself is surjective.

It f o l l o w s that in the a b o v e c a s e , the i n v e r s e

s a m e finite dimensional subspaces as

V = A~E V;% such that

S and

N c o m m u t e (3) If

X, then

is the decomposition of

S(V') ~ V ' ,

X/V,

defined in

the previous section.

Proof. Let {V A , ~ A . } spaces of V.

Since X

be the set of allfinite-dimensional X-invariant subis locally finite, V =

7--- V~.

Consider X;k = X / V ~ .

-

28

-

Then by proposition I of the previous section, X)~ = Sg~ + Ng~ ; S~, N g ~ E n d S~

semisimple, N A nilpotent; S Aand

S/V ~ = S ~ Claim: Consider

V~(/~

Define N E E n d V

N~

commute.

by N / V ~ = N ~

Define S ~ E n d V by

Vle/k.

S and N are well-defined. V~

and V~. X/W

On V~/~ V~ = W, which is invariant under X,

= S~/W

X/W

+ N~/W

S~/W + N~/W

~

are decompositions into sernisimple

3

and nilpotent parts.

Hence by uniqueness of such a decomposition~ B[~/W = N~,/W . Hence S. N = N.S.

S)~/W = S~,/W and

S,N are well-defined.

Also X = S + N.

Note that S and N

Since S ; v N 9 = N ~ . S ~ V A E A ,

This proves the proposition.

defined above are locally finite. Further, S is semi-

simple (i.e. V has a basis consisting of eigenvectors of S). N nilpotent (i.e. N

Proposition 2. phisms

V~;

is locally

is nilpotent on every finite-dimensional invariant space).

Let x EAut V

s, u EAut V

be locally finite. Then there exist automor=

such that (I) x = s.u, (2) s is semisimple, u is

locally unipotent (see (4)), (3) s and u commute,

(4) If V' is a finite f

dimensional subspace of V x/v, = S/v , . U/v,

v

!

with x(V') C V' then s(V') ~ V , u(V ) C V and

is the decomposition for x/v,.

The proposition is deduced from the previous proposition just as proposition 2 of 2.2 is deduced from proposition I of 2.2.

(The only thing to be noted is

that s has all of its eigenvalues non-zero and hence, in view of the l e m m a above, is an automorphism).

-29

Remark.

The decompositions

-

h e r e and in t h e p r e v i o u s s e c t i o n s w i l l be c a l l e d

Jordan decompositions.

2.4.

J o r d a n d e c o m p o s i t i o n of an e l e m e n t

b e an a l g e b r a i c g r o u p a n d

A= k [G].

x of an a l g e b r a i c g r o u p

In v i e w of t h e l e m m a

G.

of 1 . 8 ,

~:

Let and

N* are locally finite V x~G. x Definitions.

(i) An e l e m e n t

s EG

i s s a i d to b e s e m i s i m p l e

if ~s :A

~ A

is SO. (ii) A n element

s £G

l o c a l l y a l l t h e e i g e n v a l u e s of

Note.

s

is semisimple

X = i 0 Px_l oi. iff ~

x

rA

is said to be unipotent if ~s : A ~s

a r e e q u a l to

1).

( r e s p e c t i v e l y u n i p o t e n t ) iff s

Hence

is so. (i.e.

-!

is so.

~ x = i o fx-i o i . Thus, ~ x

Again,

is semisimple

is so.

Also,

f x is unipotent iff

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

x is semisimple

x is so.

iff

~k* is semisimple. x

x is unipotent if ~ * x

Proposition

1.

x = y.z = z.y;

Let

x E G.

is unipotent.

Then there exist elements

y is semisimple,

z is u n i p o t e n t .

y, z ~ G

such that

Such e l e m e n t s a r e u n i q u e

and a r e c a l l e d t h e s e m i s i m p l e

and u n i p o t e n t p a r t s of x.

b y x s and

x = x s.x u is called the Jordan decomposition

xu

respectively,

They are denoted

of No

Proof.

Consider

~x : A - - - ' ~ A .

H e n c e by p r o p o s i t i o n 2 of 2 . 3 , t h e r e e x i s t

G

-

automorphisms

s and

p

u

-

of A a s a l i n e a r s p a c e s u c h t h a i

= s.u = u. s ; s

%

30

semisimple,

u

unipotent.

X

It follows that w h e n e v e r

u(a" ~ E n d V).

o- corn.mutes with

This is so since any

X-invariant s p a c e a n d ~- s(v) = s . o - ( v ) .

~ x ' o- c o m m u t e s

v EV

with

s and

b e l o n g s to a f i n i t e d i m e n s i o n a l

s is a p o l y n o m i a l of

~x

on s u c h a space.

Similarly,

Q-" c o m m u t e s

w i t h u.

commute with ~* for every t

t ~G, since

~x

Hence

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

clearly does so.

s and ~

u (1)

W e n o w p r o v e a l e m m a u n d e r a g e n e r a l s e t up.

Lemma

1. L e t

A b e an a l g e b r a o v e r

or associative. finite.

Let

Proof.

L e t 0" be a k - a l g e b r a

0- = s . u

also k-algebra

k which is not necessarily automorphism

of A

be t h e J o r d a n d e c o m p o s i t i o n of ~ - .

automorphisms

commutative

which is locally Then

s

and

u are

of A .

One n e e d o n l y p r o v e t h a t

s i s an a l g e b r a h o m o m o r p h i s m .

Since

o-

is locally finite, integer nJ.

/k = ~ A~ ; w h e r e A ~ = ~ f ~ A / ( c r - - ~)n(f) = 0 f o r s o m e ~k We a l s o h a v e : s i s j u s t m u l t i p l i c a t i o n by ~ on A ~ .

Claim:

A ~.

A 0 C A~ ' ~

g6A ~

and hence V f,g t~A.

This clearly shows that For

f,g ('A,

s(f.g) = s(f).s(g)V lEA ~,

t h e f o l l o w i n g i d e n t i t y c a n be

p r o v e d b y induction:n

(~__ ~.~)n(fg) If f ~ A ~ , g ~ A ~ , f. g E A ~ ' ~ .

= ~ (i~ . ( ~ r - ~ ) 1 i=O

•

( ~

n-i

then for large enough

f) .

o_i

(¢r--]3)

n, ( ~ ' - ~ . ~ ) n

n-i

(g).

(fg) = 0 t h u s

H e n c e t h e c l a i m and t h e l e m m a .

The lemma immediately

g i v e s : If

k-algebra automorphisms

of A.

~: = s.u

(as a b o v e } , t h e n

s and

------------ (2)

u are

-

31

-

Next we prove : Any k-algebra automorphism (or endomorphism) of A which

c o m m u t e s with Consider

G is of the f o r m

x~G,

f~A.

~w f o r unique w @G.

Then ¢ ' o ~*

=

X

~*

~- . Hence

O

el(~'(~

(3)

0"( ~ * (f)) = ~k* (~(f)).

X

X

C o n s i d e r the evaluation e 1 at the identity of We have

-

X

G.

f)) = (e I o )%*)(~r'(f)) X

i.e.

(e I o r ) (

Now,

e I o ~r- : A ~

f) = (e I o

X

k, a k - a l g e b r a h o m o m o r p h i s m .

that e 1 o~" = e w by p r o p e r t y (3) of v a r i e t i e s . since

e I O2~x = ex.

ex(O-(f)).

Now,

ew o

For:

Hence 3 w C-G such

Thus,

e w ( N f) = ex(~" (f))'

x = exw = e x o ~w.

Since x EG and f ~ A a r e a r b i t r a r y ,

of w follows i m m e d i a t e l y . ~fEA.

)(0-(f)).

~w =

Hence e w ( f ) = e w , ( f ) ~ f ~ A

T h u s ex(

~: = o'.

fw' implies

implies

(f)) --

Now u n i q u e n e s s

e l ( ¢ w ( f ) ) = el(~w,(f) )

w = w'.

It follows immediately, f r o m (i), (2) and (3), that ~x

=

~y o ~z for unique

y,

semisimple in G and z, unipotent in G. H e n c e x = y. z by (3) again, since @ @ #()y o 8z , = ~yz , " Clearly y and z c o m m u t e . This proves the proposition.

Note.

(1) It follows i m m e d i a t e l y that f o r ; k : , which is a l s o l o c a l l y finite,

- ~

/~*

X

(2) F o r

o

~*

X s

is the Jordan decomposition.

Xu

gEG,

~'g c o m m u t e s with

since these a r e p o l y n o m i a l s in ~ :

~x

iff it does so with

locally.

~x s and ~ x u

Hence ZG(X) = ZG(Xs)~ ZG(Xu).

To c o m p l e t e the development, two m o r e points r e m a i n to be p r o v e d .

Proposition 2.

If G is a closed subgroup of GL(V) and x EG, then the two

Jordan decompositions of x, one as automorphism of V and the other as an

-

32

-

e l e m e n t of G, a r e one and the s a m e . P r o p o s i t i o n 3. The J o r d a n d e c o m p o s i t i o n is p r e s e r v e d by m o r p h i s m s of algebraic groups.

Both t h e s e p r o p o s i t i o n s follow f r o m :

L e m m a 2.

Let ~: G - - - ~ G L ( V )

s o m e GL(V)) and x ~ G .

be a r e p r e s e n t a t i o n ( i . e . a m o r p h i s m into

T h e n ~(x) = ~(Xs). ~(x u) i s the J o r d a n d e c o m -

p o s i t i o n of ~ (x) a s an a u t o m o r p h i s m of V.

T o get p r o p o s i t i o n 2, we apply this l e m m a to the i n j e c t i o n : Ge---~GL(V). p r o p o s i t i o n 3, a s s u m i n g ~ :G ~ some

G'

T o get

is the given m o r p h i s m , we i m b e d G' in

GL(V) and then apply p r o p o s i t i o n 2 to the r e s u l t i n g r e p r e s e n t a t i o n s of

G and of G ' .

L e m m a 2 i t s e l f follows f r o m :

L e m m a 3. E v e r y r e p r e s e n t a t i o n of G is i s o m o r p h i c to one in which G a c t s v i a r i g h t - t r a n s l a t i o n s on a s u b s p a c e of A n f o r s o m e

n, A = k [ G ] .

F o r : In view of the d e f i n i t i o n s and the fact that the J o r d a n d e c o m p o s i t i o n on v e c t o r s p a c e s i s p r e s e r v e d by d i r e c t s u m s and r e s t r i c t i o n s to s u b s p a c e s , l e m m a 3 c l e a r l y i m p l i e s l e m m a 2.

P r o o f of l e m m a 3. L e t ~ : G - - ~ GL(V) be the g i v e n r e p r e s e n t a t i o n . f v 1 ' v *2 ~ ' " 'y*n ~ be a b a s i s of the dual s p a c e of V. : V--~A

n given b y : ~(V) = (Cvlv,~ . . . . .

s t a n d s for the m a t r i x c o e f f i c i e n t x ~ v * ( v e r i f i e d that

Let

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

C v l v ~n ); w h e r e , as u s u a l , C v t v , ~ (x)(vl)).

Now, it can be e a s i l y

~ y i e l d s a G - m o d u l e i s o m o r p h i s m of V with ~(V).

This proves

the l e m m a . Remark.

The J o r d a n d e c o m p o s i t i o n m a y be c a r r i e d o v e r f r o m e l e m e n t s to

-

33

-

Abelian groups: If G i s an A b e l i a n a f f i n e g r o u p , then

G = G s . G u , a d i r e c t p r o d u c t (in the

s e n s e of a l g e b r a i c g r o u p s ) of c l o s e d s u b g r o u p s . T h e r e a d e r m a y w i s h to p r o v e this u s i n g the l e m m a of 2. I.

2.5.

Kolchin's Theorem.

We c o n t i n u e with an i m p o r t a n t t h e o r e m about u n i -

potent groups.

T h e o r e m (Kolchin). L e t G be a s u b g r o u p of Aut V (for a f i n i t e d i m e n s i o n a l space

V) 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 .

s i m u l t a n e o u s l y put in u p p e r t r i a n g u l a r f o r m 0 = V°

~

V 1 C...C

v

=v

such that V.

T h e n the e l e m e n t s of G can be ( i . e . G f i x e s a flag: is of c o d i m e n s i o n l

in

Vi+l, 0 ,~i,~n-1).

Proof.

F o r the p r o o f , we m a k e u s e of l e m m a :

Lemma 1 (Burnside). dim.)

s u c h that ¥

irreducible).

Let

S be a s e m i g r o u p of e n d o m o r p h i s m s of V (finite

is a simple

Then

S - m o d u l e ( i . e . the a c t i o n of S on V is

S c o n t a i n s a b a s i s of E n d V.

F r o m t h i s l e m m a , the f o l l o w i n g l e r n m a f o l l o w s :

L e m m a 2 (Burnside).

Let

V be an n - d i m e n s i o n a l s p a c e / k , S be a s e m i -

g r o u p of e n d o m o r p h i s m s of V s u c h that e l e m e n t s of S h a v e only r set: { t r s, s E S t be e l e m e n t s ~ r n2.

r).

V is s i m p l e S - m o d u l e .

L e t the

d i f f e r e n t t r a c e s (i. e. l e t the c a r d i n a l i t y of the

Then

S i t s e l f i s f i n i t e and the n u m b e r of

-

Proof.

34

-

B y L e m m a 1, S c o n t a i n s e l e m e n t s

of E n d V.

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

S --~

Yl' " " " 'Yn 2 which form a basis

k n2 g i v e n by :

~(S) = (tr x . y 1 . . . . . S i n c e Yl . . . . .

tr x.y 2) n

Yn2 is a b a s i s of E n d V, it f o l l o w s t h a t

t h e n xYiE S , 1 - ~ i ~ n 2.

Hence

t r xy i h a s

f i n i t e a n d IS~ = n u m b e r of e l e m e n t s in S

r

possible values. 2 ~r n .

From this, Kolchin's theorem follows immediately. series

V r . 1 ~~ . . .

V = Vr

G(Vi) C - Vi and Vi/vi.1 ~i : G --->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<(x,v) = x . v .

o<* :k[V]--~ k [GJ~) k IV] be the co-morphism. x* : k[VJ

Let

For x eG, define

> k [VJ by : (x*f)(v) = f ( ~ ( x -1, v)) = f(x - 1 . v ) .

Thus x

i s a k - a l g e b r a h o m o m o r p h i s m of k [%r] . A l s o , x*o y* = (xy)*.

we get a m a p : G ~ f £ k [V~ . space

Aut k [¥]

which is a g r o u p - h o m o m o r p h i s m .

Let.

Then a n a r g u m e n t s i m i l a r to the one in the l e m m a of 1.7 gives: The

W(f) s p a n n e d by {x~'f, x ~ G t

G* = {x'~,x EG 1 .

is f i n i t e d i m e n s i o n a l and i n v a r i a n t u n d e r

Hence it follows that G a c t s l o c a l l y f i n i t e l y .

F u r t h e r , it

can be s e e n , as in the proof of the l e m m a of 1.9, that the m a p ~:G given by

~(x) = X*/w(f ) is a m o r p h i s m of a l g e b r a i c g r o u p s .

p r o p o s i t i o n 2 of 2 . 4 , we get: X*/w(f ) is u n i p o t e n t Vx e G . it follows that x* is l o c a l l y u n i p o t e n t V x ~ G. u n d e r G.

Thus

If p o s s i b l e , l e t O be not c l o s e d .

> 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) - ¢
Hence

and ~ l ( h o ) ~ ~ o ( h o ) .

T h i s c o n t r a d i c t s the m i n i m a l i t y of r .

This proves

the l e m m a .

Note. F r o m t h i s , it follows that f o r a d i a g o n a l i z a b l e group G, X(G) is a b a s i s of k [ G ] .

It a l s o follows, a s s u m i n g H to be a c l o s e d s u b g r o u p of G,

both d i a g o n a l i z a b l e , that e v e r y c h a r a c t e r of H can be extended to one of G, and that H is j u s t the k e r n e l of a s e t of c h a r a c t e r s on G.

(Exercise: Prove

these a s s e r t i o n s . )

It can be e a s i l y v e r i f i e d that X ( G I ~ G2) = X(G1)X X(G 2) f o r d i a g o n a l i z a b l e groups

G 1 and G 2.

char, k Notation:

if c h a r . k ~ 0

Denote p = p(k) = if not.

Note. F o r an a l g e b r a i c group G, X(G) does not have p - t o r s i o n (p = p(k)

as

d e f i n e d above). For:

L e t X p = 1. ::~

Thus we have:

T h e n xP(x) = I ~ V x E G

(X(x) = 1) p = 0

V xEG ~

X = 1.

F o r a d i a g o n a l i z a b l e group G, X(G) is a f i n i t e l y g e n e r a t e d

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

-40

The

converse

Proposition

-

is also true, viz.

If X is a finitely generated

3.

abelian group with torsion prime

to p(= p(k)), then there exists a diagonalizable Proof.

A finitely generated

groups.

Since

X(GIX

following cases

only.

C a s e (a). X = ZZ.

abelian group

G2) = X(G~ xX(G2),

group

G

such that X(G) = X.

is a (finite) direct product

of cyclic

it follows that we need consider

the

In this c a s e , c l e a r l y G = GL(1) i s such that X(G} = X.

C a s e (b). X = 7Z/nZE with (n,p) = 1.

In t h i s c a s e , we can take for G

the

c l o s e d s u b g r o u p of GL(1), i . e . of k*, c o n s i s t i n g of the n th r o o t s of 1. Since

(n,p) = 1, this group is i s o m o r p h i c to ZS/n2Z.

i s o m o r p h i c to Z~/n ~

a s can be v e r i f i e d at once.

Hence X(G) is a l s o

T h i s p r o v e s the p r o p o s i -

tion.

We note that the group ZE/nZ~ of c a s e (b) is d i s c r e t e .

!

P r o p o s i t i o n 4. L e t G , G ~* : X(G') ~

Proof.

be d i a g o n a l i z a b l e g r o u p s . T h e n e v e r y h o m o m o r p h i s m

X(G) c o m e s n a t u r a l l y f r o m a m o r p h i s m ~ : G ~

Since G' is d i a g o n a l i z a b l e ,

X(G') i s a b a s i s of k ~ G ' J .

e x t e n d s to a k - a l g e b r a h o m o m o r p h i s m of k [ G ' J to a m a p

o(: G - - - ~ G '

of v a r i e t i e s .

For

G'.

into k [ G ] .

given by: e o((u) = eu 0 ~

Hence

~*

This gives rise

C l e a r l y ¢~ is a m o r p h i s m

x , y ~ G and f 6 X ( G ' ) ,

f(~(xy)) = (o< f)(xy) = (o¢~*f)(x). ( ~ * f ) ( y ) , (since o¢ f EX(G)) = f ( ~ ( x ) ) . f(a~(y)) = f ( ~ ( x ) . ~ ( y ) ) . S i n c e X(G') s p a n s k [ G ' 3 , ~ ( x y ) = ~ ( x ) . ~ ( y ) . m o r p h i s m and h e n c e a m o r p h i s m .

Hence ~

is a group h o m o -

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

-

41

-

P r o p o s i t i o n s 3 and 4 p r o v e the following t h e o r e m .

Theorem.

The c o r r e s p o n d e n c e G ~ X ( G )

b e t w e e n d i a g o n a l i z a b l e g r o u p s and

f i n i t e l y g e n e r a t e d a b e l i a n g r o u p s with t o r s i o n p r i m e to p i n d u c e s a f u l l y f a i t h ful c o n t r a v a r i a n t f u n e t o r of c a t e g o r i e s .

Proposition 5.

Let G

be a diagonalizable group.

T h e n the following state-

m e n t s are equivalent: (a) G

is connected.

(b) G

is isomorphic to s o m e

n G L I.

(e) X(G) is f r e e , i . e . h a s no t o r s i o n .

Proof.

B e c a u s e of the t h e o r e m above,

G is i s o m o r p h i c to a group of the f o r m :

n

GLIX ~/nlZZ ~-.. and then X(O) = ~ n x n=Go" Clearly G L 1

~ Z E / n r Z Z with (ni,P) = I V l ~ i

Z~/nl2 z X... ~ 2 Z / n r ~

Hence

G

~r.

"

n is connected iff r = 0 iff X(G) = 77, . This

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

Definition.

A diagonalizable group

G

is said to ~e a torus if the above set of

equivalent conditions holds for G.

A s an i m m e d i a t e corollary, w e get:

Corollary.

E v e r y d i a g o n a l i z a b l e group G is a d i r e c t p r o d u c t of a t o r u s and

a f i n i t e a b e l i a n g r o u p (with t o r s i o n p r i m e to p). m i n e d as the i d e n t i t y c o m p o n e n t of G.

T h e t o r u s is u n i q u e l y d e t e r -

-

Proposilion

6.

are dense.

For

with order

n.

(b)

If k

(a) The

elements

then there

-

of finite order

a given integer

is not the algebraic

42

n, there

closure

exists an element

x ~G

of a diagonalizable

group

exists only a finite number

of a finite field and

whose

powers

are dense

G

G

of elements

is a torus

over

k,

in G.

P r o o f of p a r t (a) follows i m m e d i a t e l y f r o m the d e c o m p o s i t i o n G ---~ GL 1 ~ 2Z/n Z ~ X ' ' " ~ Z ~ / n 7-/.1

(The Zariski

topology

infinitely many

r

GL 1 is the cofinite topology

on

and roots

of unity are

in number).

The proof of p a r t (b} (which we s h a l l not use) m a y be found on page 208 of B o r e l ' s book.

We

now

prove

Proposition

7.

X(T)(X(T) Then

~

a proposition

Let

T

is a free t~T

which

is very

useful in later discussions.

be a torus and ~l ..... ~r ZE-module).

suchthat

Let

be linearly independent

C 1 ..... C r E k*

be arbitrarily

in

given.

~i(t) = C i ~l~i~r. r

Proof. Since

Consider

o(1 . . . . .

the morphism

~r

T.

It follows

on

T.

Hence

GL[.

But

is closed

f*

of X(T)

and hence

that ~i ..... ~r is injective

f is a morphism in GL[.

GL

1

, given by: f(x} = (~:l(X) ..... ~

r

(x)).

a r e l i n e a r l y i n d e p e n d e n t in X(T), m o n o m i a l s in ~ 1 ' " ' " ~ r

are distinct elements on

f : T --~

The

are algebraically

so that of groups

proposition

are linearly independent

now

f is dominant and hence follows.

independent i.e.

f(T)

by proposition

as functions as functions is dense

in

2 of i. 13, f(T)

-

2.7.

Rigidity Theorem.

43

-

We p r o c e e d to p r o v e an i m p o r t a n t t h e o r e m .

!

Theorem. Let M: V~ H - - ~ H

be a rnorphisrn such that:

(I) II is an algebraic group in which the elements of finite order are dense. (2) II

I

is an algebraic group which contains only finitely many elements of

a given finite order. (3) V is a connected variety. I

(4) F o r a fixed v E V , the map o(v : H ~ a rnorphism of groups.

H , given by O(v(h) =o((v,h), is

Then ~ v : H - - ~ H' is the same rnorphism Vv ~ v

(i. e. ¢~ factors through P2 : Vx H ~

Proof. Let h ~H be of finite order.

H).

Consider o<(Vx h) which consists of

elernents having orders which divide the order of h. is a finite set.

Hence by (2), o((V x h)

But it is connected also, since V is connected.

(V x h) consists of a single element. order, O((v,h) = o((v',h) V v , 'polynomial' condition on H

v' ~SV.

Hence

Thus, whenever h ~H is of finite For fixed v,v'~ V, the above is a

and hence defines a closed subset F

closed subset contains the set of all elements with finite orders.

of H.

This

Hence by (I),

!

F = H, i.e. z~(v,h) = o((v',h) ~ h follows that o((v,h) = o((v ,h)

E H.

Since v,v ~ V

v,v @ V , h @H.

are arbitrary, it

Thus O(v = O(v, v,v e V.

This proves the proposition.

Corollary i . Since conditions (1), (2) are satisfied in the case of diagonalizable groups

I

H, H , the above proposition holds for these groups.

Corollary 2. Let H be a diagonalizable subgroup of an algebraic group G, Then we have the following:

-

(a}

NG(H)°

normal

(b)

= ZG(H)°.

subgroup

then

NG(It}/ZG(H)

Hence H

H = H and

~:

-

is connected

H

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

1

to be the m o r p h i s m c ~ t ( v , h )

(Since

corollary

1.

Taking

NG(H)° ~ ZG(H).

Hence

= vhv - 1 .

group

Take

Hence

v = e, vhv -1 = e . h . e -1 = N G(H) o C ZG(H) °

ZG(H) ° C_ NG(H) ° i s a l w a y s t r u e . )

NG(H)/ZG(H ) is a homomorphic The last mentioned

is a diagonalizable

1, l e t V = NG(H} °, w h i c h i s a c o n n e c t e d v a r i e t y .

VxH--~

NG(H) ° = ZG(H) °.

H

is central.

vhv "1 i s i n d e p e n d e n t of v, by c o r o l l a r y h Vv ~V.

and

is finite.

P r o o f . I n the c o r o l l a r y !

if G

44

Thus

Further,

i m a g e of N G ( H ) / Z G ( H ) O = N G ( H ) / N G ( H ) o .

is finite by proposition

1 of 1.12.

Hence

the

is proved.

Proposition.

(a) L e t G be a d i a g o n a l i z a b l e a l g e b r a i c g r o u p , a c t i n g on an

affine variety

V.

T h e n o n l y f i n i t e l y m a n y f i x e d p o i n t s e t s u n d e r the a c t i o n of

s u b s e t s of G o c c u r in V. as stabilizers

A l s o , o n l y f i n i t e l y m a n y s u b g r o u p s of G o c c u r

of s u b s e t s of V.

(b} If G is a l s o c o n n e c t e d ( i . e . if G i s a t o r u s ) , t h e n t h e r e e x i s t s an e l e m e n t xEG

such that

most

x£G

Proof.

x.v = v implies

( i . e . V G = Vx); in f a c t , f o r

this is so.

G a c t s on k [VJ

morphically,

Choose a finite set {fl ..... generates

y.v = v VYEG

h e n c e via s e m i - s i m p l e

f n t of e i g e n v e c t o r s

k [VJ a s k - a l g e b r a .

Since

endomorphisms.

of the a c t i o n of G w h i c h

fi i s an e i g e n v e c t o r f o r e a c h

x ~G,

there exists a character

X i E X(G)

s u c h that: f i ( x . v ) = X i ( x ) . f i ( v ) V x E G , v

Now, x . v o v iff qIx.v

o qIv/Vi

<since { f t . . . . .

k ~V])

iff fi(v) (Xi(x) - 1) = 0 V l ~ i

_~ n.

generates I

£V.

-

(a) L e t W C V

45

-

be the s e t of f i x e d p o i n t s of a s u b s e t S of G.

Let

J= {i/Xi/S ~ I} . Then W = [vCV/fi(v) = 0 V iCJ~. For:

From I, x . v = v Vx ¢ S

fi(v) = 0 ~ i ~ J . i, 2 ..... n~ The

Thus

. Thus

statement

about

iff fi(v)(Xi(x)- 1)-- 0 V x C S , V i

iff

W d e p e n d s e n t i r e l y on J which is a s u b s e t of such

W's

occurence

are finitely many of finitely many

in number.

stabilizers

can be proved

in a s i m i l a r way.

(b) If G i s a t o r u s , then G i s i r r e d u c i b l e a s a v a r i e t y . G i = ~xCG/Xi(x)

~ i~

X i ~ 1 on G.

Let J=

. Then

each

G i is an open

~i ~ G i ~ ~ .

Consider

set and

G i ~ ~ whenever

Since G i s i r r e d u c i b l e ,

• Gi~ # i~J

So l e t x g /~ G.. 1 iEJ C l a i m : V G = V x. Now V G C V x i s a l w a y s t r u e . fi(v)(Xi(x) - 1) ; 0 V i .

Solet

Now f o r i E J ,

a n y y t ~ G , fi(v) (Xi(Y) - 1) = 0 ~ i t~J. f i ( v ) ( N ( y ) - 1)-- 0 V i ~ J .

Corollary.

Thus

y.v:v

v~V

suchthat

x.v=v.

Hence

Xi(x) ~ 1 s o t h a t fi(v) = 0. But then f o r i ~ J , o r v £ V G.

Hence for

Xi(Y) = 1.

Hence

T h i s p r o v e s the p a r t (b).

L e t H be a d i a g o n a l i z a b l e s u b g r o u p of G.

(a) Only f i n i t e l y m a n y c e n t r a l i z e r s in G of s u b s e t s of H o c c u r .

A l s o , only

f i n i t e l y m a n y c e n t r a l i z e r s in H of s u b s e t s of G o c c u r . (b) If tt i s c o n n e c t e d , then Z G ( H ) = ZG(X ) f o r s o m e

x ~H; in f a c t , f o r m o s t

X.

Proof.

Make H a c t s on G by c o n j u g a t i o n ( i . e . h(g) = h g h - l V h

The C o r o l l a r y now f o l l o w s .

CH, g £ G ) .

Second P a r t : Q u o t i e n t s and s o l v a b l e ~ r o u p s

2.8.

Solvable G r o u p s .

The b a s i c r e s u l t h e r e i s a s follows:

Theorem 1 (Lie-Kolchin).

A c o n n e c t e d , s o l v a b l e l i n e a r a l g e b r a i c group fixes

a flag (of the u n d e r l y i n g space) i . e . can be put into an u p p e r - t r i a n g u l a r f o r m .

A proof of this t h e o r e m will be given l a t e r (see 2.11).

F o r an a l t e r n a t e proof,

which could be given now, see S e r r e ' s b o o k ' L i e A l g e b r a s and Lie G r o u p s : I

L . A 5.11 o r e l s e the a u t h o r ' s l e c t u r e s on C h e v a l l e y g r o u p s .

T h e o r e m 2. (a)

L e t G be a c o n n e c t e d s o l v a b l e a l g e b r a i c group.

Gu= ~g~GIg

unipotent~

isaelosed,

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

G c o n t a i n i n g DG = ~G, G 3 ; and h e n c e the l a t t e r is n i l p o t e n t . (b) If G is n i l p o t e n t , then G s = {g e G / g

s e m i s i m p l e t is a (closed) t o r u s

and the d i r e c t p r o d u c t d e c o m p o s i t i o n G = G s. Gu holds (i. e. the c a n o n i c a l map

m : GsX G u

* G ; m ( s , u ) -- s . u

i s an i s o m o r p h i s m of a l g e b r a i c g r o u p s ) .

(c) The m a x i m a l t o r t of G a r e c o n j u g a t e . is a s e m i - d i r e c t product.

If T i s one of t h e m , then G = T . G u

(The g e o m e t r i c r e q u i r e m e n t is that T ×

Gu-"-"~G

is an i s o m o r p h i s m of v a r i e t i e s ) . (d) If S is a s u b g r o u p of G, c o n s i s t i n g of s e m i s i m p l e e l e m e n t s , then S can be i m b e d d e d in a t o r u s .

(Hence S is a b e l i a n ) .

(e) NG(S) i s c o n n e c t e d (S a s in (d)) and h e n c e i s equal to ZG(S) (by the Rigidity Theorem).

°

47

°

H e r e the r e s u l t s i m p o r t a n t f o r o u r p u r p o s e s h a v e been u n d e r l i n e d . m a y be found in B o r e l ' s book (page 244).

/k p r o o f

It u s e s t h e o r e m 1, i n d u c t i o n on the

l e n g t h of the d e r i v e d s e r i e s , and the t i m e - h o n o u r e d m e t h o d of a v e r a g i n g o v e r f i n i t e g r o u p s which t u r n s out to be a p p l i c a b l e b e c a u s e the e l e m e n t s of f i n i t e o r d e r in a t o r u s a r e d e n s e .

Corollary.

(a) E v e r y u n i p o t e n t e l e m e n t (in f a c t , subgroup) can be i m b e d d e d

in a c o n n e c t e d u n i p o t e n t g r o u p . (b) E v e r y s e m i s i m p l e e l e m e n t (in f a c t , c o m m u t a t i v e s u b g r o u p of s u c h e l e m e n t s ) can be i m b e d d e d in a c o n n e c t e d such g r o u p .

Remark.

For

nonsolvable

connected

groups

(a) continues

to hold, but (b) fails,

as will be seen later.

2.9. Varieties

in general.

general

than the affine ones.

number

of affine varieties,

To continue, Roughly,

we have to consider

varieties,

a variety is a collection

suitably patched

together.

More

more

of a finite

precisely,

we

have: Definition.

A variety is a topological

space

V

with a finite cover

~Uil i~_n

1 of open subsets (ii Each (2)

satisfying the following properties:

U i is an affine variety.

U i f% Uj

is a principal

the identity map

of Uit% Uj

Ui•U

j (obtained from

(3)

The set of points

open set in both the affine varieties is an isomorphism

U i and

of the two affine

U i and

structures

Uj).

(x,x) •U iXUj

(x •U i ~Uj)

Uj and

is closed there.

on

-48-

Note that an affine variety is a variety. V

iff

U ~U.

is so in

U.

l

Further

U

is open (resp. closed) in

(in the Zariski topology) for every

i.

A closed

1

subset of a variety is a variety in a natural way, and so is an open one since an open subset of an affine variety is the union of a finite number of principal affine open subsets. For a variety V

(Check all of this.) V,

we write

that are defined everywhere.

k[V]

for the algebra of rational functions on

A function

Ux

sense) and

h(x) ~ 0.

itself is affine, then

with

in the old sense (by 1.13, cot. to lemm~ 1).

k[V]

Definition. (1)

f

Let

V, W

be varieties.

is continuous,

f(S) ~ T ,

the map

V

(2)

f : S -~T

x,

is said to be defined at

for some affine neighborhood If

of

f

f = g/h

A map

with

for all affine open sets

if

g,h C k[UxJ (in the old

k[V]

f : V-~W

x

as just defined agrees

is called a morphism if S~V,

T~W

with

is a morphism of affine varieties

We can construct the product of two varieties by taking the products of the constituent affine varieties and then patching them together suitably.

It and the

resulting projections satisfy the universal property mentioned in 1.6 in the affine case.

(3')

In

The condition (3) above can then be restated (check this):

V X V

the diagonal is closed.

If we were using the product topology on say that the topology on

V

V XV

(which we aren't), this would

is Hausdorff (which it isn't).

Various of the properties of an affine variety continue to hold for an arbitrary variety, e.g. the decomposition into irreducible components. see Mumford's book.

For further details

- 4 9

2. I0.

Complete

varieties

-

and Projective

varieties.

D e f i n i t i o n . A v a r i e t y V is said to be c o m p l e t e if f o r e v e r y v a r i e t y W, the p r o j e c t i o n m a p P2 : V ~ W

Remarks.

~ W is c l o s e d .

T h e affine l i n e ~

= (k, k I X ] )

is not c o m p l e t e .

In fact, a s we

s h a l l see p r e s e n t l y , a n y c o m p l e t e affine v a r i e t y c o n s i s t s of f i n i t e l y m a n y points.

P r o p o s i t i o n 1. (1) A c l o s e d s u b v a r i e t y of a c o m p l e t e v a r i e t y is c o m p l e t e . The i m a g e of a c o m p l e t e v a r i e t y u n d e r a m o r p h i s m i s c l o s e d and c o m p l e t e . P r o d u c t s of c o m p l e t e v a r i e t i e s a r e c o m p l e t e . (2) A c o m p l e t e affine v a r i e t y c o n s i s t s of f i n i t e l y m a n y p o i n t s . (3) A m o r p h i s m f r o m a c o n n e c t e d c o m p l e t e v a r i e t y to an affine v a r i e t y is constant.

Proof.

(1) C l e a r l y a c l o s e d s u b v a r i e t y of a c o m p l e t e v a r i e t y is c o m p l e t e .

Let f : V-

Claim:

> W be a m o r p h i s m of v a r i e t i e s with V c o m p l e t e .

f(V) is a c l o s e d s u b v a r i e t y of W.

Consider F

= f(v, f(v)); v e V t

If we now p r o v e that F

C V xW,

f.

is c l o s e d , then f(V) will be c l o s e d in W, b e i n g the

i m a g e of j-7 u n d e r the m a p

P2 : V x W ~ W .

that r 7 i s c l o s e d , holds f o r a n y m o r p h i s m complete).

the g r a p h of

F o r : The diagonal

is the inverse image of

~

of

(V is c o m p l e t e ) . f : V ~

W ~ W

W ( i . e . V m a y not be

is closed by

"% under the morphism

The fact

(3')

of 2.9 and

f X Id : V ~ ( W . @ W X W .

-

50

-

R e t u r n i n g to the p r o o f of the fact that f(V) i s c o m p l e t e , we see that f(V) is a $ u b v a r i e t y in its own r i g h t (being c l o s e d in W). themaps:

V~T

f n Id ~ f(V)×T-

P2

;T.

F o r a n y v a r i e t y T, c o n s i d e r

For a closed set SCf(V)~T,

P2 (S) = P2 o (f ~Id)((f x I d ) ' l ( s ) ) which is c l o s e d in T

since ¥

is c o m p l e t e .

T h i s p r o v e s the r e q u i r e d r e s u l t .

(2) L e t W be a c o m p l e t e affine v a r i e t y .

L e t A be i t s a l g e b r a of f u n c t i o n s .

C l a i m : The s e t f(W) is finite for e v e r y f ~ A . C o n s i d e r the affine l i n e

~%1 and the m a p Wm ~

s:

--~2~ .

s

P2(S) c a n n o t be the whole set A

( s i n c e 0 ~P2(S)).

and h e n c e P2(S) i s c l o s e d i n ~ 1 .

Let

,vx,,,,,. But then W is c o m p l e t e

Hence P2(S) is a finite s e t in ~k1.

It now

follows that f(W) is a l s o f i n i t e .

F r o m this c l a i m , it follows i m m e d i a t e l y that W c o n s i s t s of f i n i t e l y m a n y points.

(One can c o n s i d e r W as a s u b v a r i e t y of s o m e

k n and then c o n s i d e r

the c o o r d i n a t e f u n c t i o n s X i ,1 ~ i & n).

Note.

A s a n i m m e d i a t e d e d u c t i o n , one gets: A c o n n e c t e d , c o m p l e t e , affine

v a r i e t y c o n s i s t s of a s i n g l e e l e m e n t .

(3) Since the i m a g e of a c o n n e c t e d v a r i e t y is c o n n e c t e d (as a set), (3) i m m e d i a t e l y follows f r o m (1) and (2) above.

We now c o n s i d e r a v e r y i m p o r t a n t c l a s s of c o m p l e t e v a r i e t i e s , viz. the projective varieties.

Projective Spaces. 0 in kn + l .

We b e g i n with:

C o n s i d e r kn + l .

Let IPn be the set of all l i n e s t h r o u g h

One can e a s i l y see that IPn i s the s e t of e q u i v a l e n c e c l a s s e s

-

Ix o.....

Xn] , x i • k

iff 3 ) ~ k *

51

-

with a t l e a s t one x i ~ 0, (x ° . . . . .

Xn)~'~(Yo . . . . .

Yn )

such that xi = lYi}"

Let IPni =

{IXo . . . . .

#i : IPin-

> kn given by

and bijective.

Xn~

with x i ~ 0 } . It can be s e e n that the m a p A x0 xi xn @i ( Ix 0 ..... Xn] ) = ( - - , . , - - . , - ) xi xi xi

Thus, ion can be given the structure of an affine variety. i

fact, of /A n, the affine n-space).

k I ~ ° . . . . . . .~. .i

is well-defined (In

T h e algebra of functions is

~ ]

n ]Pn It i s e a s y to s e e that Ipn = ~J A l s o , Ip.nt'~ ]pn i= 0 i" z j i s a p r i n c i p a l open s e t in IPn a s w e l l a s in ]pn. Define U C]P n to be open 1

iff U Cl IP n i s open V 0 _~ i An. i

j

It f o l l o w s that IP n i s a v a r i e t y .

This variety

i$ c a l l e d the p r o j e c t i v e s p a c e of d i m e n s i o n n.

O n e can define, for a vector space manner, by choosing a basis.

V

of dimension

n+l, IP(V) in a similar

It is easy to check that the structure defined

on IP(V) is independent of the basis chosen.

A closed subvariety of a projective space is called a projective variety.

Our basic proposition is :

P r o p o s i t i o n 2. A p r o j e c t i v e v a r i e t y i s c o m p l e t e .

Proof. In view of the proposition I, it is enough to prove that a projective space i °n is complete. complete.

One proceeds by induction. ]pO being a point, is

Now let W be any variety and S C_IPn~(W be closed. We have

to show that P2(S) is closed, where P2 : ipn X W - - ~ W is the projection map. It is clear that one may assume the following: (1) W is affine, (2) S is irreducible. Let B be the function-algebra of the affine variety W. Let B be the function-algebra of the irreducible subvariety P2(S)

of W .

(~ is an integral domain).

Consider

S/% Ipnx W = S i, i

O,
-

If S. i s e m p t y f o r

some

52

-

i, then S ~ ( I P n - I p n ) ~ w . .

1

It is e a s y to s e e that

1

IP n - IPn

can be c a n o n i c a l l y i d e n t i f i e d with IPn - 1 .

Hence b y i n d u c t i o n , it

i

follows that P2(S) is c l o s e d .

Xi T h u s the f u n c t i o n X--j

So let Si ~ @ y o ,~ i ~ n .

is no___~ti d e n t i c a l l y z e r o V i, j (hence not i d e n t i c a l l y o0 V i, j).

C o n s i d e r the

e l e m e n t s of B a s f u n c t i o n s on S. in the obvious way and l e t Xk denote Xk 1 Xj Xj r e s t r i c t e d to Si.

B

Xo

.. •

Since

Xi --,.. "' x i

Xn

Xo

=k

""-~i

Xi .. __ ,.. '"

algebra of Ipnxw, it follows that B I Siisirreducible. Claim:

'xi

B is the function "" x i A

~ o ..... Xq __ ..... X n Xi

is that of S.. Also, I

Hence BI~'''''-Xi'''''Xi ~[~in]= C'I isanintegraldomain.

The quotient field F i of C i is independent of i. This is easy to see, -

s i n c e the f u n c t i o n

Xk

xj ¢cj

=

T h u s we have the following:

Xi/x i EF i A

B ----~, B

- - ,

P2

Let q E P2(S).

Hence eq : B ~ k

.

.

~ : R

.

.

is a k - a l g e b r a h o m o m o r p h i s m .

I e m m a i n p r o p o s i t i o n 1 of 1 . 1 3 , it follows that e algebra homomorphism

.

Xi

.....

q

e k, w h e r e R C F

By the

can be e x t e n d e d to a k is a v a l u a t i o n r i n g of F

(as obtained in the proof of the lemma). Claim:

R _D Cio f o r some

i o.

Let i o be an index (O-
n

-- ~ ~_~ Xi o

R

Xi° then -~j ~ R.

e

(R is a valuation ring). Also, k ~Jio ~

k Eft, T h u s

i0 ~ Jj - Jio which contradicts the fact that Jio has the maximal cardinal. This proves that Jio = {0, I ...n~ . Hence R ~ Cio . Hence we get ~ : Cio---~ Hence J P ~ S i o

= S /~<X

W

such that @ = ep. It follows from the

k.

-

53

-

c o m p a t i b i l i t y of ep and eq that p 2 ( p ) = q f a c t t h a t P2(S) i s c l o s e d .

or q~p2(S).

This proves

the

Hence IP n i s c o m p l e t e .

We n e x t p r o v e :

P r o p o s i t i o n 3. P r o d u c t s of p r o j e c t i v e v a r i e t i e s a r e a g a i n p r o j e c t i v e v a r i e t i e s .

Proof.

C l e a r l y , it i s enough to p r o v e that the p r o d u c t of two p r o j e c t i v e s p a c e s

is a projective variety.

So c o n s i d e r two p r o j e c t i v e s p a c e s

IPn and IP m.

C o n s i d e r the m a p : ipnx ipm

([Xo

. . . . .

IP

(n+l)(m+l)-I

Xn]'[Yo .....

given by :

= [XoYo . . . . X J m ' X l Y o

.....

XnYm].

It can be e a s i l y s e e n t h a t ~ i s a m o r p h i s m , in f a c t , a n i s o m o r p h i s m onto the image.

Hence the p r o p o s i t i o n f o l l o w s .

We now g e n e r a l i s e t h e c o n c e p t of p r o j e c t i v e s p a c e s :

We define:

G r a s s m a n n i a n V a r i e t i e s . L e t V be an n+l - d i m e n s i o n a l v e c t o r s p a c e . Gd(V) be the s e t of a l l d - d i m e n s i o n a l s u b s p a c e s of V(0 ~ d &n+l). Gd(V) i s c a l l e d a G r a s s m a n n i a n - v a r i e t y . given by: ~ (W) = (Vl~ v2 ^ . . . d-dimensional subspace

Let

Then

C o n s i d e r t h e m a p ~ : G d ( V ) - - ~ IP(]~V)

^Vd), where {vl,...,Vdt is a basis for'the

W of V.

It can e a s i l y be c h e c k e d t h a t

~ is well-

defined.

Proposition 4.

If Gd(V), @ a r e as above, then @ is injective and ~(Gd(V))

is closed in IP(tkdv).

Thus Gd (V) can be given the structure of a projective

variety (which is complete by proposition 2). Its dimension is d(n+1 - d).

-

Proof.

The proof may

54

-

be found in Borel's book

W e d e f i n e a n o t h e r t y p e of p r o j e c t i v e v a r i e t i e s

Flag-varieties.

Let

vi z.

V b e an ( n + l ) - d i m e n s i o n a l

t h e s e t of a l l f l a g s of V. of s u b s p a c e s

(page 239).

(To r e c a l l ,

of V s u c h t h a t

space over

a f l a g is a s e q u e n c e

k.

Let

~(V)

be

0 = V o C V I ~ - . . ~Vn= V

d i m V. = i V1 ~ i ~ n ) . 1

Prcposition 5. If V, "~(V) are as above, the natural map :~(V)

~ Go(V)X ...XGn(V) is injective. Also,~(~(V)) is closed. T h u s

'~ iV) can be given the structure of a (complete) projective variety.

The

proof is easy but it will be omitted

Let

2.11. Quotients.

G

(See Borel's

be an affine algebraic

book,

page 241).

group and

H C G

be closed

subgroup.

Definition.

A pair

(T~,V), w h e r e

is called a quotient for

G/H

V i s a v a r i e t y and 7f : G

1, V is a m o r p h i s m ,

if the f o l l o w i n g c o n d i t i o n s hold:

(1) T h e f i b r e s of 7l" a r e j u s t t h e c o s e t s of H in G.

( h e n c e 7~:is s u r j e c t i v e ) .

(2) "/i- is o p e n . (3)

If U C V i s o p e n , t h e n

= k

, t h e a l g e b r a of a l l

-1 f u n c t i o n s on 7 r

Example. quotient for

(U), c o n s t a n t on t h e c o s e t s of H, i . e .

C o n s i d e r two a f f i n e a l g e b r a i c g r o u p s U X V / U (P2: U x V ~

It c a n be s e e n t h a t if a q u o t i e n t

t h e f i b r e s of 7t-.

U and

V.

Then

(P2'V)

is a

V is the projection).

(TF,V) f o r

G/H

exists,

then it is unique.

The

-

morphism ~:

G ~

55

-

V is universal among all the morphisms

from

G which

a r e c o n s t a n t on c o s e t s of H.

Orbit maps. variety

V.

Let

G b e an a f f i n e a l g e b r a i c g r o u p a c t i n g t r a n s i t i v e l y

Let

vEV,

H=G v

:{

gEG/g.v=v

t h e r e e x i s t s a n a t u r a l m a p 7[-: G - - - ~ V morphism

a n d (1) a b o v e h o l d s .

below (see appendix). holds.

1.

Let

G/H, i.e.

G,V,

b e t h e s t a b i l i z e r of v.

t

g i v e n b y "W(g} = g . v .

Then

C l e a r l y , 71-is a

T h e o p e n n e s s of "~, (2) a b o v e , wiI1 b e p r o v e d

("~, V) i s a q u o t i e n t f o r

When does this condition hold ?

Proposition for

Then

on a

G / G v if the c o n d i t i o n (3)

We have:

v, H, 71" be a s a b o v e .

Then

(71",V) i s a q u o t i e n t

(3) h o l d s , i f f t h e d i f f e r e n t i a l m a p (dTIJ 1 : T(G)I-----~ T(V) v i s

surjective.

T h i s m a p i s a l w a y s s u r j e c t i v e if p(k) = 1 (Thus

quotient for

G/H

(71", V) i s a

if p ( k ) = 1).

T h e p r o o f of t h i s p r o p o s i t i o n m a y b e found in B o r e l ' s b o o k (page 180).

A s h o r t d e s c r i p t i o n of t a n g e n t s p a c e s and d i f f e r e n t i a l s i s a s f o l l o w s :

Let

V b e an a f f i n e v a r i e t y and

dual numbers ~:

A

k [ 6 J w i t h E 2 = 0.

• k[63.

a k-algebra

A b e i t s a l g e b r a of f u n c t i o n s . Consider a k-algebra

T h i s i s of t h e f o r m

homomorphism

~=

~+~.

¢

Introduce the

homomorphism

where ]3: A ~

k is

and ~ : A ~

k is a linear map satisfying:

r(a.b) --~{a}.r(b} +/5 {b).r(a) Va, b E A .

Now, ~ = ev for some v ~ V .

r ( a . b ) = a ( v ) . r(b) + b(v). r ( a ) , s o t h a t

v a k-algebra

satisfies the rules for differentiation

o, s.o

homomorphism~

t a n g e n t s p a c e to V at

r

v.

Then

ev

) is a vector space over It i s d e n o t e d by

T(V) v .

r/o.+

k and is called the

T h e s e t of a l l ~ ' s

i.e.

-

t h e u n i o n of a l l

T(V)v'S

then

~ T(W)f(v)

e l ( v ) + (r o f*). E

in a n a t u r a l w a y :

Then

In t h e c a s e of an a r b i t r a r y b y u s i n g an a r b i t r a r y

Remark.

the map

~1

Let

is a mor-

map

r ET(V) v, then (ev+r.6)o f =

homomorphism,

h e n c e r o f* ~ T(W)f(v).

df i s c a l l e d t h e d i f f e r e n t i a l of f.

v a r i e t y one can d e f i n e t h e t a n g e n t s p a c e a t a p o i n t

o p e n a f f i n e n e i g h b o u r h o o d of t h a t p o i n t .

The eondition

"71" is s e p a r a b l e " .

If f : V - - - ~ W

f g i v e s r i s e to a k - l i n e a r

is again a k-algebra

D e f i n e (df)v(r) = r o f*.

-

is j u s t t h e t a n g e n t b u n d l e .

p h i s m of a f f i n e v a r i e t i e s , (df} v : T(V) v

56

" d~I" i s s u r j e c t i v e "

in P r o p o s i t i o n

It r u l e s out u n w a n t e d i n s e p a r a b i l i t y ,

1 is often stated

which, e.g.,

prevents

/ ~ 1 , x ~ - - ~ xP(p = c h a r k ~ 0) f r o m b e i n g an i s o m o r p h i s m

even

though it is bijective.

Theorem then

1.

G/H

variety).

If G i s an a f f i n e a l g e b r a i c g r o u p and

exists as a quasi-projective

H ~G

v a r i e t y (i. e . o p e n s u b s e t of a p r o j e c t i v e

It c a n be r e a l i z e d v i a an o r b i t m a p in t h e p r o j e c t i v e s p a c e

c o m i n g f r o m a l i n e a r a c t i o n of G on V ( f o r s o m e V). then

is a closed subgroup,

G/H

IP(V),

If H i s a l s o n o r m a l ,

i s a f f i n e and b e c o m e s an a f f i n e a l g e b r a i c g r o u p .

T h e p r o o f c o n s i s t s of c o n s t r u c t i n g an o r b i t m a p f o r w h i c h t h e c o n d i t i o n s of proposition 1 hold.

(See p a g e 181 of B o r e l ' s

After these preparations,

we c o m e to a b a s i c t o o l in t h e s t u d y of a f f i n e g r o u p s .

Borel's fixed point theorem. a non-empty complete variety

Proof.

book).

A connected,

solvable affine group

G a c t i n g on

V always has a fixed point.

W e p r o v e t h e t h e o r e m b y i n d u c t i o n on d i m G.

If d i m G = 0, G = { e l

-

a n d t h e n t h e r e i s n o t h i n g to p r o v e . and

~ te],

DG C G.

57

-

So l e t

dim G 70.

Since

G is solvable

Now DG i s c l o s e d and h e n c e i s of s m a l l e r

T h u s , by t h e i n d u c t i o n h y p o t h e s i s ,

DG h a s a f i x e d p o i n t in V.

s e t of f i x e d p o i n t s of DG (W ~ ~).

Clearly,

of G.

(DG i s n o r m a l in G).

of V) t h a t

Thus,

one c a n a s s u m e

(by t a k i n g

DG a c t s a s i d e n t i t y on t h e w h o l e of V.

of v in G.

Then

G--~V

i s c o n s t a n t on t h e c o s e t s of G v. there exists a morphism bijective.

Also, Gv ~

;~(x)

W be the

= x.v

W instead

We now have a homoLet

v ~V, G v =

is a morphism

of G w h i c h

H e n c e , by t h e u n i v e r s a l p r o p e r t y of G --9, G / G v ,

~ : G/G v

-'- V g i v e n by

DG a n d h e n c e

an a f f i n e a l g e b r a i c g r o u p .

Let

W is invariant under the action

g e n e o u s s i t u a t i o n (taking a c l o s e d o r b i t of G, w h i c h e x i s t s ) . stabilizer

dimension.

~(g) = g . v .

G v i s n o r m a l in G.

~ is clearly

Thus,

G/G v

is

Now t h e t h e o r e m f o l l o w s f r o m t h e f o l l o w i n g p r o -

position.

Proposition

2.

Let

f : V1-----~V 2 be a G - m o r p h i s m

of h o m o g e n e o u s G - v a r i e t i e s

(G i s a c o n n e c t e d , a f f i n e a l g e b r a i c g r o u p and a c t s t r a n s i t i v e l y with finite fibres.

If V 2 i s c o m p l e t e t h e n s o i s

A p p l y i n g t h e p r o p o s i t i o n to t h e m o r p h i s m is complete group. fixes

(since

V is complete).

V 1.

~ : G/G v

But t h e n

on V 1 and V 2)

";. V, we g e t t h a t

G/G v

H e n c e b y p r o p o s i t i o n 1 of 2 . 1 0 , G / G v = ~ e t

or

G/G v

is a connected, affine G = G v.

Thus

G

v.

T h e p r o o f of the p r o p o s i t i o n 2 i s c o n t a i n e d in t h e a p p e n d i x .

A s a c o r o l l a r y to t h e a b o v e t h e o r e m ,

(Lie-Kolchin) Theorem.

we h a v e :

(as s t a t e d in 2 . 8 ) .

( c l o s e d } s u b g r o u p of G L ( V ) .

Then

Let

G be a connected,

G a c t s on ~ ( V )

solvable,

in a n a t u r a l w a y .

By

-58-

p r o p o s i t i o n 5 a b o v e , ¢~ (V) i s c o m p l e t e .

H e n c e the t h e o r e m f o l l o w s i m m e d i a t e l y

f r o m the f i x e d p o i n t t h e o r e m .

A p p e n d i x to 2 . 1 1 .

T h i s a p p e n d i x b r i n g s s e v e r a l c o m p l e m e n t s to the d e v e l o p -

ment so far.

P r o p o s i t i o n 1. L e t G be an affine a l g e b r a i c group; V1, V 2 h o m o g e n e o u s s p a c e s f o r G; a n d - / I ' : V 1 - - - ~ V 2 a G - m o r p h i s m

( i . e . 7 r ( g . v 1) = g . T T ( v l ) V g ~ G ,

v I ~ VI). (a) 71- i s open. (b)

If 71"has f i n i t e f i b r e s and V 2 is c o m p l e t e , then V 1 i s a l s o c o m p l e t e .

P a r t (b) i s the m i s s i n g s t e p (viz. p r o p o s i t i o n 2) of the p r o o f of the f i x e d p o i n t t h e o r e m , while p a r t (a) s h o w s , a s m e n t i o n e d a b o v e , t h a t o r b i t m a p s a r e a l w a y s open, so that in p a r t i c u l a r ,

s u r j e c t i v e m o r p h i s m s of a l g e b r a i c g r o u p s a r e

a l w a y s open.

W e s h a l l g i v e the p r o o f , a s s u m i n g

G to be c o n n e c t e d , in s e v e r a l s t e p s .

(1) We define a m o r p h i s m (of affine v a r i e t i e s ) f(U 1) i s d e n s e i n

U 2 and k [ U 1 ]

e v e r y h o m o m o r p h i s m : f*(k [U2] ) ~

f : U1-----~ U 2 to be f i n i t e if

isintegralover

f*(k[U2]).

T h e f a c t that

k e x t e n d s to k [U1] t r a n s l a t e s g e o m e t r i -

c a l l y to : f(U 1) i s c l o s e d , and s i m i l a r l y f o r any c l o s e d s u b s e t of U1; i . e . is a closed map. subset

f

T a k i n g c o m p l e m e n t s , we s e e that f(U) i s open f o r a n y open

U (of U 1) which i s m a d e up of c o m p l e t e f i b r e s .

(2) L e t f : V1-----~ V 2 be a m o r p h i s m of i r r e d u c i b l e v a r i e t i e s with f(V1) d e n s e in V 2.

Then f o r s u i t a b l e open affine s u b s e t s

1 V 21 of V 1 , V 2 r e s p e c t i v e l y , V1,

-59

we have: f(V ) C_ V 21 and f l : f / v with g f i n i t e and a r e affine.

P2 a s u s u a l .

Identify B = k

~

-

_r 1 1 can be factored: Vll ----------~/~xV g 2-------r~_ ~'2 V ,

C l e a r l y , we m a y a s s u r r e that V 1 and V 2

i v ]2

with a p a r t of A = k

[vii

via f° . W o r k i n g

in the q u o t i e n t f i e l d Q(A) of A, we see by N o e t h e r ' s l e m m a (page 4, M u m f o r d ' s book) that t h e r e e x i s t e l e m e n t s X l , . . , x r ~ Q ( B ) [ A ] , i n d e e d in A, a l g e b r a i c a l l y i n d e p e n d e n t o v e r Q(B), such that Q(B) [ A ] Q(B) Ix 1 . . . . .

Xr].

is integral over

In the e q u a t i o n s e x p r e s s i n g the i n t e g r a l i t y o v e r q(B)[Xl,...,Xr]

A/B, the

for a f i n i t e g e n e r a t i n g s e t f o r

c o e f f i c i e n t s a r e p o l y n o m i a l s in

x 1 . . . . Xr, with c o e f f i c i e n t s in Q(B), finite in n u m b e r . d e n o m i n a t o r f o r a l l of t h e s e l a t t e r c o e f f i c i e n t s . B[bl--] [Xl . . . . ,Xr] which is p u r e o v e r B I l l . we have o u r r e s u l t with

(3) We o b s e r v e that f l

L e t b E B be a c o m m o n

Then A l l

]is integral over

G e o m e t r i c a l l y , this m e a n s that

V1 V1 1 =(VI)b ; 2 = (V2)b" in (2) is open on s e t s m a d e up of f i b r e s s i n c e g is

by (1) and P2 c e r t a i n l y i s .

(4) To p r o v e (a), we m a y a s s u m e that V the obvious way; viz. g(g') For:

_

1

= G, c o n s i d e r e d as a G - s p a c e in

!

g. g . f

L e t v 1 E V1, then

V 1 ......

> V2

~NG/~ is a c o m m u t a t i v e d i a g r a m m e .

#(g) -- g . v 1 ~ (g)-- g ' f ( v l

Hence f i s open if ~ and ~

a r e open.

~ and

a r e both G - m o r p h i s m s . L e t S be an open s e t in V 1 = G.

We m u s t show that f(S) is open.

If H is

the s t a b i l i z e r of v = f(1) in G, then SH = c o m p l e t e f i b r e s , and f(SH) = f(S).

U Sh is a l s o open, c o n s i s t s of hEH T h u s we m a y a s s u m e that S i s m a d e up

1 V 21 as in (2), and V1,

I xit

of c o m p l e t e f i b r e s .

Now choose

f i n i t e in G, such

that U x i V1 = v 1 1 i

T h e n by (3), f(S/% xiV1) is open V i, whence f(S) is

-

60

-

open, as required.

(5) A s s u m e n o w a s in (b). c l o s e d f o r a n y W, s i n c e V 1i ' V 21

a s in (2) a n d

d i m V 2 and 1

It is e n o u g h to s h o w t h a t V

2

is complete

~ x i t a s in (4).

r = 0 above, so that

V1x

W -----~V 2 x W is also finite

integral over

B~)C),

1

~ V 2xW

W a f f i n e in f a c t ) .

is Choose

Since the fibres are finite, dim V 1 = ~ V12

is finite.

(if A i s i n t e g r a l o v e r

s o t h a t it i s c l o s e d .

Hence B, t h e n

A~C

is

The same holds for every

1

xiV 1XW-

Proposition Then

(for a n y

f t : Vll

1

V lxW

dim f

~xiV2)cW

2. -1

Let

so that

f : U~ V

(v) = d i m U

e.g.,

"~V2~W

is closed as required.

be a d o m i n a n t m o r p h i s m

- dimV

T h i s i s a w e a k v e r s i o n of L e m m a of o u r p u r p o s e s ,

V lx W

of i r r e d u c i b l e

f o r a d e n s e o p e n s e t of v ' s

varieties.

in V.

2 n e a r t h e end of 1. t3 t h a t i s e n o u g h f o r m o s t

f o r t h e p r o o f of P r o p o s i t i o n

2(c) of 1 . 1 3 .

For the

p r e s e n t p r o o f we m a y a s s u m e b y s t e p (2) a b o v e t h a t

f has the factorization

U

Now p21

g ~rXv

-~2

V with

g a finite morphism.

d i m e n s i o n s of c l o s e d s e t s by e x a c t l y

r

and

g-1

b r a i c e x t e n s i o n s do n o t c h a n g e t r a n s c e n d e n c e p o i n t , and to V3 we g e t

Proposition

3. L e t

preserves

degrees.

clearly raises

them since alge-

A p p l y i n g t h i s to

v, a

d i m f - l ( v ) = r = d i m U - d i m V.

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

variety.

>_,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)<~--~2 (a'). For:

(a) ~

that g B l g ' l

(a'). = B 2.

by(a),~nEN(T)

Let

B 1 , B 2 be two B o r e l s u b g r o u p s .

Choose a m a x i m a l t o r u s

C o n v e r s e l y , (a')

~

(a).

T c C c B, x £ G .

and B c o n t a i n s a m a x i m a l t o r u s bx B t q B ~ T f o r s o m e b E B . fore

~) Bn B. wE W w

Now T , T '

Hence

b ' . (bx)-I T = T for s o m e

xEB. N.B =

T'.

b ' ~ B.

' 1.

g=bl.n.b

( n b '1 . g "1) C g B l g ' l

Pick

such

T such that T C Z(T) C B1. T h e n

and b l , b ~ E B 1 s u c h t h a t

A l s o , b l T b l 1 = ( g b1" l . n - 1 ) . T

Then ~ g E G

= B 2.

Now, b l . T . b ~ I c B 1 T h i s p r o v e s (a').

T h e n by (a'), XB = xBx -1 a r e conjugate in B.

Hence

( b x ) - l T , T a r e c o n j u g a t e in B.

There-

T h u s b'(bx) -1 C N = N(T).

F o r u n i q u e n e s s , let Bn B = Bn B. w w'

T h i s shows:

Then I

bn w = n w , b ' with b , b ' E B.

Assume

t 6T, arbitrary.

Hence bnw t = n w ' b t .

T h e left side e q u a l s nwt. v (v ~ B~) s i n c e B = Bu . T ( s e m i d i r e c t ) a n d T is Abelian.

The right side is thus in B and e q u a l s n w ' t . n w ' v' ( v ' ~ Bu).

n w ' v ' E B u.

Hence

nwt =

n

t

w t , and w = w

t

as r e q u i r e d .

Thus

It would be n i c e

if s o m e o n e could supply at this stage of the d e v e l o p m e n t a s i m p l e proof of (a) o r (a'), and a l s o for the m i r a c u l o u s fact that W a c t i n ~ on X i s a l w a y s a g r o u p g e n e r a t e d by r e f l e c t i o n s .

We s h a l l have to u s e t h e s e r e s u l t s without proof.

We p r o v e the l e m m a for the c l a s s i c a l g r o u p s .

-73

-

(i) G = S L n . Let

B

be the group of upper triangular matrices.

diagonal matrices.

Let

T

be the group of

T h e n the following can easily be verified: Z(T) = T; N(T)

is the group of m o n o m i a l

matrices (i. e. each r o w and c o l u m n contains exactly

one non-zero element).

N ( T ) / z ( T ) is isomorphic to the permutation group~ I

S n.

We

b,b'EB

n o w prove that any element and

wEN(T).

Choose

gEG

b~B

can be written as

such that the total n u m b e r

appearing at the beginning of the r o w s of b. g is m a x i m a l . seen that in this case, the n u m b e r s

b.w.b

with

of zeros

It can be easily

of zeros at the beginning of the various

r o w s of b. g are all distinct (since otherwise w e could add a multiple of one r o w to a higher one, i.e. multiply on the left by s o m e element of B, and increase the n u m b e r

of zeros) and hence are

0, 1 ..... n-1 (in s o m e order).

Again by suitably multiplying on the left by an element order can be m a d e

0, 1 . . . . .

n-l.

!

w

It is clear that w . b . g

of N(T), the above is in B.

I

that g = b l W l . b I for s o m e

i : Z(T)\N(T)/z(T)-.-- ~

bl,b 1 ~ B; w 1 ~ N(T).

B\G/B

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

that b . w . b' = w' for b, b' E B. i . e . define Supp x = ~ ( i , j ) / x i j ~ 0 t ; x t

b

I

T h u s the m a p

Next, let w, w ' e N(T) such

= w'l.b'l.w

= (xij).

I

.

F o r a m a t r i x x,

In the above c a s e , it c a n be e a s i l y

#

c h e c k e d that Supp b ' = Supp ( w ' l b - l w ') ~ Supp ( w - l w ') (since b -1 diagonal).

This proves

But then b'

a t e l y follows that w - l w

is s u p e r diagonal and w - l w ' is m o n o m i a l . !

is d i a g o n a l i . e . g T.

is s u p e r It i m m e d i -

This p r o v e s the i n j e c t i v i t y of i.

This c o m p l e t e l y p r o v e s the l e m m a for SL n.

(ii) G = SP2 n

Let

J denote the n X n m a t r i x

I° l 0

.

1

0

L SP2n = I A C S L 2 n / A M A t

=MI.

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

Let M =

I°:I

. Define

SP2 n i s t h e set of fixed points

-

in SL2n of the a u t o m o r p h i s m

74

-

o - : SL2n

~ SL2n, given by ~" (A) -- M. (At) -1.

M -1" Now, in SL2n, the B r u h a t l e m m a can be r e f i n e d to: E v e r y e l e m e n t of BnwB is u n i q u e l y e x p r e s s i b l e in the f o r m U~-I(u, Uw

B,N,T

M. ( x - l ) t. M -1.

I

W

. b with b G B and u E U w = Uf~n w-

respectively the unipotent upper triangular, lower triangular groups).

Now ~" k e e p s

(of SL2n) i n v a r i a n t , e . g . l e t x ~ B .

Now x - l ~

j Zt.J

I~'(x) .~

u.n

B again, so let x =

Then .

0-(x) =

Then

_j. yt Jl jxtj

0

~ B again.

S i m i l a r l y f o r N and T.

T h u s if

xESP2n, x =U. nw.b in SL2n .. ($)~ ~hen x -- ~"(x) = cr'(u). ~'(nw). ~(b).

Now f r o m u n i q u e n e s s of ~, we get : f ( n w) 6 Tn w, Gr-(u) = u, a"(b) = b (rood T). T h u s the B r u h a t d e c o m p o s i t i o n in SL2n l e a d s to one in SP2n.

If we l a b e l the

b a s i s e l e m e n t s of the s p a c e on which SP2 n i s a c t i n g by i n d i c e s n , n - 1 . . . . l , -1 . . . . .

-n, then the W e y l g r o u p of SP2 n w o r k s out to be the g r o u p of t h o s e

p e r m u t a t i o n s "/l'(on 2n s y m b o l s )

such that-K(-i) = -~'(i) Vi,

i . e . the

octahedral group.

(iii) If we r e p I a c e

- J by J in M a b o v e , we obtain a p r o o f f o r SO2n, and

with a s l i g h t m o d i f i c a t i o n , one f o r SO2n+l

as well.

Hence the B r u h a t l e m m a

i s p r o v e d f o r the c l a s s i c a l g r o u p s .

Remarks.

SL

n

a c t s on IP n - I

sional vector space).

and ~ ( V )

A s i m p l e x ~ - i s an o r d e r e d s e q u e n c e

l i n e a r l y i n d e p e n d e n t p o i n t s in IPn - 1 . a simplex fpl..,

in a n a t u r a l way.

(V i s an n - d i m e n f p 1. . . . .

pnt

of

A f l a g 0 = W o C _ W l _ C . . . C W n = V and

pn 1 a r e s a i d to be i n c i d e n t if t h e r e e x i s t s a p e r m u t a t i o n -)T of

n symbols such that Wi is generated by Ip~T(1),..,pT~(i)}V l - ~ i ~ n. Now the Bruhat decomposition in form (a') for SLn can be stated geometrically as follows: Any two flags are incident with some simplex.

In fact, this is the

-75

f o r m in w h i c h t h e B r u h a t 1 e m m a w a s o r i g i n a l l y p r o v e d . to c o n s t r u c t a p u r e l y g e o m e t r i c

p r o o f of t h i s s t a t e m e n t .

The reader may wish

C h a p t e r III R e d u c t i v e and s e m i s i m p l e

algebraic grouPs,

r e g u l a r and s u b r e g u l a r

Let

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

s o l v a b l e , n o r m a l s u b g r o u p of G. clearly unique).

R

3.1. Definitions.

elements

Let

R denote the maximal,

connected,

(R e x i s t s f o r d i m e n s i o n r e a s o n s a n d is

is c a l l e d t h e r a d i c a l of G.

(1) G is s a i d to be r e d u c t i v e if R u = {1 I

not contain a non-trivial unipotent element), ¢

(2) G i s s a i d to be s e m i s i m p l e

or, equivalently,

(i.e.

R

does

if R

is a t o r u s .

9

if R = ~ 1 ) , o r , e q u i v a l e n t l y ,

if G d o e s n o t

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

Remarks.

(1) If G i s r e d u c t i v e ,

t h e c e n t r e of G.

(Since

to t h e o r e m of 2 . 7 ,

then the radical

R is n o r m a l ,

a decomposition

Examples. spaces these

G = R . G 1, w h e r e

(1) G L

n

V i on t h e u n d e r l y i n g s p a c e

G is reductive.

vidual

I

In f a c t , s i n c e

R

V.

R

R

V i s , h e n c e t h e r e is o n l y one s u c h

V i.

of it h a s e i g e n -

is normal, Since

G permutes G acts irredu-

is d i a g o n a l i z a b l e ,

G is connected, Thus

G has

(Levi decomposition).

Vo invariant. Thus

By c o r o l l a r y 2

(this is c l e a r ) and

The radical

Since R

is in

is central.)

G 1 is semisimple.

V.' s a n d h e n c e k e e p s t h e i r s u m 1 V o = V.

Hence

is s e m i s i m p l e

is a reductive group.

c i b l y on V, it f o l l o w s t h a t and

NG(R) = G (= N G ( R ) ° ) .

Z G ( R ) ° = N G ( R ) ° = G.

(2) F o r a n y ( c o n n e c t e d ) g r o u p G, G / R

R , w h i c h is a t o r u s ,

hence a torus,

G m u s t fix t h e i n d i R

c o n s i s t s of s c a l a r

-

matrices.

It is easy to see that

as in remark

(2)

GL

77

n

-

= R.SL n

In t h e n e x t 3 s e c t i o n s ,

Let

Main Theorem

T

be a maximal

Definition.

on s e m i s i m p t e

x~

t.x

(2)

R

a maximal

relative

to

W.

T. T.

groups

xM : Ga(k) ~

to

G (Ga(k)

T)

if

is the

(~ (t)r)Vt ET,

Hence

X~,

which is an unipotent

T.

X(T)

W

Clearly,

r Ek.

additively (till 3.4).

Let

= dim T.

For a semisimple

of a u t o m o r p h i s m s . under

by

of d i m e n s i o n

torus

i s c a l l e d a r o o t of G ( w i t h r e s p e c t

onto the image

b e t h e s e t of r o o t s of G .

W = N(T)/T.

of G

normalised

~

group.

such that the following conditions are satisfied:

we s h a l l w r i t e

space over

algebraic

groups.

of a l g e b r a i c

( r ) . t -1 = x

Henceforth,

Let

~ of T

is an isomorphism

group,

to that above works.)

torus in G.

A character

a d d i t i v e g r o u p of k)

(A p r o o f s i m i l a r

G will denote a connected semisimple

there exists a morphism

(1)

of G L n

(2) a b o v e .

SL n, SP2n, SOn are semisimple,

3.2.

is the decomposition

group

V =X(T)~)2Z Identify G,

X(T)

~"

Then

with

V is a vector

X ( T ) ~} 1.

it can be proved that

Let

Z(T) = T

for

is a finite group and is called the Weyl group W a c t s on X ( T )

a n d h e n c e on

V as a group

Choose a positive definite inner product on V, invariant

(Such an inner product exists,

since

W

is finite}.

Denote it by

-

( ,

).

78

-

T h e n the f o l l o w i n g r e s u l t s a r e t r u e :

Theorem.

R C V forms a root-system

w h i c h is r e d u c e d . X ~ < a n d X _ ~

a s u b g r o u p w h i c h is i s o m o r p h i c to SL 2 o r element w

.~).

The Weylgroup

Choose a basis for the root system {X,,~>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<(i,j } (r) = I + i s p o s i t i v e if j m i .

r. Eij

given by is the

Here, U-B

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

A n y s u c h p r o d u c t h a s the

n o t e q u a l to z e r o ~

(1 ,~i ,~ n).

i xi

m i n o r in the u p p e r l e f t - h a n d c o r n e r

Conversely, any such matrix may be uniquely so

f a c t o r e d , t h e f a c t o r s b e i n g p o l y n o m i a l s in t h e c o - o r d i n a t e s a n d the r e c i p r o c a l s

-79

of t h e s e m i n o r s .

( T h e s e f a c t s m a y be v e r i f i e d by i n d u c t i o n on n. )

A s h o r t d e s c r i p t i o n of r o o t - s y s t e m s dimensional vector-space.

o( to

- @C• A r o o t s y s t e m

(2)

so< (R) C_ R

system

If (

,

{s

so( w i t h r e s p e c t to a v e c t o r

in V is a f i n i t e s e t of n o n - z e r o

so( f o r e a c h @( £ R

f~-s

,~ER~

so<

(~)

such that

if ~ ,

t~CR

~

t = +- 1.

The group

i s a f i n i t e g r o u p and is c a l l e d the W e y l g r o u p

R

is a s u b s e t S of R

2(~,x).e(,

(~,~)

x EV.

s u c h t h a t (1} S i s a b a s i s

for the vector space

V and (2) a n y ~ E R = ~L

with the same sign.

A b a s i s a l w a y s e x i s t s and f o r a n y two b a s e s

such that

m

are integers S,S'

of

w(S) = S ' .

A c o m p l e t e d e s c r i p t i o n m a y be f o u n d in J - P . semisimples

gener-

is an i n t e g r a l m u l t i p l e of o( .

i s g i v e n by : s o((x) = x -

for a root system

R, 3 wEW

~ ~ 0

) i s a p o s i t i v e d e f i n i t e i n n e r p r o d u c t on V, i n v a r i a n t u n d e r

then the symmetry

Abasis

R

R i s s a i d to be r e d u c e d

W g e n e r a t e d by of R .

V be a finite

V.(ER,

For each ~, ~CR,

Aroot

Let

of V w h i c h k e e p s a h y p e r p l a n e p o i n t w i s e f i x e d and

ators along with a symmetry

(i)

is a s f o l l o w s :

A symmetry

in V i s an a u t o m o r p h i s m takes

-

complexes

( C h a p i t r e V).

found in B o u r b a k i ' s b o o k .

Serre's

b o o k : A l g e b r a de L i e

A comprehensive

treatment

S e e a l s o t h e a p p e n d i x of L e c t u r e s

m a y be

on C h e v a l l e y

groups.

3.3

Some Representation

Let G - ~

Theory.

GL(V) be a representation of G (V finitedimensional). For a

W,

- 80

-

characterAE X(T), define V) -{v£V~(t).v- ~(t).v V t C T I . c a l l e d a weight of the r e p r e s e n t a t i o n zable,

V = ~--

V~

~ if V ~

~ is

~ { 0 } . Since T is d i a g o n a l i -

(in f a c t , a d i r e c t s u m ) .

X ~ X(T) P r o p o s i t i o n 1. E v e r y c h a r a c t e r of T i s a weight f o r s o m e r e p r e s e n t a t i o n of G.

Proof.

Given

~£X(T),

choose fEk[GJ

s u c h that f / T = 0 4 .

Since T a c t s

s e m i s i m p l y on k [ G ] , t h e r e e x i s t f i ( l ~ < i ~ r ) ~ k [ G ] and 0(i, ( l ~ i ~ r ) r such t h a t f = ~ fi and t*f i ; ~ i ( t } . f i V t ~T, l~i~r. Now i=l r r T ~ ( t ) = f ( t ) = ( t ' f ) ( 1 ) = i~'=l

(t*fi)(1) = i~1 =l fi(1)'~i(t)"

6 X(T)

Hence o ( = i=l ~ fi(1)~i"

By the l i n e a r i n d e p e n d e n c e of d i s t i n c t c h a r a c t e r s , ~ = ~ . f o r s o m e 1

i.

This

Let ~

be a

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

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

Let G - ~

GL(V) be a r e p r e s e n t a t i o n of G.

w e i g h t of ~ and v E V ~ . L e t ~ be a r o o t of G (as d e f i n e d in 3 . 2 a b o v e ) . oo i Then X o ( ( r ) . v = ~--- r . v i, the s u m b e i n g f i n i t e , v i I~ V;~ + i ~ i n d e p e n d e n t of i;O r E k , and v = v. o Proof.

Since x ~ ( r ) . v

is a p o l y n o m i a l in r , with c o e f f i c i e n t s in V, we have:

oo i x ~ ( r ) , v = ~" r . v i , a f i n i t e s u m . i=0 We h a v e to show t h a t v i e V ~ +i¢( co i t . ( x ~ ( r ) , v) = ~ r . i v i. A l s o , i=0

(Vr~k).

and vo = v. L e t t ~ T .

Then

t. (xe~(r). v) = (t. xo((r), t "1) (iv)

= x~t). GO = ~o((t)

Hence

r) ~ (t). v a s i

i

,r,2%(t).v

v £V)~ i .

i=0 co (]o i i i ~ ( t ) . ~ o((t) . r . v i = ~ r . ( t v i ) . V r E k ,

i=O

i=O

t~T.

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

-

%(t).0((t)i.v i = t . v i V t ~ T ,

Lemma. o(~R Proof.

Let G --~

v=w =

Hence by definition,

. Setting r = 0, we get V = V o .

GL(V) be a r e p r e s e n t a t i o n of G.

(W~1 t w w

-

i.e. viEV~+i

( i . e . o( be a root of G . ) t.w

81

Then w

. V£Vw~ (~).

).V=Wo(.~(w-1.

(A) (t). w

V V

wo( . v ~ V w

L e t v (V~k and

t. we().v

t ET.

(A)"

As an i m m e d i a t e c o n s e q u e n c e of this l e m m a , we have:

Vg~ and Vw(A) have

the s a m e d i m e n s i o n f o r w ~ W.

P r o p o s i t i o n 3. L e t ~ E X ( T ) Proof.

and o ( ~ R .

Choose a r e p r e s e n t a t i o n

~: G

e " (This e x i s t s by p r o p o s i t i o n 1). under

Xo( and X

Then

2(A,~) (w,.<)

~- GL(V) such that ~k is a weight for

By p r o p o s i t i o n 2,

and hence under

is an i n t e g e r .

~ V, is i n v a r i a n t i ~ 77, ,~*i¢(

wo( by the t h e o r e m of 3.2.

to the group generated by X 0< and X . 0( " ) In particular,

(w~

w~. vEL

belongs ~+io(

i6zz for v ~ 0 ~ V follows that w w

()l) = A

(Such v exists.) But w (A) is of the f o r m

- 2(A,~)

.~

= ~+io(

.v ~ V w

A + i~for some (for s o m e

If we denote the lattice

f ~/

position 3 s i m p l y says:

X = X(T) C L~(R).

2(/~,~)

(A) by above l e m m a . i ~77.. H e n c e

i~ZZ).

~ 77. V~E

d

It

This proves the result.

*

by L (R), then p r o -

If L(R) denotes the lattice

g e n e r a t e d by R i . e . the set I ~" n .o(, n 6 2ZI then we have: L(R)C_XC_L~(R). m(£R Definitions. (I) G (2) G

is said to be simply connected if L*(R) = X. is said to be ad~oint if L(R) = X.

-

L e t tI~t

.....

82

-

Ofn} be a b a s i s f o r the r o o t - s y s t e m

2(~i,0(j) (~j,0(j) = ~ i j

t~( i ~ n ,

l _ < j ~ n.

R.

Define ~i E V by

It can be e a s i l y checked that { ~ t . .". .

is a b a s i s of V and L*(R) is just the lattice g e n e r a t e d by it.

~n}_

~1 . . . . . ~ n a r e

called the f u n d a m e n t a l weights with r e s p e c t to the b a s i s fO
(To p r o v e that )~.E L (R), we have to u s e the fact that a b a s i s of the dual r o o t - s y s t e m

(

R ~ of R).

7

P r o p o s i t i o n 4. If G, R , V , tj~Ai}

(~., ~<

3

, 1.< j ~

is

3

a r e as above, then the following 1 ~
conditions are equivalent: (I) G is simply connected. (2))~i~X,

l~
(3) There exist ~I . . . . . ~n in X such that wc~j(~i)= ~i" ~'ji~j 1~ i, j,
F~rther ~*i = ~ i

±~ (3).

The proof is clear. Remark. The following condition is equivalent to the above 3 conditions: If : G' ~

G is an isogeny (i. e. having a finite kernel) with G

connected

such that 71- is an isomorphism on connected unipotent subgroups, then TC itself is an isomorphism.

P r o p o s i t i o n 5. Proof.

SL n is s i m p l y connected.

We choose R, ~Mi } as given in 3.2 above.

is of the f o r m o<(i,j) with i < j ) . ti . . . . ti 1 ~ i ~ n - 1 .

~t i E x ( w )

by ~i (diag ( t t ~ . . . , t n) =

(~i is indeed c h a r a c t e r , ) It is e a s i l y checked that

w°f(J,J +1)" (~i) = ~i " ~J i ~ ( j ' j + l ) connected.

Define

(To r e c a l l , a positive r o o t

V j • Hence by p r o p o s i t i o n 4, SL n is s i m p l y

-

Remarks.

The g r o u p s

83

-

SP2n, Spin n a r e a l s o s i m p l y c o n n e c t e d .

SO2n+l, PSL n a r e a d j o i n t , while the g r o u p s

The g r o u p s

SO2n a r e n e i t h e r .

The g r o u p s

G 2, F 4, E 8 a r e s i m u l t a n e o u s l y s i m p l y c o n n e c t e d and a d j o i n t .

We now s t a t e the f u n d a m e n t a l r e s u l t on the c l a s s i f i c a t i o n of s e m i s i m p l e g r o u p s .

Theorem.

G i v e n an a b s t r a c t root s y s t e m

R and any l a t t i c e X b e t w e e n L(R)

and L*(R), t h e r e e x i s t s , up to i s o m o r p h i s m , a u n i q u e s e m i s i m p l e group o v e r a n y a l g e b r a i c a l l y c l o s e d field k s u c h that R and X a r e r e a l i z e d a s above ( r e l a t i v e to a n y m a x i m a l t o r u s . )

3.4.

Representation Theory (vontinued).

D o m i n a n t Weights.

Let G , X , R ,

V be a s in 3 . 3 .

An e l e m e n t AEV i s s a i d to be d o m i n a n t if (A , ~ ) r o o t s ~ (o(* = 2 ~ )). (,~ ,0( ;k 1 . . . . .

~

~EC =

~" a i A i , a i >" 0).

n

It follows that such ~ ' s

Choose a b a s i s

S to R.

>~ 0 f o r a l l p o s i t i v e f o r m a cone C with

(the f u n d a m e n t a l weights) a s a b a s i s . (i. e. e v e r y e l e m e n t

Define an o r d e r >~ on V by : A ~

iff A - ~

can b e c h e c k e d that >~ i s a p a r t i a l o r d e r .

is a s u m of p o s i t i v e r o o t s .

M o r e o v e r , given AE C ~ L (R),

,

t h e r e e x i s t only f i n i t e l y m a n y ( F o r : If ~ > ~ t h e n

It

$

~ s, a l s o in C Pt L (R), s u c h that 9~>~.

(A,~) - ( ~ , ~ ) = ( A + ~ t , A - ~ x ) > ~ 0 .

T h u s ~ is confined

to a bounded p a r t of s p a c e as well as to a l a t t i c e . )

We now p r o v e a b a s i c r e s u l t -

F u n d a m e n t a l T h e o r e m of R e p r e s e n t a t i o n T h e o r y . Let G be a s e m i s i m p l e

-

84

-

group, T a m a x i m a l t o r u s and B the B o r e l s u b g r o u p c o r r e s p o n d i n g to the p o s i t i v e r o o t s (with r e s p e c t to a b a s i s ) .

T h e n we have:

(a) If (7~, V) is an 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 of G, then t h e r e e x i s t s a unique l i n e D which is fixed by B, the c o r r e s p o n d i n g c h a r a c t e r A is u n i q u e l y d e t e r m i n e d and is d o m i n a n t , and a l l o t h e r weights of this r e p r e s e n t a t i o n a r e of s t r i c t l y l o w e r o r d e r ( r e l a t i v e to the above p a r t i a l o r d e r ) .

(b) Two 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 s a r e i s o m o r p h i c iff the c o r r e s p o n d i n g dominant characters,

called highest weights, are equal.

(c) G i v e n a d o m i n a n t c h a r a c t e r A on T, t h e r e e x i s t s an 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 (which is unique up to i s o m o r p h i s m b e c a s e of (b)) (71, V) such that /% the c o r r e s p o n d i n g highest weight i s

Proof.

(a) L e t

~t.

(~, V) be a n 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 .

By B . F . P . T . ,

fixes a flag.

Hence t h e r e e x i s t s a line D which is kept i n v a r i a n t by B.

v E D , v ~ 0.

Let ;% be the c o r r e s p o n d i n g c h a r a c t e r on T. ( i . e . t . v

Since V is i r r e d u c i b l e , it follows that V is s p a n n e d by G . v . cell' U - B U - . v (as

is dense in G, it follows that V B a c t s a s s c a l a r s on v).

By p r o p o s i t i o n 2,

x

~((C~). v = v +

Since the 'big

is spanned by U ' B . v

C i~ . v i with v i ( V

Choose

= A(t).v~t£T).

i.e. by

C o n s i d e r a n e l e m e n t x_ ( C ~ ) ~

B

of U ' .

_io ( .

It now

i>~i

-

follows that f o r a n y u ' E U ' ,

u-.v = v+w

where

V = k . v . (~)( ~0 n < . ~ , Note that V A = k . v = D.

w ~ ~)V, .

n

~ 0).

Thus, -

(*)

If D' is a n y o t h e r line fixed by B, then a r g u i n g in

the s a m e way a s h e f o r e , we get a d e c o m p o s i t i o n (*) of V, with s o m e weight ~' i n s t e a d of ~ . It now follows that ~ k ' ~ k

and ~ > ~ ' .

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

= <~G.v>

= ~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

+~<,~

+x

-

(**).

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

G = SL n.

Let ~.£X(T)

be a s d e f i n e d in p r o -

1

p o s i t i o n 4 of 3 . 3 .

We can r e a l i z e

~i

in A i (kn) with the u s u a l a c t i o n

-

g . ( v l A . . . A v i) = g v l A . . . then e l A . . .

Agv i.

87

-

If {e I . . . . .

en}

^ e i i s a h i g h e s t weight v e c t o r and the o t h e r s t a n d a r d b a s i s

v e c t o r s of Al(kn) a r e o b t a i n e d by p e r m u t a t i o n s . Symm ~.

is the s t a n d a r d b a s i s of kn ,

on T and hence X A (g) is j u s t the

1

Hence, in this c a s e , X A i =

ith e l e m e n t a r y s y m m e t r i c p o l y -

• 1

n o m i a l in the e i g e n v a l u e s of g.

(This is c l e a r if g E T

and hence a l s o if g £Gr

arbitrary. )

D e f i n i t i o n . f E k [ G ] i s s a i d to be a c l a s s - f u n c t i o n if it i s c o n s t a n t on the c o n j u g a c y classes.

The set of a l l c l a s s f u n c t i o n s is denoted by C [ G ] . e . g . Any c h a r a c t e r

X ~ , c o r r e s p o n d i n g to a r e p r e s e n t a t i o n ~ , is a c l a s s - f u n c t i o n (being a t r a c e function).

T h e o r e m 2.

Let G be a s e m i s i m p l e a l g e b r a i c group, T, a m a x i m a l t o r u s and

W, the c o r r e s p o n d i n g Weyl g r o u p . t

(a) The r e s t r i c t i o n to T of the X ~ s (for d o m i n a n t c h a r a c t e r s ~ ) f o r m a

linear basis of k [9?] w • w h e r e a s the X l s t h e m s e l v e s f o r m a l i n e a r b a s i s of C[G]. ( ? (b) If G is simply connected and l~i~

then {X ;ki/TI

are the fundamental weights, l~i.~n W (respectively{X;~il 1.i.~n ) freely generate k IT]

iSi~n (respectively C6G3) as k-algebra. P r o o f . We f i r s t p r o v e the s t a t e m e n t s c o n c e r n i n g T and then deduce the c o r r e s p o n d i n g r e s u l t s f o r G.

(a) Since S y m m ~ and X A

are interrelated, by the equations (*). (**)above,

iris sufficientto prove that ISymm

isabasis of kiT3 w

Let f ¢ k [T 3 w

Since T is diagonalizable, the characters of T span k [9?], So let

-

f =

~-"

Cx.X.

Hencefor

88

-

wCW, f=wf

= ~" C x . W 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<), Co(E k.

and

Choose

e v e r y s i m p l e r o o t v( .

-

XsET , xu EU.

t ET

Now x u is o f t h e f o r m :

s u c h t h a t ¢< (t) = C, p r e g i v e n in k*, f o r

(Such t e x i s t s by p r o p o s i t i o n 7 of 2 . 6 ) .

Now t x t "1 = t. Xs. X u . t ' l

= X s . t Xut-1 : x . - [ 1 - xa ( C htc< oC¢.-) . w h e r e

ht 0(=

s ¢~>0

~ni'

¢~ = ~-" ni~i' (¢~i..... O
x -i . clS

=

el

s

x-i s

S.

Thus, "[I- x((cht~co() E ~>0 C ~ k . Take any f E k [ G ] such

This is true for every

that f(cl x -I S) = 0. The m a p s

map, which is zero for every In particular, f(e) = 0, also.

C ",,-~f(-[]- x ¢(cht°(co()) is a polynomial ~>0 C E k $. This m e a n s that it is identically zero. It now follows that e Ecl(x "I S) = x "I cl S. s

Hence

3.5.

x s E cl S.

Let

This proves the l e m m a and hence the corollary.

Regular elements.

algebraic

s

In t h i s s e c t i o n

G will denote a connected,

reductive

g r o u p w h i c h m a y o r m a y n o t be s e m i s i m p l e .

G be a (reductive) group.

w h i c h i s c e n t r a l in G.

Let

R

We also have, G = R.S,

group uniquely determined

as the commutator

t h e r o o t s y s t e m of G ( v i z . t h a t of S). R.T ' where

be i t s r a d i c a l .

Then

where

R

is a torus

S is the semisimple

subgroup.

One c a n n o w t a l k of

A m a x i m a l t o r u s of G is of t h e f o r m

T ' i s a m a x i m a l t o r u s of S.

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

ZG(RT') = R. T ' .

N o w , t h e r e s u l t s p r o v e d in t h e p r e v i o u s s e c t i o n s c a n b e u s e d h e r e (with s l i g h t modifications,

if r e q u i r e d ) .

Definition. x EG

i s c a l l e d a r e g u l a r e l e m e n t if ZG(X ) h a s t h e m i n i m u m d i -

mension (among the centralizers

of e l e m e n t s of G)

t h e c o n j u g a c y c l a s s of x, h a s t h e m a x i m u m

or, equivalently,

dimension.

if C(x),

-

Clearly,

94

-

regular elements exist for dimension reasons.

Proposition

1.

T h e m i n i m a l d i m e n s i o n a b o v e is j u s t

r , t h e r a n k of G i . e .

t h e d i m e n s i o n of a m a x i m a l t o r u s .

Proof.

Fix a maximal torus

T.

Choose

t gT

s u c h t h a t Z G ( T ) = ZG(t)(= T ) .

{Such an e l e m e n t e x i s t s by c o r o l l a r y to p r o p o s i t i o n o f 2 . 7 ) . (1) T h e n

d i m T -- r = d i m ZG(t) >~ m i n i m a l d i m . For any

sition.)

aBoretsubgroup.

Let

torus and dim

xEB,

x

6:G, d i m

(2) N e x t we p r o v e :

ZG(x) ~ r . Wehave,

U i s t h e u n i p o t e n t p a r t of B.

B / [ B , B~ ~ d i m T = r .

CB(X) in

B >~r, s i n c e

Since

(This proves the propo-

B =T.U,

T

isamaximal

( B , B ) C_ U, it f o l l o w s t h a t

Now, d i m ZG(X) >~ d i m ZB(X ) = c o d i m e n s i o n of

C(x).x'lC_

[B, B J .

This proves the required result.

T o p r o v e (2), we g i v e an a l t e r n a t e m e t h o d : D e f i n e

xG/x,y

lie

~G~G,

given

S2 = f(x,y)6G %.

in a c o m m o n t o r u s ~ Then by

.

Fix a torus

T.

S 2 is t h e i m a g e of G ~ T x T

@(x,t,t')=(xtx'l,xt'x'l).

Hence

of S 2 in G k G .

Then

Pl:GXG

Since semisimple

~

G.

under the map

S2 is irreducible.

G 2 is a l s o i r r e d u c i b l e .

x is s e m i s i m p l e ,

fEk[G],

f(x.y) = f(y.x) V(x,Y) E S

Since words,

for

2

such that x£G,

at

d i m f i b r e at

Consider the projection G and

(x, 1)E S 2

For any function

x.y =y.x).

Now t h e f u n c t i o n

i s z e r o on S 2 a n d h e n c e on G 2.

x . y - - y . x V ( x , y ) 6 G 2.

x i s c o n t a i n e d in

ZG(X).

t = d i m G 2 - d i m G.

d i m Zg(X) > ~ 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:

(a<(t).c) = 0<x-(c)~c ~ k, it follows that Hence ~ (t) ~ 1 f o r e v e r y root o(

Z G ( t ) ° / ~ U.-B C_T.

Let x E Z G ( t ) ° ~

U:B.

Now x is of the f o r m : "17-~Y0x . ~ ( c ~ ) . t ' . ' ~ - ~ > 0 xa~(d ), with c ~ , d 0 ( E k , Since, x EZG(t), x = txt - I = t( ~ - x o<(c~)).t'. ('71-- x

#

t'~T.

(do()).t'l

='~- x (0((t)-Ic).t', " ~ X (¢<(t).d). a~p0 - 0~ o~;*0 ~ Hence by uniqueness, o~(t)'l.cee = c and o((t).da( = do< ~ ] ~ 0 . Since

o((t} ~ 1 for anyo~ER, it follows that co(= d ~ = 0 V ~

. Thus x = t'~ T. O

N o w U.=B is open in G (by theorem of 3.2). Hence ZG(t) /~ U: B is open in ZG(t)°. Since T C ZG(t)°f~ U : B and ZG(t)° irreducible, it follows that ZG(t)°f~ U.'B --T is the whole of ZG(t)°. This proves (b)o (b) ~

(a) is obvious.

This proves the proposition completely.

We now give a c o m p l e t e p i c t u r e of ZG(t) f o r a s e m i s i m p l e e l e m e n t t E G.

-

P r o p o s i t i o n 4.

98

-

Let t ~T, a maximal torus.

with respect to T.

Then

L e t R be the r o o t s y s t e m of G

ZG(t) is the group G 1 generated by T, by those !

XI s

such that

o((t) = 1 and by t h o s e

n s (w ~ W - - N ( T ) / T )

s u c h that w(t) = t

W

(i.e.

nw tn'lw = t).

The identity component

ZG(t) °

is g e n e r a t e d by T and the

?

X

s

(such t h a t D((t) = 1) and is r e d u c t i v e .

Proof.

By B r u h a t l e m m a , G =

Uw : U f l n suchthat

W

. U ' . n "1.

It is known that

W

o(>0

and

[J Bn B, a d i s j o i n t union. n w 6 N(T) w

w-l~<0.

U

W

(Notethat

Define

is the g r o u p g e n e r a t e d by X ( s n w . X o ( n -wl = Xw(~ .

~ e m a k e u s e of

a l e m m a w h i c h w i l l be v e r y u s e f u l l a t e r on.

Lemma.

The map

~ : U w X B - - - ~ BnwB g i v e n by

an i s o m o r p h i s m of v a r i e t i e s . uniquely written as

P r o o f of t h e l e m m a . Further,

W

.b

with u E U w , b E B .

Since n

W

.T.n

U = -~"

= T, it f o l l o w s that Hence, whenever ~0,

It now f o l l o w s that s u c h

( R e c a l l that

is in f a c t

H e n c e , in p a r t i c u l a r , any e l e m e n t of BnwB is

X <.n w =nw. Xw_l(~).

C nw.U. in B.

u.n

~(u,b) = u . n w . b

X

X'~

s

Bn w . B = U . n w . B . w ' l ( o < ) ;> 0, X o < . n w

can be ' p a s s e d ' o v e r n w and a b s o r b e d

f o r e v e r y o r d e r of p o s i t i v e r o o t s . )

Thus

>0

Bn B = U . n w . B = U w . n w . B . W

=

-1 nw'U - "nw

for some

T h i s p r o v e s the s u r j e c t i v i t y of ~.

u- E U-

"

Thus

~(u2,b2), u i £ U w , b i E B, i = 1 , 2 . i . e .

nwU

h 2 for s o m e

( u - , b ) -- u ' . b sequently

u;, u2

.

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

u I = u 2.

Hence

u . n w = n w .u .

Now, l e t

U l n w . b 1 = u 2 . n w b 2.

Thus u ; b I it f o l l o w s that

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

.b 2

W

W

u EUw,U

~(Ul,b 1) =

H e n c e nw.U [ . b 1 =

S nce

.B,

u i = u~ and b 1 ° b 2.

It is e a s y to s e e that

an i s o m o r p h i s m of v a r i e t i e s s i n c e the n a t u r a l map: U ' X B - - ~ U - B / G = ~ U . n . B , a d i s j o i n t union. nw

For

Con-

~ is in f a c t is.

Thus,

-

99

-

C o m i n g to t h e p r o o f of t h e p r o p o s i t i o n , we o b s e r v e t h a t the g r o u p G 1 i s c o n t a i n e d in Z G ( t ) . suchthat x =u.n

Now l e t x ~ Z G ( t )

X~Uw.nw.B. .b.

be arbitrary.

Hence there exists

H e n c e t h e r e e x i s t u n i q u e u ~ U w, b ~ B

Since X~ZG(t),

x = t x t -1

suchthat

=tut'l.tn

W

nw

t - l . t h t -1. W

Now t n wt -1 ' = n w . n w" l t n w t ' l

=nw't

' . t -I

, t' = n w1 t n w E T ,

A l s o , u is of the

~(%<).

form: "TV oc>O

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<(t) = 1 w h e n e v e r

H e n c e b y u n i q u e n e s s , rut -1 = u , t ' . b t -1 = b. )T

c0( ~ 0.

Further,

t ,,. u o = t ,. t ,,. u .t-I = t',t".t'l(tuot'l). O

let b =t

. u o , u o E U.

Again, by uniqueness, t

Then

= t and

u ° = tUot - 1 " d o ( ~ 0.

W r i t i n g u ° = - [ i - x ~ (do<)" o n e s e e s t h a t oK(t) = 1 w h e n e v e r Q(>0 A l s o , t ' = t g i v e s n w . t . n w 1 = t . It i s n o w c l e a r t h a t x ~ G 1. H e n c e

ZG(t ) = G 1. Further,

whenever

o((t) = 1, ( w ' l ~ ) ( t ) = ~ ( n w . t . nw1) = o((t) = 1.

a n y e l e m e n t of G 1 i s of t h e f o r m by T and X~

g2" nw" w h e r e

s a l o n e a n d n w s u c h that

s u b g r o u p of f i n i t e i n d e x i n G 1. immediately follows that

nw.t,n

Hence

g2 ~ G 2 ' t h e g r o u p g e n e r a t e d w

= t.

Clearly, G 2 is a

S i n c e G 2 is c l o s e d a n d c o n n e c t e d , it

G 2 = G o1 = ZG(t)o. !

F o r t h e r e d u c t i v i t y , one m a y s e e ' S e m i n a i r e C h e v a l l e y , Vol. 2 .

T h e r e i t is

I

a l s o s h o w n t h a t t h e ~ s , f o r w h i c h o((t) = 1, f o r m a r o o t s y s t e m f o r G 2 a n d T

that

T a n d t h e Xo¢ s w i t h ~ >0, ~ ( t ) = 1 g e n e r a t e a B o r e l s u b g r o u p (of G2).

We o b s e r v e t h a t

Remark.

ZG(t) ° a n d G h a v e the s a m e r a n k s i n c e

t is regular

iff ZG(t)

O

= T.

of (a) a n d (e) i n t h e e a r l i e r p r o p o s i t i o n .

T C_ Z G ( t ) ° .

This clearly shows the equivalence

-

Corollary.

100

-

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 in a r e d u c t i v e g r o u p f o r m a n open

set whose c o m p l e m e n t has c o d i m e n s i o n 1.

Proof.

Choose a m a x i m a l t o r u s

T.

Let R be the r o o t - s u s t e m r e l a t i v e to T.

Let fo = - ~ - (C<-l). Since w EW p e r m u t e s R, it f o l l o w s t h a t f o E k [ T ] w . o<ER Now by t h e o r e m 2 of 3 . 4 , f e x t e n d s to a unique c l a s s f u n c t i o n f £ C [ G ] . We o

claim: S -- fx/f(x

p 01 is the set of regular semisimple e l e m e n t s

Clearly

f r o m (e) of p r o p o s i t i o n 3, S c o n t a i n s the s e t of a l l 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 . Let x £S.

We can a s s u m e

x £ B (= T . U

in u s u a l n o t a t i o n ) .

g a t i n g by a s u i t a b l e e I e m e n t of B, one can a s s u m e the J o r d a n d e c o m p o s i t i o n of x.

Since x ~ ZG(s) , x

x = s.u,s~T,

Since f £ C [G] , f(x) = f(s).

to t h e o r e m 2 of 3 . 4 . ) i . e . ~ ( ~ - l ) ( s ) ~<ER p r o p o s i t i o n 3, s is r e g u l a r . Hence

A g a i n , by conj u u~U

is

(See c o r o l l a r y 1

~ 0, i . e . o((s) ~ 1 k/~ER.

Hence by

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

is s e m i s i I n p l e and h e n c e r e g u l a r (since u = 1).

Thus S

is an open d e n s e s e t of G, whose c o m p l e m e n t has c o d i m e n s i o n 1 s i n c e it i s defined by a s i n g l e equation.

(Check the l a s t i m p l i c a t i o n . )

P r o p o s i t i o n 5. x E G is r e g u l a r iff x u £ Z G(Xs) o is r e g u l a r .

Proof.

Let

ZG(Xs )° = G 1.

Then ZGl(Xu) = Z G ( X ) ~ G 1 ~ ZG(X) ° ( s i n c e o

ZG(X ) C_ ZG(Xs)), Also, ZGI(Xu )° C_ ZG(X ) /'%G 1 ZG(X) ° .

=Z

o

G(x) . Hence

ZGI(X u)

o

=

F u r t h e r , w h e n e v e r x E T , a m a x i m a l t o r u s , T ~ZG~(Xu )O ( c G1). t

Hence T is a m a x i m a l t o r u s in G 1 as well.

The p r o p o s i t i o n now follows

immediately.

3.6 Unipotent classes, To continue, we must know'that regular unipotent elements exist. In case of characteristic 0, this was proved by Dynkin and Kostant,

-

using Lie algebras.

101

-

For arbitrary characteristic,

in P u b l i . Math. I . H . E . S .

# 25 (1965) (§4).

t h i s w a s p r o v e d by the a u t h o r

U s i n g e x p l i c i t c a l c u l a t i o n in the

Lie algebra, Springer proved this result for 'almost all' characteristics.

Here,

we s h a l l give an a c c o u n t of R i c h a r d s o n ' s m e t h o d ; he p r o v e d that u n d e r s o m e c o n d i t i o n s , a r e d u c t i v e g r o u p G c o n t a i n s only 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 . ( F r o m t h i s the e x i s t a n c e of r e g u l a r u n i p o t e n t e l e m e n t s w i l l follow).

We s t a r t with s o m e p r e l i m i n a r i e s :

L i e a l g e b r a of an a l g e b r a i c g r o u p G.

We r e c a l l that the t a n g e n t s p a c e a t a p o i n t

p• G i s d e f i n e d to be t h e s e t of a l l k - a l g e b r a h o m o m o r p h i s m s into the a l g e b r a of d u a l n u m b e r s

k [ C ~ E 2 = 0).

In s h o r t , ( W G ) p = { r : k [ G ] - - * - k / r

i s k - l i n e a r and r(f. g) = r(f). g(p) + f(p). r(g), f, g C k [ G J t . L i e a l g e b r a of G, to be the t a n g e n t s p a c e

ep+ e . l " of k[G]

We d e f i n e ~ , the

(TG) 1 of G at the i d e n t i f y e . ~ can

b e c a n o n i c a l l y given the s t r u c t u r e of a ' L i e a l g e b r a ' (i. e. a b r a c k e t o p e r a t i o n with s o m e p r o p e r t i e s ) .

F o r u s , ~ w i l l be r e g a r d e d a s a k - l i n e a r s p a c e only.

Since k [GJ i s f i n i t e l y g e n e r a t e d , it f o l l o w s that oj i s f i n i t e d i m e n s i o n a l . fl .....

fn g e n e r a t e

k [G]

as k-algebra.

(r) = (r(fl)~.,~r(fn) ). Then A s an e x a m p l e , c o n s i d e r [1],

D = determinant.

)~ij = T(Xij) E k.

~ : ~ - - - - ~ k n by

~ is k-linear aud injecfive.)

G = G L n. For

Define

(Let

We h a v e ,

k[G 3 = k[Xll,X12 .....

TC(TG)I, definea matrix

Xnn~ -

T--(~kij) g i v e n b y

It i s e a s y to s e e that t h i s s e t s up a o n e - o n e c o r r e s p o n d e n c e

between

(TG) 1 and Mn(k) , the a l g e b r a of a l l n x n m a t r i c e s .

algebra

~1 n of G L n i s i d e n t i f i e d with Mn(k).

Hence the L i e

Now, t h e L i e a l g e b r a of a p r i n c i p a l open s u b g r o u p G 1 of G can be i d e n t i f i e d with that of G i t s e l f .

T h e L i e a l g e b r a of a c l o s e d s u b g r o u p i s a s u b s p a c e of

-

t h e L i e a l g e b r a of G.

A surjective morphism

G,G'

braic groups

-

We h a v e a n a t u r a l a c t i o n of G on ~ ,

in a n y r e p r e s e n t a t i o n , c o m i n g f r o m

Isogenies.

102

(Adx)(X) = xXx -1

x(1 + ~.X)x "1 = 1 + ~ . x X x "1.

f: G ~

G'

of c o n n e c t e d , r e d u c t i v e a l g e -

is said to be an i s o g e n y if it h a s a f i n i t e k e r n e l .

It is e a s y to s e e that in this c a s e , G is s e m i s i m p l e iff G'

is so.

Further,

we

h a v e t h e following: (a) G i v e n a s e m i s i m p l e g r o u p G ' , t h e r e e x i s t s a s i m p l y c o n n e c t e d ( s e m i s i m p l e ) a l g e b r a i c group

G and an i s o g e n y f:G ~

G'.

(b) T h e k e r n e l of any i s o g e n y f i s d i s c r e t e (finite, in f~ct) and n o r m a l . S i n c e G i s c o n n e c t e d , t h e k e r n e l of f i s c e n t r a l . (c) f s e t s up a b i j e c t i n n of c o n j u g a c y c l a s s e s of u n i p o t e n t e l e m e n t s . because

(This is

the k e r n e l of f c o n s i s t s of s e m i s i m p l e e l e m e n t s . )

We now p r o v e an i m p o r t a n t r e s u l t :

Theorem (Richardson).

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

be i t s L i e a l g e b r a ( ~ ~ of

~ln).

A s s u m e that t h e r e e x i s t s a s u b s p a c e

~1 n s u c h that

(i)

no

}

(2) ~tt

is stable under Ad(G)

Then every conjugacy class of GLn meets G in finitely many conjugacy classes of G.

Proof.

L e t G 1 = GL n and ~1

Consider

C 1 f~ G.

gi n .

Let

C 1 be a c o n j u g a c y c l a s s of G 1.

E v e r y c o n j u g a c y c l a s s of CII3 G, b e i n g i r r e d u c i b l e , is

c o n t a i n e d in s o m e i r r e d u c i b l e c o m p o n e n t of Clt~ G and c l e a r l y e a c h c o m p o n e n t

-

is a union of c l a s s e s .

Since C 1/% G

103

-

has finitely m a n y c o m p o n e n t s , the t h e o r e m

is p r o v e d if we p r o v e that each c o m p o n e n t Z of C 1/% G is a single c l a s s of G.

So let Z be s u c h a component.

Let g EC. Claim:

L e t C be a c l a s s of G such that C C Z .

C o n s i d e r the m a p f : G 1 - - ~ C l g ' l

(a)

given by f(x) = x g x - l g -1.

(df)1 = 1 - Adg.

(b) (dr)1 is surjective. W e have, (df)1 : (GI) 1

~ (Clg-l)l.

(a) can be verified in a straight forward

m a n n e r (One m a y use: (I + ~ X ) .

g.(l + E x ) - l . g -I -- 1 + 6 ( I - (Adg)(~)).

Also, dim (df)l ( ~i ) = dim = dim

~i " dim ker (df)1 ¢~i " dim Z

l(g) by (a).

Since ZGI(g) is an open s u b s e t of Z ~ l(g), c o n s i s t i n g of the invertible e l e m e n t s , it has the s a m e dimension. Hence dim ( d f ) l ( ~ l ) = dim G 1 - dim ZGI(g) = dim C 1 = dim C l g

-1 = d i m T ( C 1g.1)l.

F r o m this (b) follows. Hence

T(Cg "1} 1

C._ T ( Z g - 1 ) I C T ( C l g ' l ) l f %T(G)I = (1-Adg)(a~l)/~ ~ (by c l a i m (b) above) = (1-Adg) (~)

by (*)

C T ( C g - 1 ) l " s i n c e f(G) = Cg -1 It follows that T ( C g ' I ) 1 = T ( Z g ' I ) I . Now f o r a v a r i e t y V, d i m (TV) all v ~ V

dim Z.

>~ 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¢<(c~), t h e n

G = To.S , where

co< ~ 0 f o r e v e r y s i m p l e r o o t ~ .

Proof.

Let

T o is the radicatand

group.

T o is a torus and it is central.

S = [G,G],

a semisimple

T O is contained in every Borel subgroup.

-

Iii

-

!

If y = t o . Y in S.

with t o e T o , y ' & S , then y i s r e g u l a r in G iff y '

is regular

It i s now c l e a r t h a t in p r o v i n g the e q u i v a l e n c e of the a b o v e s t a t e m e n t s ,

one m a y a s s u m e

G i t s e l f to b e s e m i s i m p l e .

(b) ~

(c) i s o b v i o u s .

(c) ~

(d).

r o o t o( o.

L e t x = - ] ' [ - x (c~). if p o s s i b l e , l e t c = 0 for some simple o~ >'0 °Co Now { c<>0, ~ o(° t O {- e~° 1 i s a g a i n a s e t ° f p ° s i t i v e r ° ° t s (with

r e s p e c t to s o m e

ordering,

Hence X.~ ° normalizes

s i n c e wc< ° p e r m u t e s p o s i t i v e r o o t s Uc( ° =-[~-

X

o~>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 .

(Xc<(t), x•(u)) = -~-- X r ( P r ( t , u ) ) , w h e r e P r ( t , u ) r>0 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.
-

above formulae show: U'~ EU,U] abelian.

A l s o , we n o t e t h a t

C o m i n g b a c k to the c l a i m , c~

112

-

, hence U' is normal in U and U/U'

is

c o d i m U (in U) = r .

"TT-

xo = I| xo((c~) and x = ] | x (d) with ~0 ,~ 0 0, d ~ ~ 0 f o r e v e r y s i m p l e r o o t ~ . By p r o p o s i t i o n 7 of 2 . 6 , c h o o s e

t~T

such that ~(t).d~

let

= c~

for all simple roots ~ .

Hence

x ' = txt "1 = 7 ] - x o ( ( ~ ( t ) . d ), ~>0 , Now i t i s e n o u g h to s h o w t h a t x and

x

are conjugate.

O

,

Clearly

I

Since C (Xo).X: I C_ U', dim Cu(Xo). xo-l-'~ dim U = dim U - r.

x x

-I

O

'

~ U .

Also,

dim ZU(Xo) ~ dim ZG(Xo) = r, since xo is regular. Thus dim Cu(Xo) ~ d i m U - r .

Thus

d i m Cu(Xo) . xo 1 = d i m U ' .

A l s o , C u ( x o)

i s c l o s e d , b e i n g a c l a s s in u n i p o t e n t g r o u p (by c o r o l l a r y to p r o p o s i t i o n of 2 . 5 ) . Since

U'

is irreducible

and

-1 CU(Xo). x °

as U' , it follows that Cu(Xo) "x-lo = U' '

We h a v e : 1

x = UXoU

(a) ~

X'xol e U . -1

.

Co).

Hence

~ ueU

i s c l o s e d and h a s t h e s a m e d i m e n s i o n

i

such that

UXoU

-i

-1

x°

= X'Xol

or

This proves the claim.

Let

x be regular.

c o n d i t i o n (d).

Hence

of 3 . 6 , x and

x

O

Choose a

xo is regular

are conjugate.

XoE U s u c h t h a t

(as p r o v e d e a r l i e r ) .

xo satisfies the

However, by theorem

A l s o , t h e p r o o f of t h i s t h e o r e m

x ° i s c o n t a i n e d in a u n i q u e B o r e l s u b g r o u p .

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

3 shows that

x is also contained

in a u n i q u e B o r e l s u b g r o u p .

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 .

Corollary. (For."

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

x E U, t h e n

ZG(X)° ~_ U a n d h e n c e i s u n i p o t e n t .

In t h e p r o o f of (d) = ~ (a), t h e i n e q u a l i t i e s a l l b e c o m e e q u a l i t i e s a n d h e n c e

d i m ZG(Xo )° = d i m Z u ( x o )°. )

3

-

113

-

We a g a i n c o n s i d e r U i a s d e f i n e d e a r l i e r . where

(<Xxi, X

L e t P i = (T. < X , e i, X . ~ . > ). U i, 1 < > ) i s the g r o u p g e n e r a t e d b y X ~ . I and X . ~ ¢ i . Both T

and <X~I,. X . ~ i >

normalize

U.1 {because of the c o m m u t a t i o n f o r m u l a e ) .

Here, Pi is a rank 1 - parabolic subgroup,

U i i s i t s u n i p o t e n t r a d i c a l and

T.<X~.,

X ~> i s a L e v i c o m p o n e n t ~f P i o 1 1 NOW, d i m T . < X ~ i , X_~i"2. = r + 2 . (The o v e r l a p of T a n d ~ X ~ i , X

h a s d i m e n s i o n 1) .

(i>

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 <~ d i m G - (,'+2). i T h i s p r o v e s the u n p r o v e d p a r t of s t a t e m e n t (b) of t h e o r e m 3. I n c i d e n t l y , e a c h V i d e f i n e d a b o v e is c l o s e d , s i n c e G / P i i s c o m p l e t e .

This

s h o w s once a g a i n t h a t the s e t of 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 i s c l o s e d .

T h i s a r g u m e n t can b e c a r r i e d out in c a s e of a r b i t r a r y i r r e g u l a r e l e m e n t s of G. We s t a r t with a l e m m a :

L e m m a 1. F o r e a c h

o
B i = T i . U i ( B i h a s c o d i m e n s i o n 2 in B).

L e t Ui, P.1 be a s b e f o r e ,

T h e n e v e r y i r r e g u l a r e l e m e n t of G

-

114

i s c o n t a i n e d in a c o n j u g a t e of s o m e

For the proof,

From

see I.H.E.S.

-

B i.

C o n v e r s e is a l s o t r u e .

# 25 (§ 5).

this, w e get: The irregular elements of G

f o r m a closed subset of G,

each of whose components has codimension 3. In particular, regular elements f o r m a dense open subset of G.

T h e proof of this statement is like that just

given, but with B i inplace of U i. W e have c o d i m p i

Lemma

2.

Among the irregular

Proof.

For each

i, s e t

elements,

Ji = ker ~i

B. = 3. i

the s e m i s i m p l e

- U k e r c~

.

ones are dense.

Consider

J i . Ui, w h i c h

I

is open in B..

O u r claim is that J..U.

1

i

consists of semisimple elements. So

1

let x = t.u E Ji'Ui" t E Ji o u E U i. N o w by conjugating b y a b o of Bi, w e h a v e : easy to see that Since

x ' --b ° xb o"I = t.u' with u' E U i and

u' = 1 so that

x'

is semisimple.

the s e t of i r r e g u l a r

elements.

W e now c h a r a c t e r i z e

Theorem

2.

Let

t . u ' - - u .'t .

Hence

(J J .1U 1 is d e n s e in L ] B i , (J U g. J i . U i . g - 1 1 g

suitable element Itisnow

x is also semisimple.

i s d e n s e in

U gBig g,i

-1

= Ir,

This proves the lemma.

the r e g u l a r e l e m e n t s

of G

in a n o t h e r w a y .

x b e a n y e l e m e n t of G (G r e d u c t i v e ) .

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

x is f i n i t e .

Then

x i s r e g u l a r iff

( T h i s n u m b e r is 1 if

x i s u n i p o t e n t and e q u a l to [WI if x s e m i s i m p l e ) .

F o r t h e p r o o f , we r e q u i r e a l e m m a .

Lemma

3.

Let

t E G be s e m i s i m p t e .

s u b g r o u p of G, c o n t a i n i n g

Set

GO = ZG(t)°.

Then each Borel

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

-

Conversely,

each Borel

115

s u b g r o u p of G

-

contains

O

t and is contained in finitely

m a n y Borel subgroups of G.

Proof.

Let

of p r o p o s i t i o n X~ s

4 of 3 . 5 ,

we s e e t h a t

t

Then from the proof

B/~G ° (the group generated

B.

Conversely,

let

s i n c e t i s i n t h e c e n t r e of G

c o n t a i n e d in

B o.

by

T

and all

g r o u p s of G

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

Iwl

can thus contain

Let

O"

It i s a l s o m a x i m a l

it is contained in exactly

in G

t i o n of x.

B.

the lemma.

This proves

let

At most

As such

IW~Borel

sub-

x = t. u b e t h e J o r d a n d e c o m p o s i -

subgroup 5 of 3 . 5 ,

It c o n -

t o r u s of G o ,

since rank G = rank G o .

B o.

Also, by proposition

ZG(t) ° = G o say.

be a maximal

O

of G .

It follows that for aBorel

u are in

T

Borel subgroups

C o m i n g b a c k t o t h e p r o o f of t h e t h e o r e m ,

in

t.

s u c h t h a t ~ > 0 , ~ ( t ) = 1) i s a B o r e l s u b g r o u p of G o , c l e a r l y t h e u n i q u e

o n e c o n t a i n e d in tains

G, c o n t a i n i n g

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

B

of G , x E B

x is regular

The proof is now immediate

iffboth

in G t f f u

t and is regular

from the above lemma

and the

theorem.

Remark.

If W o d e n o t e s t h e W e y l g r o u p of G o , t h e n a c l o s e r a n a l y s i s

that the finite number with

in the lemma

IWl /IWo I (and accordingly

in the theorem

t = Xs).

Theorem

3.

If G i s a r e d u c t i v e

b i j e c t i o n of t h e r e g u l a r

Pronf.

is

shows

(i)

Surjectivity,

s i t i o n 4 of 3 . 5 , which is regular,

ZG(S) °

classes

Let

group,

onto the semisimple

s

be a semisimple

is reductive.

unipotent.

then the map

Now

classes.

element.

Hence by theorem

x = s.u

x ~,--~x s yields a

Then by propo-

3 a b o v e , "~ u ~ Z G ( S ) °

is the Jordan decomposition

to

x

-

116

-

a n d b y p r o p o s i t i o n 5 of 3 . 5 , x i s r e g u l a r s i n c e x

= s.

u i s s o in

ZG(S)°.

Also,

This proves the surjectivity.

S

(2) I n j e c t i v i t y . conjugate.

Let

x,y

be regular elements

By c o n j u g a t i n g

So l e t Now

ZG(S) ° , it f o l l o w s t h a t

g. Xu.g gxg

w i t h o u t l o s s of g e n e r a l i t y ,

ZG(S)°, which is reductive.

e l e m e n t s in

-I

-1

-- Yu" g ~ Z G ( S )

= g. x s. Xu.g

-i

Xs and

x by a s u i t a b l e e l e m e n t (in f a c t by

g X s g ' l ~ y s ), one m a y a s s u m e , Consider

such that

o

Since

that

Ys a r e

g, w h e r e x s = Ys = s.

Xu, Yu a r e r e g u l a r ,

Xu,Y u a r e c o n j u g a t e in

unipotent

ZG(S) ° ( T h e o r e m

3).

'

= g. sxffg

-i

= s° g,Xug

-i

= s. Yu -- y"

This proves the injectivity.

3.8.

Regular elements

in s i m p l y c o n n e c t e d ,

In t h i s s e c t i o n , G i s a s s u m e d

semisimple

groups.

to b e a s i m p l y c o n n e c t e d s e m i s i m p l e

algebraic

group.

Let

T

be a f i x e d m a x i m a l t o r u s and

R be t h e r o o t s y s t e m r e l a t i v e to it.

recall that the fundamental weights I~il

1 _~ i ~ n

characters

(by d e f i n i t i o n of s i m p l y c o n n e c t e d n e s s ) .

characters

on G b y X i (X i = X X . ) .

in t h i s c a s e ,

We

a r e in f a c t

We d e n o t e t h e c o r r e s p o n d i n g

Consider the map

p : G-->/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 <~ i.
Ei, i+llti ~k]"

If n = 2, then an e l e m e n t

For

hi, w e c h o o s e a m o n o m i a l m a t r i x a s b e l o w .

It i s now c l e a r t h a t f o r a r b i t r a r y

n, a n e l e m e n t

JI

t 2 -I

I

=

| tl

t

-t2... In-~

[ 0:1 10] [ :]

g of C is of the f o r m :

1

t1 .. u w

.

--

1

.

g of C i s of the f o r m :

]

0

o

]

-t 2 .... )

Ivt-~

0 "'" 0

"

|

t..

{ (-l)n-I

I 0

I~_,_ --

-n w

/

,J

,i.e.

-

121-

it is in J o r d a n n o r m a l f o r m (one b l o c k - see p r o p o s i t i o n 2(c) of 3.5)).

Also,

f o r a n y k(1 _~ k , < n - 1 ) , SL - - - ~ E n d (Ak (kn)) is the 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 n

f o r the f u n d a m e n t a l weight ;% . k k-minors.

For

It is e a s y to s e e that Xk(g) -- ~ p r i n c i p a l

g 6 C , the only n o n z e r o p r i n c i p a l k - m i n o r that a p p e a r s i s that

in the u p p e r left c o r n e r and in this c a s e the c o n t r i b u t i o n is t k. a n d T h e o r e m 5 is v e r i f i e d in t h i s e a s e .

H e n c e X k = t k,

T h e proof in the g e n e r a l c a s e i s

s i m i l a r , a s we now show.

To p r o v e t h e o r e m 5, we r e q u i r e the following d e v e l o p m e n t :

Definition.

Let ~1 . . . . .

~ r be the f u n d a m e n t a l weights.

Define 4 . ~A. if 1 j

t h e r e e x i s t s a d o m i n a n t weight ~ such that ~i>,,~ ( i . e . A i - A is a s u m of p o s i t i v e roots) and ;k. is in the s u p p o r t of ~ .

(Supp ~k = t~ k

such that

J

~k= Z n i ~ i i

with n k ~ Ot).

It can e a s i l y be checked that this is indeed a w e l l - d e f i n e d ( s t r i c t ) p a r t i a l o r d e r . S o m e t i m e s , e . g . for type A r , it is v a c u o u s .

Main l e m m a .

Let y E C ,

y = Yl . . . . Yr'Yi = X o ( i ( t i ) ' n i "

T h e n for each I

i, Xi(Y ) = c i.ti + polynomial in earlier (with respect to 2/ ) t js, where is a non-zero constant, independent of

Proof.

t s. i

pace be the 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 of G with ~ . a s the A 1

L e t V = V.~i 1

highest weight.

Ci

Then

Xi(g) = Trv(g) , g E G.

Now, by the fundamental theorem of representation theory, V --~ ) V A , where each ~ is of the f o r m • i

-

n ko< k with n k ~ 0.

Also,

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

Let

0 ~ vCV%, where

~ = ~ J

mj~j,

rnjEZ~.

By the l e m m a of 3 . 3 ,

-

m~0

m

.v

m

-

Hence by proposition 2 of 3.3, x~j(c),n..] v =

n j , v E V wj.;%= V% -m.c(j'] c

122

(finite sum) with v m £V)~.mjo(j + m@(j • independent of c and

v o = nj. v. It follows that if y = Y l ' ' " Y r ' Y j = % ( t j ) ' n j ' then k1

y,v = Zt

I

... t

kr

where k i > 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 <(u~).T[__ t:.l -[]-~)0x.(v~).p [" of 3 . 2 .

Let K = (Gin(k))

We m a y c o n s i d e r X

t_ is

at x,

Hence

s

x ~

, s = number

given by: ~((ti), (u0(), (v•))

T h i s is an i s o m o r p h i s m of v a r i e t i e s by t h e o r e m to be a f u n c t i o n on K (via ~).

Since ~ is an

i s o m o r p h i s m , we m a y p r o v e II f o r K.

Any t a n g e n t Y to K (at a p o i n t y) c a n

be u n i q u e l y written as

Y1 is the t a n g e n t to

YI + Y2' w h e r e

(Gin(k))

r

and Y2

is a t a n g e n t to ~ s . Claim. For:

(dX) x (Y) = (dX) x (Y1).

(We identify ~(y) with y. )

X i s a p o l y n o m i a l of the f o r m

P1 + P 2 ' w h e r e

P1 is a p o l y n o m i a l , each

t e r m of which c o n t a i n s a t l e a s t o__ne of {u~t,(40 U { v ~ t ~ > 0 and P2 is a p o l y n o m i a l m tI s alone.

Since X is a c l a s s f u n c t i o n , X(tyt "1) = X(y) f o r each t ET.

-

125

-

If y = ((tl), (u~), (%)), then tyt -I = ((tl), (o((t).u~),(#(t) .v~ )). It is now easy to see that every term of P1 :uKt
must contain at least two of

. T h u s , e v e r y t e r m of Y ' P 1

eontainsatleastoneof

to>0 :U,I U

:% 1

Since x is givenby

F u r t h e r , it is c l e a r that

((tl), (0),(0)), it follows that (Y. PI)x=O.

(Y2" P2) x = 0, s i n c e

P2 is a p o l y n o m i a l in t'ls alone.

Hence, (Y.X) x = (Y(P1 + P2))x = (Y' P2}x = ((Y1 + Y2 )" P2)x = (YI" P2)x"

Also,

(Yl.X)x = (YI.(PI + P2)) x = (YI.P2)x, since (Y1.PI)x = 0. Hence

(dX) x (Y) = (Y.X) x = (YI.X)x = (dXx) (YI).

II follows immediately.

II Shows that it is sufficient to p r o v e that dX 1 l i n e a r l y dependent.

Now, dXi/t = d(Xi/T).

. . . . .

dXr, r e s t r i c t e d to t , a r e

In o t h e r w o r d s , we a r e c o n c e r n e d

with the functions X i r e s t r i c t e d to T only. On T,

X. -- • :

+ ( t e r m s which a r e s m a l l e r than ~ i)

(*)

i

(See t h e o r e m 2 of 3.4). r

r

L e m m a . q - [ dX. = f, -~" (~-1 dA ) ( e x t e r i o r product) : i i i=l i=l where, f = ~" wCW

(det w) . (w~) (= ~kew ~ , by definition) !

with

r

~ = ( ]To( }2 = ~ hi" ~> 0 i=l

P r o o f of the l e m m a .

We o b s e r v e f i r s t that ~ - A "I d ~ is not z e r o . i i i F o r : the h t s a r e a l g e b r a i c a l l y independent g e n e r a t o r s of k [ T J , so that t h e r e exist v e c t o r fields vj with vj(~i~ = ~ j i ' i . e . ,

with (d~i)(vj) = ~ij"

(a) f is s k e w i , e. w . f = det w . f f o r w E W . For:

w. dX i = d(w.Xi) = dXi, since X i is a c l a s s function.

Hence w. "[]-dX. =-~-w. dX, = -[i- dX . . . . . . i I i : : Let w. ~ = ]T~.nij (written multiplieatively. ) i j J

-

(A)

126

-

-i Then w( )%i

-

d ~ i) = ~ l ) d ( w A i )

d(]..[ ~niJ)

~-1,

W

j

J

= ('~fnij).j j ( Zj {kZ ~. :ik}.nij~j'nij-1, d~k.), =

Z

n...

j ij

~ -ldA

•

j

j

Now, w.(f.'[]-A-I. dA) = wf,'[]-w(A-ldA) i i i i i i = wf.~-i ( ~

n... ~:1 .dA )

j

~

j

j

= (wf) . d e t w. - ~ - ~ - 1 k k This is so since follows that (b) F r o m

d e t w -- d e t e r m i n a n t

f = ( w f ) . d e t w.

of

k T h u s f r o m CA) and (B), it

(nij).

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

(B)

dA

w~ = d e t w . f (det w = :~ 1).

*, dX. = d A + ~_, d ~ 1 i A~ i = A (~

-1

.dA) + Z

A . ( ~mij

r

~:

1 dA

.}, w h e r e ~ = ' ~ , ~

rn.. 13.

1

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

f = " ~ - ~. + l o w e r t e r m s , i . e . f = ~ + l o w e r t e r m s . J=l 1 S i n c e f i s s k e w , a n a r g u m e n t s i m i l a r to one in the p r o o f of t h e o r e m g i v e s : f = s k e w ~ + ~-

skew

$'<~-

.

Consider such a

~

2 of 3 . 4 ,

Since

~' d o m i n a n t -

itf o l l o w s

= s u m of p o s i t i v e r o o t s , T

suchthat (~°, ~

i

(~-~,

#

~.

1

I

} > 0.

} i s an i n t e g e r .

that det w.w

~_!

Wennwhave

Hence

(~,~

+ d e t (w. w i ) ° w . w i ~

c a n c e l out in p a i r s , lemma.

that there exists a simple root ~.

(This argument

giving skew

!

1

: 1 = (~',~*}>(~,~)~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- ~<-1).

Since x

is s e m i s i m p l e and i r r e g u l a r , ¢K(x) = 1 f o r s o m e ~¢ >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~
Proof.

which v a n i s h T h u s the

is n o r m a l . F o r a n y v a r i e t y , the s e t of s i m p l e p o i n t s is d e n s e in it.

this holds f o r F .

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

But by t h e o r e m 3 of 3, 6, the s e t of r e g u l a r e l e m e n t s i s open

-

and d e n s e in F .

128

-

Hence t h e r e e x i s t s a r e g u l a r e l e m e n t which is s i m p l e .

Since

a n y two r e g u l a r e l e m e n t s in F a r e c o n j u g a t e , it follows that a l l the r e g u l a r e l e m e n t s in F a r e s i m p l e .

We s h a l l p r o v e l a t e r that the c o n v e r s e of the

a b o v e s t a t e m e n t is a l s o t r u e .

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

(b) follows i m m e d i a t e l y f r o m t h e o r e m 3 (b) of 3.6.

(c) and (d) : Choose a r e g u l a r e l e m e n t x C F .

By t h e o r e m 3, (dX1) x .

are linearly independent.

Hence

(X1-C 1) . . . . .

( X r - C r) f o r m a p a r t of a l o c a l

c o - o r d i n a t e s y s t e m at x.

Also,

F has c o d i m e n s i o n r.

in a l g e b r a i c g e o m e t r y shows that the i d e a l of F {(X 1- C 1) . . . .

(X r - Cr) t .

Since F

. . . .

(dXr) x

Now, a g e n e r a l r e s u l t

i s , in fact, g e n e r a t e d by

is i r r e d u c i b l e , the i d e a l of F is p r i m e .

F r o m this it can be s e e n that F

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

Since I ( X 1 - C 1) . . . . .

g e n e r a t e the i d e a l of F , it follows that the

(Xr - Cr) ~

t a n g e n t s p a c e to F at a p o i n t

x(CF)

is given by : { Y£(TG) x / _(dX i) x (Y)= 0 V i ] .

Hence d i m (TF) x = C o d i m e n s i o n of the s p a c e

L g e n e r a t e d b y ~ ( d X i) ~ X)l
T h i s p r o v e s that !{(dXi) x J~

Hence by t h e o r e m 3, x is r e g u l a r .

Remark.

3.9.

are linearly independent. l.~i .
The above a p p l i e s , in p a r t i c u l a r , to the v a r i e t y of u n i p o t e n t e l e m e n t s .

V a r i e t y of B o r e l S u b g r o u p s .

In view of T h e o r e m 7 -(a) of 3 . 8 , o u r

a t t e n t i o n is f o c u s s e d on the 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 i . e . on the s i n g u l a r i t i e s of the v a r i e t y V.

The g e n e r a l p r o b l e m , to which we now t u r n , i s to study

t h e s e s i n g u l a r i t i e s , e s p e c i a l l y the one at

1.

-

129

-

As a simple example, consider the group G = SL 2. A~ unipotent element is

oftheform

[:

V= { I :

bl d

with a + d - - 2 a n d ,

2-bal/(a-l)2 = - b c ~ .

of c o u r s e , a d - b c = 1.

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

This is clearly a cone in k 3 with vertex at

(I, 0, 01, It is now clear that this is the only non-simple point of V.

Thus V

has e x a c t l y one s i n g u l a r point, the v e r t e x of a q u a d r a t i c cone.

It is pleasant to be able to start with a nice desingularization of V.

Let ~

denote the set of all Borel subgroups of a group G.

variety-structure on ~

We introduce a

in the following way: Let G act transitively on a

variety V such that (i) the stabilizers are just the Borel subgroups of G. ( It is enough to assume that one stabilizer is a Borel subgroup) and (2) for any v @V, the orbit map G

~ G . v ~ V is separable (i.e. the differential map is

surjective). It now follows that the elements of 0~ and V are in one-one correspondence (Gv = Gv, ::~ v -- v', since a Borel subgroup is its own n o r malizer).

We introduce a structure of a variety on ~ via this correspondence.

We note that V = G/B ,B is a fixed Borel subgroup, satisfies the above conditions.

It is easy to see that any V with above conditions is isomorphic to

one such variety (and hence to any such variety).

T h e o r e m 1.

Thus, we may write

Let V be the v a r i e t y of unipotent e l e m e n t s in G.

(closed) subset of ~ x V ,

defined by: W = { ( B ' , x ) / x C B ' }

.

Let W be the

(If B is a fixed

B o r e l s u b g r o u p and we have identified tT~ with G / B , then W --f(gt3, x ) / g - l x g ~ U ~ . ) Then W is a d e s i n g u l a r i z a t i o n f o r V.

(We shall define this t e r m p r e s e n t l y . )

Proof. A desingularization of a variety VI is a pair (V2,~) where V2 is a variety and ~ : V2---~V 1 is a morphism such that (I) each of the points of V2

-

s is simple and (2) ~ : ~-1 (VI)

1 3 0

-

:~ V s1 is a n i s o m o r p h i s m

(V s1 is t h e s e t of

simple points in VI). Consider the projection P2 : W

> V°

Our claim is that (W, P2 ) is a desin-

gularization of V. (I) W

is irreducible.

0: G ~ U - - - > G / B ~ V ; (2) W

(It is the image of G ~ U under the m o r p h i s m 0(g,u) = (gB, gug'l).)

is non-singular (i.e. each of its points is simple).

cell U ' . B

which is open in O.

is openin W.

Let W ' = {(gB, x ) ~ W / g

Consider t h e m a p

~ : U-K U

m W'

Consider the big ~U- I . Then W'

given by~(u', u) =

-I (u-B,u-.u. (u') ). It is easy to see that ~) is a m o r p h i s m and in fact ~Ln isomorphism of varieties.
The product of non-singular varieties

It now follows that W'

is non-singular.

Since W'

is open,

I

a point which is non-singular in W lates of W'

cover W.

remains so in W

as well. Also, trans-

It follows immediately that W

itself is non-singular.

(3) As seen in theorem 7 of 3.8, the simple points of V ones.

-I Let V r be the set of these points. Let W r = P2 (Vr)" W e must prove:

P2 : w r - ' ~ V r

is an isomorphism.

characteristics.

W e give the proof only in case of good

Fix an element x E vrf~ B.

t h e - ~ r e m s h o w s t h a t the m a p 7I: G ~ V map.

are just the regular

(The d i f f e r e n t i a l i s s u r j e c t i v e ,

G/ZG(X)

r

Then the proof of Richardson's

g i v e n byTi'(g) = gxg -1

is a q u o t i e n t

s i n c e T ( C g -1) = (1-Ad(g)) ~ . ) H e n c e

is i s o m o r p h i c to V r v i a T ' .

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

~ : G

) Wr 1

g i v e n b y : ~ ( g ) = (gB, g x g - 1 ) . = g'B.

(Since x is r e g u l a r ,

If g = g ' g o

with g o E ZG(X) , t h e n gB = g , g o . B

it b e l o n g s to a u n i q u e B o r e l s u b g r o u p a n d h e r e we

i~ave x = go" x. gol~ goBgo I. ) Hence the m o r p h i s m ~ is constant on cosets of ZG(X) and gives rise to a m o r p h i s m

O : G/ZG(X)

• W r. Since G/ZG(x)

131

-

-

is i s o m o r p h i c to V r , we have an i n v e r s e V r .... r~W r isomorphism.

of P2' which is thus an

T h i s p r o v e s the t h e o r e m ,

R e m a r k s . (1) F o r

G = SL2, the d e s i n g u l a r i z a t i o n of V (the v a r i e t y of u n i -

potent e l e m e n t s ) i s i s o m o r p h i c to

/AI×Ip1.

The picture is

~-,----~

the s i n g u l a r p o i n t b e i n g blown up to a p r o j e c t i v e line (G/B).

(2) A g e n e r a l i z a t i o n of the above t h e o r e m i s given by the following t h e o r e m of G r o t h e n d i c k : T h e following d i a g r a m is a r e s o l u t i o n of the s i n g u l a r i t i e s of all the fibres of the m a p X

q~G

p:G

>/~r.

where, X =

I(Bl,X)@~×

GIBl~X 1

y

!

and the m a p s q, p , q Let

(BI,X)~X

Define

a r e o b t a i n e d as follows: q((Bl,X)) = x ; q ' ( ( B , t ) ) = p(t).

then B 1 = gBg -1 f o r s o m e

that g ' l x s g g B . = B 1.

!

Hence ~ b E B

In o t h e r w o r d s , ~ h ~ G !

p ((Bl,X)) = (B,t).

morphism.

g EG.

suchthat b suchthat

Since x s EB 1, it follows

-1 -1 ~ .xsgbET.

A l s o , gb. B b - l g "1

B 1 = h B h "1 and h - l x h = t C T . s

It i~ e a s y to check that p

!

is w e l l - d e f i n e d and is a

A l s o , p(x) = p(x s) = p ( h ' l x s h) which gives the c o m m u t a t i v i t y of the

above d i a g r a m .

We turn to the study of the fibres of the above desingularization of Over a fixed U,

or,

u

in

V

we have the variety ~u

on r e p l a c i n g ~

of fixed points of

u

by

G/B,

the variety

V.

of all Borel subgroups containing (G/B)u : (gB g G/BIu " gB = gB}

acting on the "flag variety", hence a projective variety.

P r o p o s i t i o n 1. (G/B)u is c o n n e c t e d .

(u is an u n i p o t e n t e l e m e n t , )

-

Proof.

132

Recall that for a simple root ~,

b y B and X

.

It can b e s e e n t h a t

is a projective line,

F o r : If Go< = ~ X

-

P ~ d e n o t e s the s u b g r o u p g e n e r a t e d

P~ = (T.<X , X_~>

,X

and B ~ = B t ~ G

P ~ / B ~ G ~ / B - ~ S L 2 / S u p e r d i a g o n a l ~- p r o j e c t i v e l i n e . {xP~ /B, x ~G fixed) CG/B

> ).U~ .

P~/B

, then

A set

i s c a l l e d a l i n e of t y p e ~ .

It i s a c o n n e c t e d s e t .

It i s e a s y to s e e t h a t two l i n e s of the s a m e t y p e a r e e i t h e r i d e n t i c a l o r d i s j o i n t ( s i n c e c o s e t s a r e so).

We s h a l l m a k e ue~ of t h e s e f a c t s while p r o v i n g the

p r o p o s i t i on. L e t u 6 B t 3 B 1.

We show that B and B 1 can b e c o n n e c t e d in ~ u

s e q u e n c e of a r c s of ( p r o j e c t i v e ) l i n e s of v a r i o u s t y p e s (*).

bya

This clearly proves

the p r o p o s i t i o n . Let B 1 --~B, g~G.

ByBruhatlemma,

gEBnwB.

Let w = Wl...

w. i s the r e f l e c t i o n with r e s p e c t to the s i m p l e r o o t ~ ' . 1

1

w k, w h e r e

and k i s m i n i m a l with

r e s p e c t to t h i s p r o p e r t y .

( i . e . w = w 1. . . w k i s a ' r e d u c e d ' e x p r e s s i o n f o r w . )

We p r o v e (*) b y ~ d u c t i o n

on k,

prove. b EB.

Let k>~l.

If k = O, then g t ~ B and t h e r e i s nothing to

W i t h o u t l o s s of g e n e r a l i t y , we m a y a s s u m e

L e t g l = b n w l ' " " nWk-l"

T h i s l i n e c o n t a i n s gB and t a i n e d in ~ u "

gll!l.

C o n s i d e r the l i n e gl" P ~ k / B

j o i n e d b y a l i n e in ~ u

of t y p e ~ k "

can be j o i n e d by l i n e s in ~ u "

Since

Hence by induction ~rl]~ and

W e now p r o v e the c l a i m : (1) By a s t a n d a r d p r o p e r t y of r e d u c e d e x p r e s s i o n s , w 1. . . W k . l ( ~ k ) > 0. ucgB.

Hence ~ u'~U

Write u'-- -N-x Ic ). ~>0 w(~)<0

gB a r e

of t y p e ~ k (viz. the l i n e g l P ~ k / B ), the r e s u l t f o l l o w s

immediately.

(2) W e h a v e ,

.nwk,

O u r c l a i m i s that t h i s l i n e i s e n t i r e l y c o n -

G r a n t i n g the c l a i m , we s e e t h a t u ~ B ~ g l L .

h y p o t h e s i s , B and g l ~

g = b.nwl

x ld/ /~>0 w~)>0

suchthat

u =gu'.

133

-

nwu' = - ~

Hence

) . 77-

(c'

~>o

-

x

Xw(~) w(~) /~>o

w(~) <0

w(/~)

(d'w(~))suchthat

w(~ >0

!

c = Oiff co( = 0 (and also w(a) Also, nwu' ,, b 'ul 6 U . w(v(k) = w I.

**

' ~ = 0 iff d ~ = 0). (Here nw = nwl • .nwk). dw

Hence Cw(a)

0Vw(o(),f0, o(>0.

Wk_ l . w k ( ~ k ) < 0 , it follows that C ~ k

normalizes

u'
P~k u'EUo(

k"

=

0.

Since

Since

T h u s u E V O
Hence P°(kg~lu

=

P~k g-I

U

=

P~k , u ' C U ~ k C B "

Hence u Cgl" P ~ k B.

General problem.

Study the o t h e r p r o p e r t i e s of the f i b r e s , e . g . t h e i r d i m e n -

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

sions, n u m b e r of c o m p o n e n t s etc.

We now p r o v e a p r o p o s i t i o n which links the d i m e n s i o n s of ZG(X) and ~x" More p r e c i s e l y , we have: P r o p o s i t i o n 2. L e t x ~ G

Proo For

be unipotent.

Le, C be the

o, x

Then d i m ZG(X)>~r+2.dim (5 x.

Lo, S

{'BI B2

w E W , define S w = {(gB, gnwB, y ) ~ S } .

We claim:

U S w, a disjoint union. C o n s i d e r (B1,B 2 , y } 6 S. Then t h e r e wE W gi -I , exist g l ' g 2 ~ G such that B.1-- B, i : 1, 2. B y B r u h a t l e m m a , gl g 2 ; b n w ' b "

w EW. S = -i

S =

Let g = gl.b.

~J S . W wEW

nw " g

-I

,

Then gB = g l B , gnwB = g2B.

It now follows that

Next, let (gB, gn WB• y) = (g , B, g , n w ' B , y } .

" g "nw' 6 B.

,

Hence nw, = b . n w . b

l e m m a , ~ z, z ' £ Z(T) such that n Z(T) = T so that w = w'. follows that dim S ~ dim S

w

f o r s o m e b , b EB.

. z . In our c a s e (G reductive), W

This p r o v e s that S w/~ W

Now b y B r u h a t

!

= z.n W'

Then g - l g g B and

Sw!

= ~ for w ~

!

w

.

It now

f o r all w 6 W and the equality holds f o r at l e a s t

134

-

one

w ~

(I).

Considering lemma

the projection

1 of 2 . 1 3 ,

(The image is Fix

wEW,

S

identification

C~U

t~

Hence

dim

of

~xC

C and arguing in the same way as

we get: dim S = dim C + 2.dim ~

can be regarded

w

~B

x

.

~

i s ~3x ~ x .

)

x

a s a s u b s e t of G / ( B

( g B , gnWB, y ) ~ - ( g ( B

nw

onto

C, w h i l e t h e f i b r e a b o v e

d i m S w = d i m (G is

-

n

f~ W B ) , y ) .

/~ n w B ) + d i m (C f l U f l

{II)

~nwB)

Projecting

nwU).

onto the first factor, B flnwB

U.)

S w = dim

G

T h e e q u a l i t y h o l d s iff I, II a n d I I I ,

Hence

via the

(The fibre above

- dim

(B t~nwB) + dim

(C /% U t~nwu)

~< d i m G - d i m (B /% n w B ) + d i m (U t%nwu)

From

X C

C f% U f ~ n w u

dim C + 2.dim ~

i s d e n s e in x

U /% n w u .

= dim S = dim S

d i m ZG(X) = d i m G - d i m C ~ r + 2 . d i m

= dim G - r.

w

-(III)

for some

w ~dim

G- r.

1~ . x

This proves the proposition.

Corollary.

T h e e q u a l i t y h o l d s in p r o p o s i t i o n nW U /% U.

in s o m e

(i.e.

The proof is clear from

C r~ u f ~ n w u

For each

U (~ n w u

and for elements

Conjecture

The conjecture

1.

i s d e n s e in s o m e

w, t h e r e a l w a y s e x i s t s s o m e in this class

This gives rise to a problem: G).

C, t h e c l a s s of x, i s d e n s e U fl

nw

U.)

(III) in t h e p r o p o s i t i o n .

Remark.

x and

2 iff

C with

Ct~U

t%nwu

C, t h e e q u a l i t y of d i m e n s i o n s

Whether the above equality always holds?

is that it happens i.e.

d i m ZG(X ) = r + 2 . d i m { ~ x ~ x C G .

dense in

above holds.

(for any

-

1 3 5

-

We n o t e t h a t t h e a b o v e f a c t is e a s i l y v e r i f i e d f o r r e g u l a r e l e m e n t s , element.

It i s a l s o k n o w n to b e t r u e in c a s e of c l a s s i c a l g r o u p s .

the identity It w o u l d be

e n o u g h to p r o v e it f o r u n i p o t e n t e l e m e n t s .

C o n j e c t u r e 2. C/~U Anwu

For a unipotent class i s d e n s e in

U ~

n w

C, t h e r e e x i s t s a

Since

dense class,

U /% n w u

is irreducible

these conjectures

is finite, at most

such that

U.

A s s e e n in t h e c o r o l l a r y a b o v e , c o n j e c t u r e s elements,

w£W

1 and 2 a r e e q u i v a l e n t f o r u n i p o t e n t and h e n c e c a n c o n t a i n at m o s t one

w o u l d i m p l y t h a t t h e n u m b e r of u n i p o t e n t c l a s s e s

IWl.

A r e l a t e d c o n j e c t u r e is: Conjecture

3. F o r a n y c l a s s ,

dim C

is even.

T h i s f o l l o w s f r o m c o n j e c t u * ' e 1,

s i n c e we h a v e t h e f o l l o w i n g : d i m C = d i m G - d i m ZG(X ) = d i m G - r - 2 . d i m

X

= d i m U - . U - 2 . d i m {B

X

and this

is even. However,

c o n j e c t u r e 3 i s k n o w n to b e s e p a r a t e l y t r u e f o r g o o d c h a r a c t e r i s t i c s .

A s an e x a m p l e ,

we c o n s i d e r

true since by writing a s in t h e c o r o l l a r y . as follows:

For 1

for

thena

write i>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<)~ R + - R+p

.

L e t K w denote this set.

and

Since W(Rp) > 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<"

(3) If x EU~ - V ¢ ~

then one of the following possibilities occurs:

(i) Every yB~Po
Y zB6P~/B, is a complete set of r e p r e -

Let ]~ simple (~ o() such that (~,¢,) = 0. ((1) is not applicable).Le~ x='~-Xr(Cr)EUa(. r~

r~

In this case,

n~

= 0. Hence x CV: ~ iff ~ no y B ~ Po
Since (~,cx) = 0, Wc4(~) -- ~ and hence the only positive roots which have support in { ~ ' ~ 1

are ~ and ~ .

It follows that G , and X~ commute elementwise.

It is now easy to see that for y ~ P ~ , y "lxy~U/$ iff x eU~

above, x e v

Hence from

iff x Su~ i. e ~f o~ ~ 0 This show~ that V~. ~ is open

and dense in U . Further, if x ~ V ~ , ~ , then (i) of (3} holds. We now consider a simple root /5 ~ o~ such that (~, c<) ~ 0. Let U< ~ , ~ ; = ~ r>O

X r.

This is a subgroup and is invariant under the conjugation

s~pp r ~ { , . ~ }

by elements of P~ and % . we may consider U~

Since our interest lies in actions of P~ and 9 ,

/U<,~,~ > instead of

product extension of % ~ , ~ > . ) 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/$+~<(b}. x~(t),n no4= x ~5(b+at). x~+o~ ( m o . a ) ; A l s o , x = x~s(a).x~+~(b)CU~5

m O : a c o n s t a n t depending on no( .

iff a = O. Hence if a = 0, 1. B £ P ~ / B

fies (*) w h e r e a s in c a s e a ~ 0, y = xo~(- b~ ).no~ s a t i s f i e s (*). T o p r o v e (2), we claim:

x EV

iff a ~ 0 o r b ~ 0.

satis-

Hence (1) follows.

(This c l e a r l y defines

an open n o n - e m p t y and hence dense subset of Uo(. ) L e t x be such that a ~ 0 o r b ~ 0.

tf a ~ 0, then yB = x o ( ( - b ) . n

.B

is the unique solution of (*).

a = 0 then b ~ 0 and yB = 1,B is the unique solution of (*). Conversely

If

Hence x e V ~ ,

if a = b = 0, then e v e r y yB EPo
This p r o v e s the claim and a l s o the s t a t e m e n t (i) of (3). c a s e (ii). !

Here, n~,~

RQ<,~ is of type B 2 and o( is the l o n g e r toot. =

1, U¢<

In this c a s e , P

=

Xf~.X#+oc . X 2 ~ + o < .

keeps X 2 ~ + o < i n v a r i a n t .

Hence this c a s e r e d u c e s to c a s e (i)

and is p r o v e d in a s i m i l a r fashion.

case (iii). Ro~,~ is of type B 2 and o4 is the shorter root Here, n~

=2

if c h a r G ~ 2 a n d =

1 otherwise.

Uo~= X/b.X~+0~.X~+2¢ ~.

B e f o r e p r o c e e d i n g to the p r o o f , we take a c l o s e r look at the c o m m u t a t i o n formulae. L e t x (t) "l.x~(a).xc~(t) = x (a). x +o (mat). x~+20 < { r e ' a t 2)

and x ~{t) "1 .x~b+oe(b). x (t) = x +~(b).x +2 (m"b t).

.

-

143

-

T h e n it can be p r o v e d b y i t e r a t i o n that m m " = 2 m '

and m ~ 0, m " ~ 0.

(A

d e t a i l e d a c c o u n t of such r e l a t i o n s m a y be found in the a u t h o r ' s ' L e c t u r e s on C h e v a l l e y g r o u p s , Yale U n i v e r s i t y ' . ) By c h a n g i n g the p a r a m e t r i z a t i o n s of %+~/

x(~+2~ , we m a y a r r a n g e

m = 1, m ' = 1, so that m " = 2.

Hence c h o o s i n g na~ p r o p e r l y ,

n

-1

-1

x (t)

~'~

x (a).x

~+~

"~

(b) x

" ~+2~

(c) x (t). n

"

= x0(at2 + 2bt + c). % + ~ ( a t + b). % + 2 0 ¢ (a) !

= x , say

I.

Since w (~) = o4 + ~ it follows that

n~l.X

(s)'lo X ' . x (S). n

= x.~(at2+ 2bt+ c). xo4(at + b). x ~ +2 (a) ._----- (II).

We now t u r n to the proof of c a s e (iii).

Let x -- x ~ (a).x~+ m).x ~+2 (c)c u~ If a = 0, then I.B where

satisfies (*). If a ~ 0, then x x(to).n .B satisfies (~),

to is a root of at2 + 2bt+ c = 0 (which exists.) Hence (I) is satisfied.

Next, we c l a i m :

x ~L,~

if, ~ = b 2 - a c ~ 0.

(This c l e a r l y p r o v e s (2).)

We f i r s t c o n s i d e r the c a s e when c h a r G ~ 2. Let a = 0.

Now A ~ 0 if, b ¢ 0.

Hence we have to show:

x C V 0 < / 5 i f f b ~ 0. /

It is e a s y enough to s e e : l . B and x .(-c ) . n o ( . B

a r e the two s o l u t i o n s of (*).

2~ Also,

x~V~,~----~b = (9 and (•)has

i n f i n i t e l y m a n y s o l u t i o n s o r only one

solution according as

c = C) o r

c ~ 0.

In the f i r s t c a s e , (i) of (3) h o l d s .

the s e c o n d c a s e , 1 . B

i s the u n i q u e s o l u t i o n and y

-1

. x . y = x.

In

Since x ~ X ~ + 2 ~ !

and 10+

+. 0 a r e not roo*s,

x/s)'l

x x0(s) n0

u0 for all s

so that (ii) of (3) holds. Let a ~ 0. Then

at2 + 2bt + c = 0 has two distinct solutions if, /k~ 0. It now

-

x eL,~

f o l l o w s that

h a s a unique s o l u t i o n t

= _b a

-

Further, x~V~,~__ ~ = 0 and at2+ 2bt+ c =0

iff A ~ 0. O

144

Since a t ° + b = 0, (tI) shows t h a t (ii) of (3)

h o l d s in t h i s c a s e .

C o n s i d e r the c a s e when c h a r G = 2.

T h e d i s c u s s i o n i s s i m i l a r to one in c a s e

' '/5 = 1 and the equation at 2 + 2 b t + c (a ~ 0) h a s c h a r G ~ 2 e x c e p t that no¢ j u s t one r o o t ,

c a s e (iv).

This proves this case completely.

R ~ , / 5 i s of t y p e G 2 and o( i s the l o n g e r r o o t .

I

Here, n~,~ Also,

w(

versa.

= 1, U04 = X . X ] 3 + o 4 . X 2 / 3 + ~ . X 3 / 3 + o d

X3~+20(.

k e e p s X 2 ~ + ~ i n v a r i a n t , t a k e s X 3 ~ + o 4 into X3 /5 +2o< and v i c e -

Hence t h i s c a s e i s s i m i l a r to c a s e (i) and is p r o v e d in a s i m i l a r

fashion.

c a s e (v).

Ro( "/3 i s of t y p e G 2 and o4 is t h e s h o r t e r r o o t .

He~e~ no4,/3 = 3 if c h a r G ~ 3 and 1 o t h e r w i s e . X

Uo(

.Xfl+o(.X~+2a ( .

.X

T h i s c a s e i n v o l v e s c a l c u l a t i o n s which a r e of the s a m e n a t u r e a s c a s e (iii). T h e method used there is applicable here also.

We l e a v e the d e t a i l s to the r e a d e r .

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

Corollary.

If ~ i s a n y s i m p l e r o o t , then GU~ = U g U g-1 ggG 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 . (G s i m p l e ) .

Proof. root ~.

By t h e o r e m 1 of 3 . 7 , x 6 U

i s i r r e g u l a r iff xCUf5

i s the s e t of

for some simple

Since G is s i m p l e , t h e r e e x i s t s a chain of s i m p l e r o o t s

o( = ~ o , ~ 1 . . . . .

~
(o(i,o~i+1) ¢ 0 ~ 0 ~ < i 5 k - 1 .

a b o v e l e m m a r e p e a t e d l y , x c a n b e c o n j u g a t e d into U .

B y a p p l y i n g the

Since a n y u n i p o t e n t

-

145

-

e l e m e n t i n G c a n b e c o n j u g a t e d i n t o U, it f o l l o w s t h a t 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 in G.

L e m m a 2.

Let G,B

Define W ~ ~_ L

GU¢~ i s t h e s e t of a l l

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

b e a s a b o v e , o( b e a s i m p l e r o o t .

by the following: x ~ L

(i) x ev~,p V~ simple ~ o(

iff

.

(ii) For each yB EP~
T

t

are simple roots adjoining o( and yB,y B ~P~/B such that

y-lxy~%t%U~,y'-Ix.y'~%/%U , , _ _

then yB ~ y'B.

Then W ~ is an open dense subset of Uo<. Proof.

We g i v e t h e p r o o f f o r t h e c a s e

c a s e s to t h e r e a d e r .

A r a n d l e a v e the d e t a i l s of t h e o t h e r

U s i n g t h e s a m e n o t a t i o n a s in l e m m a 1, l e t x -- x~(a).xjKe~(b),,. /

w h e r e ~ i s a n a d j o i n i n g r o o t of ~ .

/

T h e n c o n d i t i o n (ii) c a n b e s e e n to be e q u i !

v a l e n t to b ~ 0, w h i c h o b v i o u s l y g i v e s a n open s u b s e t of Uo<. two a d j o i n i n g r o o t s of o ( , t h e n x g L !

x~,(a

may be writtenas:

If ~ ~ ~

x = x/5(a).x

are

+¢<(b).-

!

). x ,+

(b)...Then

#

and ab' ~ a'b.

the c o n d i t i o n (iii) i s e q u i v a l e n t to : (a ~ 0 o r a ' ~ 0)

T h i s a g a i n g i v e s a n o p e n s u b s e t of U(X .

W E i s a n open n o n - e m p t y s e t of U

Theorem

1.

Let G,B

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

(and h e n c e i s d e n s e ) .

be a s a b o v e .

(a) T h e s e t of 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 in G i s a c l o s e d , i r r e d u c i b l e s e t of c o d i m

r+2

in G.

(b) T h i s s e t c o n t a i n s a u n i q u e d e n s e c l a s s Further,

f o r a n y s i m p l e r o o t v<, C A U

class under

Proof.

c<

C of the s a m e d i m e n s i o n a s i t s e l f . i s d e n s e in U

0(

and forms a single

P~ .

F i x a n y s i m p l e r o o t c<.

We a p p l y R i c h a r d s o n ' s T h e o r e m

(of 3 . 9 - to b e

-

146

-

r e f e r r e d to as "the t h e o r e m " in this proof) to the p a r a b o l i c s u b g r o u p P ~ G.

In this c a s e , U p = U~ .

to l e m m a 1 above give (a).

of

T h e p a r t (b) of "the t h e o r e m " and the c o r o l l a r y (We note that d i m P~/U0( = r+2. ) A s s u m i n g G

to have only a finite n u m b e r u n i p o t e n t c l a s s e s (e. g. i n good c h a r a c t e r i s t i c s ) , p a r t (c) of "the t h e o r e m " g i v e s (b) above.

H o w e v e r , one can a p r i o r i p r o v e

that GUo( c o n t a i n s a c l a s s of codlin r + 2, by a proof to be given e l s e w h e r e . Now in the p r o o f of "the t h e o r e m " , the f i n i t e n e s s of the n u m b e r of u n i p o t e n t c l a s s e s is u s e d only to p r o v e the e x i s t e n c e of such a c l a s s . follows even if G has i n 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 .

Hence (b) above

T h i s p r o v e s the

theorem,

Remarks.

(1) The c l a s s

C above is the s e t of a l l s u b r e g u l a r u n i p o t e n t e l e -

ments. (2) T h e two c o r o l l a r i e s to R i c h a r d s o n ' s T h e o r e m hold in the s e t - u p of t h e o r e m 1 above.

We now give a d e v e l o p m e n t which h e l p s us to give a f i n a l c h a r a c t e r i z a t i o n of subregular elements.

We r e c a l l that a ( p r o j e c t i v e ) l i n e of type ~ (¢<simple) is a set

g P ~ / B , g ~.G.

(See P r o p o s i t i o n 1 of 3 . 9 . )

P r o p o s i t i o n 1. (a) T h r o u g h any point of G / B p a s s e s a u n i q u e line of type ~ .

(b) If u is an u n i p o t e n t e l e m e n t , then 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 : (1) u fixes (2) u ~ U

P~/BPointwise. .

-

(c) If ~ , ~

147

-

a r e d i s t i n c t s i m p l e r o o t s , then the i n t e r s e c t i o n of a l i n e of t y p e

~ and a l i n e of t y p e ~ c o n s i s t s of a t m o s t one e l e m e n t of

G/B.

T h u s a line

of t y p e ~ i s d i s t i n c t f r o m a n y l i n e of t y p e / ~ .

(d) If u i s a s u b r e g u l a r u n i p o t e n t e l e m e n t , then

ZG(U) a c t s t r a n s i t i v e l y on

the l i n e s of t y p e 0<, f i x e d p o i n t w i s e b y u.

Proof.

(b)

(a) i s c l e a r .

(2) ~ ( I ) (1) ~

is obvious since Po( normalises U< .

(2). It is given that u.pB = pB Vp EPa( . In particular,

u.no(B = no(B i.e. n (c) L e t ~ , ~ Y)/B"

-i

.u.n

~B.

be two d i s t i n c t s i m p l e r o o t s .

This clearly shows u EUo(. C o n s i d e r the l i n e s

xPo(/B

and

L e t XPlB , x P 2 B ( P l , P 2 e P ) be two p o i n t s in t h e i r i n t e r s e c t i o n . !

fhen there exist

!

P l ' P 2 ~ P/5

-1 P2 P l ¢ P f~ P/3 '

s u c h that xPiB -- yp:B,1 i = 1, 2.

Hence

U s i n g B r u h a t l e m m a , it can be p r o v e d t h a t

P f~ P

•-1

-- B.

/5

Hence P2 P; E B which s h o w s that XPlB = xP2B.

T h i s p r o v e s the r e q u i r e d

result.

(d) T h i s f o l l o w s f r o m T h e o r e m 1 (b) of 3 . 1 0 and T h e o r e m 2, Cor. 2 o f

3-9.

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

Let

u b e an 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 in G.

of 3 . 9 , (G/B)

T h e n a s s e e n in P r o p o s i t i o n 1

i s a c o n n e c t e d union of l i n e s of v a r i o u s t y p e s .

The subregular

U

unipotent element

u w i l l now b e c h a r a c t e r i z e d b y the s t r u c t u r e of (G/B} • U

Definition.

A Dynkin c u r v e i s a n o n - e m p t y union of l i n e s of v a r i o u s t y p e s s u c h

t h a t a l i n e of t y p e c~ m e e t s e x a c t l y n'~ , ~

l i n e s of t y p e 8 "

(~, ~

are any

-

d i s t i n c t s i m p l e r o o t s and n ~' , ~

Remarks.

148

-

is as d e f i n e d in l e m m a I).

(a) T h i s notion and a n u m b e r of i t s b a s i c p r o p e r t i e s a r e due to

J . Tit,~ (unpublished). (b) It is e a s y to s e e that the a b o v e s p e c i f i c a t i o n s d e t e r m i n e , f o r e a c h o<, the 2 t o t a l n u m b e r of l i n e s of type v< v i z . [5 I , w h e r e ~ m i n is a r o o t of

12

[¢<min m i n i m u m length.

T h e r e a d e r m a y wish to d r a w p i c t u r e s of the Dynkin c u r v e s of

type A r , B r , C r (r = 3 is t y p i c a l ) , D r ( r = 4 o r 5), E 6 , F 4 , G 2.

If he d o e s so, he

wiLl find that, e x c e p t f o r the l a b e l i n g of the l i n e s , the p i c t u r e s f o r G 2 and D 4 a r e the s a m e , as a r e t h o s e f o r F 4 and E6, and s i m i l a r l y f o r the o t h e r c a s e s w h e r e r o o t s of d i f f e r e n t l e n g t h s o c c u r .

P r o p o s i t i o n 2. L e t u be a s u b r e g u l a r u n i p o t e n t e l e m e n t .

Then

(G/B)

is a U

Dynkin c u r v e .

Remark.

T h u s , Dynkin c u r v e s e x i s t .

T h i s i s t h e d i r e c t a n a l o g u e of o u r e a r l i e r r e s u l t (see 3.7) t h a t

(G/B) u

i s a p o i n t if u is r e g u l a r .

P r o o f of p r o p o s i t i o n 2.

It can be c h e c k e d that

(G/B)gug.1

= g.(G/R)

•

Hence

U

one i s a Dynkin c u r v e iff the o t h e r one i s .

F i x a s i m p l e r o o t o< .

Then

G

c o r o l l a r y to l e m m a 1 a b o v e ) .

Uo< = Set of 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 (see the

Further,

s u b r e g u l a r e l e m e n t s in U ~ f o r m a

single class under

Pe¢ and this c l a s s is d e n s e in 1 ~ .

of l e m m a 1 a b o v e .

Since V¢~,#

contains a subregular element. u itself.

T h u s we h a v e :

other simple root ~,

We u s e the n o t a t i o n

is open and d e n s e in U , it f o l l o w s that it One m a y f u r t h e r a s s u m e that t h i s e l e m e n t is

u is a s u b r e g u l a r e l e m e n t ~ Uo< s u c h that f o r a n y !

there exist exactly n ~ ~ -many points

yB C P ~ / B

with

-

y'luyg~U~,

in (G/B) u i n t e r s e c t this line in a point pB,

P'C P(5 such that pB = g p ' B .

p,-1

-

Since u ~ U ~ • P ~ / B C_ (G/B) u by (b} of p r o p o s i t i o n 1 above.

Let gP~/B exists

149

Hence y = g p ' g P

up (Note

u g

p E P~.

Now t h e r e

and ( g p ' ) ' l . u ( g p ' ) --

gPetB

his

s e t s up a o n e - o n e c o r r e s p o n d e n c e between lines of type /5 in (G/B) u which intersect

P~/B

that P ~ / B

and y B ~ P o f / B

suchthat

meets exactly n'~,~

y'luyEU

~ U~.

lines of type ~ in (G/B) u.

It now follows L e t gP¢~ /B be

any o t h e r line of type ¢(6(G/B} • By p a r t (d) of the p r o p o s i t i o n 1 above, we m a y U

assume

geZG(u}.

intersects

P~/B"

Aline

hP~/B

of type ~

intersects

It now follows that gP~ /B a l s o m e e t s e x a c t l y n , , p

meets exactly n v( is a r b i t r a r y ,

!

if g - l h p ~ / B

Since geZG(U), h P ~ / B g (G/B) u iff g - l h p ~ / B 6 !

p r o v e s that f o r all

gPa(/B

(G/B) u.

lines of type ] 5 .

This

u s u b r e g u l a r unipotent, e v e r y line of type o( in (G/B) u lines of type /5 in (G/B) u ( ~ any s i m p l e root ~ v().

Since

our p r o p o s i t i o n is p r o v e d .

We will p r o v e l a t e r that the c o n v e r s e of the above p r o p o s i t i o n is a l s o true. c h a r a c t e r i z e s the s u b r e g u l a r

This

unipotent e l e m e n t s in G.

P r o p o s i t i o n 3. Any two Dynkin c u r v e s a r e t r a n s l a t e s of each o t h e r by e l e m e n t s of G. Proof.

The idea of the p r o o f is as follows:

We f i r s t give a s t a n d a r d Dynkin c u r v e

and then show that any other Dynkin c u r v e is a t r a n s l a t e of this c u r v e . c l e a r l y p r o v e s the p r o p o s i t i o n .

We give the proof in c a s e G is of type A r.

Using it, p r o o f s in o t h e r c a s e s m a y be given. p r o o f in c a s e case (i).

G is of type D r o r E r.

L e t G be of type A r.

This

As an i l l u s t r a t i o n ,

we give a

-

Let

m(1 . . . . .

150

-

oft be the s i m p l e r o o t s such t h a t

C o n s i d e r the following s u b s e t

of G / B :

P~3/B U...~n~l...nc(r.l.P~r/B," Dynkin c u r v e .

(~(i,¢
S = P~I/B

U no( 1- Po<2/B On0(1.no( 2-

It i s e a s y to c h e c k t h a t S i s i n d e e d a

L e t S 1 be a n y o t h e r Dynkin c u r v e .

Since S 1 is a non-empty

union of l i n e s of v a r i o u s t y p e s , it f o l l o w s that S 1 c o n t a i n s e x a c t l y o n e l i n e of t y p e c~ 1, s a y g P ¢ ~ l / B . T r a n s l a t i n g the l i n e

P¢~1/B,

S 1 by g-t

T h u s we m a y a s s u m e t h a t S 1 i t s e l f c o n t a i n s

t h e r e e x i s t s p r e c i s e l y one l i n e of type ~ 2 its points.

we find t h a t g - l s 1 c o n t a i n s P~I/B"

Now

which i n t e r s e c t s t h i s l i n e at one of

By a s u i t a b l e e l e m e n t of P ~ I " the p o i n t of i n t e r s e c t i o n m a y be

t r a n s l a t e d to n ~ i B

and t h i s e l e m e n t d o e s not c h a n g e the line

P¢~l/B .

Hence,

a g a i n , we m a y a s s u m e that the line of t y p e o(2 o c e u r i n g in S 1 i s n ~ ] . P o f 2 / B . T h i s m e e t s a unique l i n e of t y p e 0< 3 in a p o i n t n ~ l . X ~ 2 that t h i s p o i n t c a n n o t b e n ~ l B .

) T r a n s l a t i n g by b --

t h i s point i s m o v e d to n ~ l . n ~ 2 B . kept invariant.

A l s o , the l i n e s

(t). n ~ 2 B.

n~l"

P~I/B

(We note

x~2(t)-l.n

~

1

,

and no(1Pm~2/B

are

H e n c e we m a y a s s u m e t h a t the unique l i n e of type °(3 o c c u r i n g

in S t i s nM~ n ~ P ~ 3 / B .

P r o c e e d i n g in t h i s way, we s e e t h a t S 1 can be

t r a n s l a t e d to the s t a n d a r d Dynkin c u r v e S.

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

this case. case(ii).

L e t G b e of t y p e E r

o r Dr . I

Let ~o

be the (unique} b r a n c h point.

!

|I

tl

ofk, o(1 . . . . . e
....

s t a n d a r d Dynkin c u r v e f o r defined curves for the f o l l o w i n g s u b s e t ndo.S , where

,

Io( 1 . . . . .

....

'I

~k'

L e t o( 1 . . . . .

and o( k and

l

. . . . . ( s e e c a s e (i)).

L 1

.....

S O of G / B : S o = P ~ o / B U n ~ o S

0 ~ t l , t 2 and t 1 ~ t 2.

i s i n d e e d a Dynkin c u r v e .

"}

~k"

L e t S be the Let S ,S "

be s i m i l a r l y

respectively.

Consider

U Xo(o(tl).n~o s'UX~o(t2).

It i s now f a i r l y e a s y to c h e c k that S o

L e t S 1 be a n y o t h e r Dynkin c u r v e .

We m a y a s s u m e

-

151

-

that the line of type o{ o o c c u r i n g in S 1 is P ~ o / B .

C o n s i d e r the points of !

TI

i n t e r s e c t i o n of this line with the lines of type ~ 1 ' ~'1' ~1

(occuring in S1).

C l e a r l y , these a r e t h r e e distinct points and hence can be t r a n s l a t e d onto any other t r i p l e t of t h r e e distinct points, by an e l e m e n t of P~o" Pc~o/U~ ° on Pc(o/B

is just that of the p r o j e c t i v e group on the p r o j e c t i v e line. )

T h u s we m a y a s s u m e that t h e s e t h r e e points a r e n ~ o B , n [ o. B ~ ( * ) .

(The action of

x ~ , o ( t l ) . n ~ o B , X~o(t2).

Now by c a s e (i), the p a r t of S 1, c o n s i s t i n g of lines of type

a( 1 . . . . . ~(k is of the f o r m g. S, g ~; G.

Also, no(oB ~ g n l B .

(A line of type o( 2

does not m e e t a line of type c< ° in S 1.) Again, the line of type ~ 1 P~o/S

is n o ( o P ~ l / B -- gPo(1/B.

It can n o w b e p r o v e d that g = no( .b with k

b~B.

L e t b = b l . b 2, w h e r e

meeting

o

b 1 =i=~lxo(i(di)

and b 2 = -~-r>0Xr(Cr)" Then

r¢~ n~o.bl

-1

-1 "nc
i = 1,2 i n v a r i a n t .

1

T r a n s l a t i o n by this e l e m e n t k e e p s

P ~ o / B , n o B , X~o(ti)n~oB

(e.g. n~o.b;1.n-I .Xofo(tl),na(oB = n0(o.no(o. -i Xo( (tl).n o.b %

as n~lo.xa(o(tl).n0( ° c o m m u t e s with

o

b l) Also, %b;ln C¢o1 gSo%b 1 !

b l . b 2.S -- n~ob2S. b2S = S.

Since b 2 does not involve any of the xo(.s, it follows that 1 T h u s by a suitable t r a n s l a t i o n , the Dynkin c u r v e c o r r e s p o n d i n g to

I~l .....

~ k t is n ~ o . S

and the condition ( * ) i s unchanged.

f o r the p a r t s c o r r e s p o n d i n g to

{-: i} ....

,

and

{<

Similar arguments

....

show that

the whole c u r v e m a y be t r a n s l a t e d onto S o . T h i s p r o v e s this c a s e . c a s e (iii).

G is of type B r. !

L e t o(o,O
o(i+1 =

to Io(1 . . . . .

O(r. 1 be the s i m p l e r o o t s such that

1 V 1 ,< i ~ r - 2 . ~r.11

. Let So= Pc~o/BVno(o.S~x

G is of type C r.

= 2 and

L e t S be the s t a n d a r d Dynkin c u r v e c o r r e s p o n d i n g

S O is the s t a n d a r d Dynkin c u r v e in this c a s e . c a s e (iv).

n~o~l

o(to).no(oS, t o ~ 0 .

Then

1B

-

152

-

Let O(o, ~1 . . . . . O(r, I be the simple r o o t s such that n ~ i , ~ , + . = 1 V 0 , < i , < r - 2 . 1

L e t S be the standard Dynkin curve c o r r e s p o n d i n g to f 4 1 So = x ~ 1 (tl).no( 1 PO(o/BD XC(l(t2).no( 1 P ~ o / B U S

1

. . . .

=(r-1 t " T h e n

where t 1 ~ t 2 , 0 ~ t l , t 2,

is the standard Dynkin c u r v e . c a s e (v). G is of type G 2. !

L e t o<,~ be the simple r o o t s such that nc~,/5 = 1 (i.e. o< is the l o n g e r r o o t ) . T h e n S o = P ~ / B t) n{%PO(/B U x ~ ( t l ) . n ~ • P~X/B U x ~ ( t 2 ) . n ~ Pe
case (vi). G is of type F 4. Let V~l,0(2,% , v44 be the s,:raple roots such that t

nv41,o< 2

=

!

I, ne<2, o< 3

=

!

2, ne43,c:<4

-_

1. Then

So = n~ 2 P~I/B U P~2/BUXc~2(tl)n~ 2 P~3/BUX~2(t2).n~iPo<3/B

U -

xo(2(tI ).n~2. no(3.Pc<4/B U x~2(t2), n~2. n~3 P~4/B , where tI ~ t2, is the standard Dynkin curve.

Exercise:

Work out the p r o o f s f o r the c a s e s ( i i i ) - (vi).

We a r e now in a position to give a final c h a r a c t e r i z a t i o n of s u b r e g u l a r elements.

Let G be a simple group.

Choose a simple root o< as follows:

(i) arbitrary if G is of type A r(2) the short branch point otherwise.

(Omit "short" if all roots have the

-

153

-

s a m e length. )

Theorem 2.

Let G,B be as above and u be an irregular unipotent element.

Then the following s t a t e m e n t s a r e equivalent: (a) u is subregular i . e .

dim ZG(U) = r + 2.

(b) (G/B)u (i. e. the variety of Borel subgroups containing u) is a Dynkin curve.

~f~2

(c) (G/B) u consists of a union of lines, ( ~min

(c') u belongs to the unipotent radical of

simple.

1/512

parabolic subgroups of

fPminJ2

type /~, ~ ~ s i m p l e . u

~

i mlnl

is a r o o t of m i n i m u m length. )

(~) (G/B)

of type

is a finite union of lines of v a r i o u s types.

(d') u is contained in the unipotent r a d i c a l of a finite n u m b e r of r a n k 1 p a r a bolic subgroups. (e) (G/B)u has dimension 1. (f) If ~ is chosen as above and u E U , then u @Wo( (i.e. the conditions (i) - (iii) in Lemma 2 hold.)

Proof. (a) ~

(b) ~

(c) ~

(b) is precisely proved in Proposition 2 above.

(d) ~

(e) are trivial.

(c') and (d') are reformulations of (c) and (d) respectively.

(e) => (f). Let, if possible, condition (i) of L e m m a simple ~ ~ ~ .

2 fail. Let x # v

If (~,0() = 0, then u fixes Po(" P~/B

"~ for some

which has dimension 2.

This contradicts (e). If (~,o() ~ 0 i.e. ~ is adjacent to ~', then by lemma I, either y-luy~ U ~ f ~ %

V yE~

or ~ Y 4 ~

suchthat z'ly'luyzEU~/~ U#

k]z ~P~ . In either case, (G/B) u contains a subset of dim 2, which is again

-

a c o n t r a d i c t i o n to (e).

(a).

c a s e (i).

-

T h u s (i) does not f a i l .

(ii) and (iii) a l s o do not f a i l .

(f) ~

154

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

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

T h e p r o o f i s g i v e n e a s e by c a s e .

G is of type D r o r E r .

H e r e , ~ i s the b r a n c h point.

Let ~ , r , ~ ' - be the a d j a c e n t s i m p l e r o o t s .

L e t u = x ~--( a ) , x ~ + ~ ( b ) . x r ( a ' ) . x r + ~ ( b ' ) . x $ ( a

. ).x~.+ . . .

( b ) . u ' w h e r e u' does

not i n v o l v e a n y of the above r o o t s . Now condition (i) is t r a n s l a t e d as: If u' =Vx@(c{}) then c@ ~ 0 for a l l s i m p l e g (not a d j e c e n t to ~ ).

!

!

tt

1~

A l s o , ( a , b ) , (a , b ), (a , b ) a r e p o i n t s of the p r o j e c t i v e

l i n e IP' ( i . e . (a,b) ~ (0, 0) etc. ) on which G a c t s .

Condition (iii) is t r a n s l a t e d

as: The above t h r e e p o i n t s of p r o j e c t i v e s p a c e a r e d i s t i n c t . Since a n y t r i p l e t of d i s t i n c t p o i n t s of IF' c a n be t a k e n onto a n y o t h e r t r i p l e t !

of d i s t i n c t p o i n t s by an e l e m e n t of SL 2, it follows that we m a y a s s u m e b=0 = a , a ~ t b" ~ 0 and a ~ 0, b'~ 0. F u r t h e r , by c o n j u g a t i n g by a s u i t a b l e e l e m e n t t ~ T , we m a y a s s u m e that a = 1 - - b ' = a

= b ,c@

1~8

simple,

o u r e l e m e n t u looks like: x~ (1). Xr+~(1). x ~ ( 1 ) . x g + ~ ( 1 ) . u ' with cg = 1 f o r

0

. Thus

where u ' = J'~xtt (c e ) g

{} s i m p l e , (@,~) = 0 ~ ( * ) .

Now by l e m m a 2, WE i s d e n s e in Uo(. e l e m e n t s a r e d e n s e in Uo(. b e l o n g s to WE

(@,~)=

A l s o , by T h e o r e m 1, the s u b r e g u l a r

Hence t h e r e e x i s t s a s u b r e g u l a r e l e m e n t v which

i . e . s a t i s f i e s c o n d i t i o n s (i) - ( i i i ) .

v has the s a m e a p p e a r a n c e a s g i v e n in (~).

Now one m a y a s s u m e that

We c l a i m that u is c o n j u g a t e to v

which p r o v e s that u i t s e l f i s s u b r e g u l a r . Let

U' = @>0 "~- Xg,

where

I e x c l u d e s a l l the s i m p l e r o o t s and ~ + ~ , r + ¢ , ~ + ~

@EI It follows that U~/U'

[U¢~, UM] C U' (by c o m m u t a t i o n f o r m u l a e ) .

is abelian.

T h i s shows that

A l s o , s i n c e u and v has the a p p e a r a n c e as given in (*4

-

uv-16 U'.

155

-

Consider CU (v).v "I which contains e and is clearly contained

in U'. Since v is subregular, we have: codimu~ Cuo((v) = dim ZU (v) dim ZG(V) = r + 2. Also, codimu~ U' = r+2. CUa((v) is closed, being an orbit of an unipotent group (see proposition of 2.5). irreducible).

It now follows that CU (v).v "I = U' (U' is

This gives: uv -1 = xvx"1v -I for s o m e x ~ U .

are conjugate.

T h i s p r o v e s the i m p l i c a t i o n (f) ~

O t h e r c a s e s a r e s i m i l a r l y d e a l t with.

Remark.

In the c a s e of D r

b e an e q u a l i t y so that

Henceu

(a) in this c a s e .

We o m i t the d e t a i l s ,

and E r j u s t d e a l t with, the l a s t i n e q u a l i t y m u s t o

ZG(U) C U~ , u s u b r e g u l a r .

T h i s r e s u l t is t r u e f o r a l l o t h e r c a s e s except A r

Hence

and B r .

ZG(U)

o

is unipotent.

In t h e s e e a s e s ,

ZG(U)° p i c k s up a o n e - d i m e n s i o n a l t o r u s along with the u n i p o t e n t p a r t . a l w a y s has

ZG(U}°C - P~ (u s u b r e g u l a r in t ~ )

o( is s h o r t .

andv

and even

ZG(U ) C ~ (

In the l a t t e r c a s e , for e x a m p l e , ZG(U} n o r m a l i z e s

P

One in c a s e

, hence

is p a r t of it.

Final remark.

The reader wishing to study conjugacy classes further should

consult (2) of the bibliography below, especially Part E.

-

Appendix.

156

-

The connection with Kleinia n singularities.

s u b g r o u p of SU 2 ( c o m p a c t f o r m ) a c t i n g on (~2/F, a surface

¢ 2.

Let

F

be a f i n i t e

We f o r m the q u o t i e n t v a r i e t y

S with an isolated, "Kleinian", singularity at the origin 0,

coordinatized by the algebra of F-invariant polynomials on

(E 2 .

In each ease

there is a generating set of three polynomials subject to a single relation.

The

possibilities, up to i s o m o r p h i s m , are as follows :

F

C y c l i c of o r d e r

a s SO

x r + l + xy2 + z2 = 0

2r

x4+y3+

Binary octahedral

x3y+y3+ z 2

= 0

Binary icosahedral

x 5 + y3 + z 2

= 0

¢3

z 2 --0

A r

Dr+2 E6 E7 E8

2 e l e m e n t s of d e g r e e 2 on (~ ,

t h e s p a c e of s y m m e t r i c

on ~t 3, t h e k e r n e l b e i n g ± id, t h e f i r s t c o l u m n m a y be d e d u c e d f r o m

the c o r r e s p o n d i n g book "Symmetry". t h e n be f o u n d . ( r + l = 1)

u,v

if F

in W e y l ' s

i s c y c l i c , g e n e r a t e d by d i a g (~, 6 "1)

are the coordinates, satisfy

then

xy = z r+l.

The third column comes about thus.

desingularization,

, as given, e.g.,

In e a c h c a s e the b a s i c i n v a r i a n t s and t h e r e l a t i o n m a y

For example,

and

~3

c l a s s i f i c a t i o n of SO 3 on

g e n e r a t e a l l i n v a r i a n t s and harder.

= 0

Binary tetrahedra!

SU 2 a c t s on 3

r+l

xy+z

r +1

D i h e d r a l of o r d e r

Since

Name

S

t h e n the f i b r e

p'l(0)

of a union of "lines" (i.e. curves) exactly that of the Dynkin curve

x = u r+l • y=

If p : S ' - - - - ~ S i s a m i n i m a l

above the singular point consists

w h o s e intersection pattern is

A r,

z=uv

The other cases are somewhat

with the s a m e fable.

this desingularization for the type

vr+l,

Here

S'

W e illustrate is the surface

-

in ¢ 3 ~ { I p 1 ) r

157

-

g i v e n b y x - - u l z , u 1 = u 2 z . . . . . Ur_ 1 = UrZ, Ur y = z (which i m p l y

xy = z r + i ) , and p i s the p r o j e c t i o n into the s p a c e of the c o o r d i n a t e s x , y , z. T h a t S' i s n o n s i n g u l a r and i r r e d u c i b l e m a y be e a s i l y p r o v e d .

We m a y r e v e r s e

p on the n o n s i n g u l a r p a r t S - {01 of S, e . g . on x ~ 0 b y s o l v i n g s u c c e s s i v e l y for Ul,U 2 ...... so that p: S' - Ip'l(o)I---~Sw e have a desingularization.

{0}

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

W h a t does p-l(o) look like ? If u I ~ 0, we can

solve successively for u2,u3,.. (all values co), so that we have a line, say LI, joining (a~,co ..... (30), or briefly oor, to (0, oor-l). then u3,u4,.., in succession all have to be

L 2 from

(0, co r - l ) to

(02 , cor - 2 ) to of type A r .

(02, c o r ' 2 ) .

If u I = 0 and u 2 # O,

oo, so that w e have a second line

S i m i l a r l y we pick up a l i n e L 3 f r o m

(03, oor-3), and so on u n t i l we have for p ' l ( 0 ) a Dynkin c u r v e E x p l i c i t d e s i n g u l a r i z a t i o n s f o r the o t h e r t y p e s , e s p e c i a l l y f o r E8,

a r e c o n s i d e r a b l y m o r e c.omplicated, and quite ad hoc.

What P r o p o s i t i o n 2

above (in c o n j u n c t i o n with T h e o r e m 1 of 3.9) m e a n s is that this need not be so. Each K l e i n i a n s i n g u l a r i t y and its d e s i n g u l a r i z a t i o n is r e a l i z e d n a t u r a l l y in the c o r r e s p o n d i n g a l g e b r a i c group, via a " r i d g e " of s i n g u l a r i t i e s on the unipote~t variety along its s u b r e g u l a r subvariety.

T h e r e i s a f i n a l c h a r a c t e r i z a t i o n of

s u b r e g u l a r e l e m e n t s , a n a l o g o u s to

T h e o r e m 3 of 3 . 8 f o r r e g u l a r e l e m e n t s , which we s h a l l m e n t i o n . on the n o t i o n of a d e f o r m a t i o n of a s u r f a c e

S and a s i n g u l a r point 0 on it.

T h i s is a m o r p h i s m (in s o m e c a t e g o r y ) of p o i n t e d s p a c e s such that the f i b r e

This depends

p: (V, v o) ~

( p - l ( t o ) , V o) above t o i s i s o m o r p h i c to

f i b r e s r e p r e s e n t s t a g e s of the d e f o r m a t i o n a s

(S,0).

(T, t o) The o t h e r

t v a r i e s in the b a s e s p a c e .

notion of u n i v e r s a l d e f o r m a t i o n can then be defined in an obvious way.

The

Brieskorn

has p r o v e d the following within the c a t e g o r y of g e r m s of a n a l y t i c s p a c e s (see his talk at Nice, 1970).

Let G be s i m p l e , s i m p l y c o n n e c t e d / ( ~ , hence a Lie

-

158

-

g r o u p , u a s u b r e g u l a r u n i p o t e n t e l e m e n t of G , V a s u b v a r i e t y of G (of d i m r+2) t h r o u g h u t r a n s v e r s to C(u), so that l o c a l l y G i s the p r o d u c t of C(u) and V.

F i n a l l y , let p:G - - - ~ T / W (T m a x . t o r u s , W Weyl group) be defined by

p(x) =

s e t of e l e m e n t s of T c o n j u g a t e to x s.

Theorem.

L e t e v e r y t h i n g be a s j u s t s t a t e d .

(a) p : (V,u)

~ ( T / W , 1) is a (the) u n i v e r s a l d e f o r m a t i o n f o r the c o r r e s -

ponding K l e i n i a n s i n g u l a r i t y . (b) C o n v e r s e l y ,

u is s u b r e g u l a r u n i p o t e n t if t h e r e e x i s t s a f a c t o r i z a t i o n

(G,u)~ Pl ~/Jx'x°) ( T / W , 1) with q r e g u l a r and r

the j u s t m e n t i o n e d u n i v e r s a l d e f o r m a t i o n .

T h u s we s e e that the u n i v e r s a l d e f o r m a t i o n t a k e s p l a c e n a t u r a l l y within the c o r r e s p o n d i n g a l g e b r a i c group with the b a s e j u s t the s e t of s e m i s i m p l e c l a s s e s b y C o r o l l a r y 2 (a) to T h e o r e m 2 of 2 . 4 .

Consider. for example, the group SL 2.

Here u = I, V is SL 2 itself, and

p(x) is b a s i c a l l y j u s t the t r a c e of x. If x = [ : d ] , then ad - b c = 1 s o t h a t p(x) 2 ~,~2 - b c = (a - - - ) + (1-~). At x = 1, p(x) = 2, we h a v e a cone with the 2 s i n g u l a r i t y of type A 1 at the v e r t e x .

N e a r b y p(x) ~ 2, the s u r f a c e is an

e l l i p s o i d , and the s i n g u l a r i t y has d i s a p p e a r e d .

-

159

-

REFERENCES (1)

Armand Borel

(2)

: Linear Algebraic Groups. (W.A. Benjamin Inc. 1969.) et al. : S e m i n a r on A l g e b r a i c G r o u p s and R e l a t e d Finite Groups. (Springer-Verlag Series No. 131. )

(3)

Nicolas B o u r b a k i

(4)

Claude Chevalley

: S e m i n a i r e C h e v a l l e y Vol. 1, 2.

(5)

David Mumford

: I n t r o d u c t i o n to A l g e b r a i c G e o m e t r y . (Harvard University Notes. )

(6)

Jean-Pierre

: L i e A l g e b r a s and L i e G r o u p s . ( H a r v a r d U n i v e r s i t y N o t e s , W . A . B e n j a m i n Inc. 1965. )

Serre

(7) (8)

(9)

(10)

G r o u p e s e t A ~ g e b r e s de L i e - C h a p i t r e 4, 5 e t 6. ( H e r m a n n , P a r i s 1968. )

: A l g e b r e s de L i e S e m i s i m p l e s C o m p l e x e s ( W . A . B e n j a m i n Inc. 1966. ) Robert Steinberg

: R e g u l a r e l e m e n t s of S e m i s i m p l e A l g e b r a i c G r o u p s . ( I . H . E . S . P u b l i . M a t h e . No. 2 5 , 1 9 6 5 . ) : L e c t u r e s on C h e v a l l e y G r o u p s . U n i v e r s i t y N o t e s , 1967.)

(Yale

: E n d o m o r p h i s m s of L i n e a r A l g e b r a i c G r o u p s . (A. M . S . M e m o i r s No. 80, 1968. )