Endomorphisms of Linear Algebraic Groups: Number 80

MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY Number 80 ENDOMORPHISMS OF LINEAR ALGEBRAIC GROUPS by ROBERT STEINBERG ...

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MEMOIRS OF THE

AMERICAN MATHEMATICAL SOCIETY

Number 80

ENDOMORPHISMS OF LINEAR ALGEBRAIC GROUPS by ROBERT STEINBERG

Published by the

American Mathematical Society Providence, Rhode Island 1968

DD
by ROBERT STEINBERG

o.

IntroductioB. In tb1s paper

we

are concerned With endo

morpb1s. of linear algebraic groups and their fixed points.

'lbe

development is diT1ded roughly into three parts, one deal1ug With autolllOrphis., another With the case in which the set of fixed points is fin1te, and the third With prepatory material about reflection groups proved in abstract form.

In IIOre detail the

main contents of the paper are as follows.

Throughout the

d1scuss1on

G

Wl11 denote a l1near algebraic group.

It,

mean that an algebraically closed field been prescribed, and that

G

Dl1s Wlll

fixed forever, has

is a subgroup of SOlie

G� (It)

and

at the same t111e an algebra1c set in the atfine space determ1ned by the

n2

matr1c coefficients.

'lbe Zar1sk1 topology, in which

the closed sets are the algebraic sets, Will be used. In add1t1on rr

Wlll denote an endomorphism of

onto

G

group of fixed points.

G,

and

Grr

its

Atter SOlie general1t1es (and det1n1t1ons) about algebra1c

§6 ,

groups in

subgroup of in case

rr

G,

it is proved in §7 that

al_ys t1xes a Borel

and also a max1mal torus of that Borel subgroup

is a s8lll1s1mple autollOrpb1sm. In 84 this is used to

prove that in case tr

rr

G

is simply connected and s8lll1simple and

is a sem1s1mple automorphism then

Grr

is connected; in

Reoeived b7 the editors August 10, 1967 and, in revised fON l'ov8Jl})er 10, 1967. 1

Library orc� C.taIoc N ..... M� COPYRIG HT C

1988

by

Lb. Ammran

NRhanatial9cJcliet.y

PrutiIIaI u. IIw UrUIaI Srotn 01 A IIIB'ICD

2

ROBERT STEINBERG

G

part1 c ular, the centraUser of e very sem1s1 mple element of connected.

1s

Also used 1n the proof 1s a (t w1sted vers10n ot a )

result which states that 1 t a reflect10n group acts e ffec t1vely on an algebra1 c torus then the stab1l1 ser of each p01nt 1s aga1n a reflect10n group prortded the latt1ce ot one-parameter sub groups 1s generated b y elements 1n the d1rect1ons of the re flect10ns . terred 1n

'lb1s 1s proved 1n

§S

&

tor ord1nary t Ori, and trans

to algebra1c tor1 b y a dev1ce due to T. A. Spr1nger.

A co rollary ot the co nnectedness theorem abo ve , under the assumpt10ns there, 1s that e very element of conta1ned 1n a Borel s ubgro up f1 xe d b y the ass umpt10n that

G

�.

G In

t1xed b y

�

1s

ne1ther case can

1s s1mply connec ted by e nt1 rely Om1tted,

b ut for the theorem 1t can be relaxe d to : on the fundame nta l group o t

G,

wb1 ch

�

1s f1xe d-p01nt-free

1n a sens e 1s best

pos s lble (see 9.7), and tor the corollary even turther (9.14). These and related matters are d1sc ussed ln §9 where t he exact range of poss lb lUt1es for the dlsconnectedness of

a

1s not s lmplY connected 1 s determined (9.1).

a�

1n cas e

!he resul ts are

tormulated not tor sell1s1mple automorph1sms , b ut , s Ughtly mo re generally, tor t hose t hat tlx a Borel s ubgro up and a maximal torus thereot, and apply equally well to complex Lte sroups (we ta lCe

K

to be t he coaplex fleld) and to compact Lte groups

(with co nnectedness talCen 1n the us ual sense ), s1nc e Borel h••

[2]

proved the bas1c c onnec �ednes 8 theorem ln that case. In

SIU

..

be,ln the study ot the case ln whlch

rlnl�. by pr ov lftK

�he ..1n (c ohomolo glcal) tool:

1t

a

�

G

ls ls

MEMOIRS OF THE

AMERICAN MATHEMATICAL SOCIETY

Number 80

ENDOMORPHISMS OF LINEAR ALGEBRAIC GROUPS by ROBERT STEINBERG

Published by the

American Mathematical Society Providence, Rhode Island 1968

ENOOMJRPHISMS OF LINEAR AWEBRAIC GROUPS connected and where

1

-

�

�

1s f1n1te, then

(1

1s def1ned by

�)G

•

G (here and else

(T) x .. x � x -l ) .

This extends a

Lang [ 15 ] .

We obta1n as by products that 1s an automorph1sm then G 1s necessarily

well known result of S. �

1n case also

-

(1 -

3

solvable (19 . 12 ) , a result proved by W1nter [33] by a d1fferent method, and that there ex1sts a d1chotomy ( 10.13 ) : 1f s1mple exactly one of the following holds: morph1sm� (b)

§ll

we

�

1s

1s an auto

G� 1s fIn1te.

Assume henceforth that In

(a)

G

G

G�

1s sem1s1mple and

obta1n a (Bruhat) cellular decompos1t1on for

fin1te. GO',

1dentlty the poss1bIl1t1es (see 11. 6; each group turns out to be essent1ally a dIrect product of Chevalley groups twisted a nalogues

[16, 24 ,

[8]

and the1r

29 ] ) , and present a new general

formula for the order (11.16).

The formula, a twisted vers10n of

Chevalley's, 1s 1n terms of the act10n of 1nvarlants under the Weyl group.

�

on the algebra of

It expla1ns, for example, why,

If k Is a fleld ot q elements, one can convert the order n (n l) - Tf ( q 3 1 ) ( 2 So 3 � n) for S � (k) 1nto that for SU (k) q n by s1mply replac1ng q3 - 1 by q3 + 1 whenever 3 1s odd _

( 11. 20 ) . N( 1: t w),

Its proof depends on the evaluat10n of the ser1es In which

t

1s a var1able, w

runs

over all elements

of the Weyl group (relat1ve to a sU1table torus) f1xed by and

N (w)

(T,

1s the number of pos1t1ve roots made negat1ve by

Th1s Is done 1n §2, by a method used by untwisted case

(�- 1 ) .

L.

Solomon

[22]

w.

In the

The above serles 1n the (untwisted)

affine case (the reflectlons a re then relatlve to aff1ne hyper-

BOBERT S'l'EINBERG

4

planes) has ar1sen In the .oft of

R.

Bott

[4]

on the topolol1'

of compact Lie groups (he evaluates lt as a by-product of deep topologlcal conslderatlons) and also In the .orIt of I. Iwahor!

and. H.

MatsWllOto

[14]

on selD1s1mple groups over local fields. We

have taken the opportUD1ty In

83

to give an elementary der1vatlon

of Bott's formula, In a twisted form because of potentlal appl1catlons to noncompact Lie groups and to twisted verslons of groups over local flelds, by an extenslon of Solomon's _thode In

8 12

the "topologyn of

Gcr

(stlll supposed finite) ls studied,

the subgroup generated by the Unipotent e18ll8Dts pla.J1.D& the role of the ldentlty cOlllpOllent.

ls slmply connected then

G(7"

elements, hence ls "connected. n

In

lf'

G

In partlcular lt ls sbowD (12.4) that ls generated by 1ts Unipotent

813

.. have taken the

G

opportUD1 ty to ref'ormulate, In terms of

and

prinClpal results about the representatlons of proved In [26]. a beautlful way

In partlcular, 13.1 (see also

(we

(7"

J

G

the

and.

13.2)

G(7"

comblnes In

th1D1c:) t.o of the main results there.

'!'he

develoPllent In these three sectlons ls to some aztent expoSltory presentlng a Unificatlon and. slmpl1flcatlon of kDowD results. Flnally, In the last t.o sectlons we derive fOl'll1Uae f'or the number of selD1s1mple classes, the number of max1mal tOr!, and. the number of' Unipotent elements (all character1stlc values have to be

1) flxed by

tT

(see

14. 6, 14. 14

J

and

1S.1).

'.l'he f'lrst

:two formulae depend on the evaluation of certain averages over the Wey1 group, and the third on the constructlon of' a certain representatlon f'or

,GtT

(15 . 6 ) .

In case

G

(stlll se�slmple)

ENDOMORPHISMS OF LINEAR ALGEBRAIC GROUPS

5

i s defined over a field k of q elements, so that � may be taken as the qth power map 1n the above considerations , the fo rmulae yield , for t he nWlbers of ob jects of the above types G redefined over k, qr ank G , q dlm G - rank G , q dim -rank G , spectively.

Why these formulae should be pure powers of

q and

not involve terms o f lower degree is not clear to t he author, exc e pt in t he first case since the semlsi mple c lasses of

G

correspond to the regular classes and the latter can , at least when

G

is si mply connec ted, be given the structure of ordinary

affine (rank G)-dimensional s pace (see

[26] for all this ) .

QUi te

likely the other l'orml1lae reflect interesti ng properties of t he varieties of maximal tori and unipotent elements of

A speci al

G.

case o f the last res ult states that with k as above the number of unipotent elements of S Ln ( k) is qn (n-l), a fact already prove d b y several e arlier authors (see

[12] for a proof and

references to e arlier articles ). Throughout the work the notation duced above will be used .

In addition,

G

�

and

1 -

�

intro-

GO will denote the

Gx and ix ' re spectively, the centralizer of and the inner automorphism by the

identity component of the group element

x

of

G.

Finally,

cardinality of t he s et

1.

Is I

G,

and

will always denote the

S.

Pre liminaries on reflection groups .

Our purpose is to

develop the bas iC properties of reflection groups in a fo rm sui t able for o ur later purposes .

The results are for the most part

6

ROBERT STEINBERG

not new (but see 1.26 and 1. 29 ) .

V

Throughout the section

w111 denote a real inner-product space relative to a symmetric

(, )

bilinear (but not necessa rily nonsingular) form

IT a

V,

set of vectors in

subset of

are

1 . 1.

The elements of

E

1. 2.

u:

than

a

E

is in

I.).

The elements of

Tr

1. 4.

Each element of

E

TT

1. 6 .

wav . W

The group

n

a group of auto

nonisotropic.

-a

is also but po other

are linearly independent. is a linear combination of the

such that all coefficients have the same sign.

n, 2(v,a

For each a

defined by

W

a !a.

multiple of

1. 5 .

and

a

V such that:

morphisms of

elements of

E,

E

,

v

-

e

the corresponding reflection

onto its§U.

is generated by the reflections W

wa

wa(a

is our prime obJect of study.

e

TT).

The systems

E

and

W

can be any group with a set of involutory generators

are used mainly to facilitate this study. Abstractly,

(va,Wp prescribed.

The elements of

E

w aW'P

are

will be called roots, those with

positive coefficients in 1.4 pos1t1ve, and those 1n

TT

s1mple.

The s1mple roots are character1sed among the posit1ve roots as those which are not pos1t1ve linear combinations of others. support of a vector that

ca +

o.

v

-

E

ca a ( a

The

7

ENOOK>RPHISMS OF LINEAR AIDEBRAIC GROUPS 1 . 7.

�

For each

othe r than

e

11 ,

w�

permutes the positive roots

�.

Since

changes each vector by a multiple of

w�

�,

this

follows from 1.4. w c

!2£

1 . 8.

W �

roots made negative by

denote the number of positive

N (w)

L (w)

and let

w,

denote the minimal

length of an expression N -

Then

L.

For the proof

we

use the following easy consequenc e of 1.7.

c

IT

and

1.9. .!!: 1 2£ - 1

�

accord1§ 1f

Now fix and

w "" w1w2

N (w1w 2 some and

• .•

• • •

w.1 + 1 )

.1 < t

w1 +1'"

C

AS

W.

wt

e

Neww� ) - Nh' ) w � is positive or negative. w

By 1.9, New) sLew) . as above with

< N(w1wi'"

w.1 )

eguals

W, �

�, -

and

L ew) .

wlw2

N (w)

Assume

•••

<.

L (w)

By 1.9 , w �.1+1 .1

<.

0

for

� +1 > 0 , we have wi w1 +1 • •• w � j +l < 0 .1 .1 W � +1 > 0 for some 1 � .1 , whenc e j .1 Since

by

"1 +1··· W j �.1+1-�

1 . 7.

But then

w1 +1 ···" + 1-"1 ··· w.1' .1

and substituting the right side for the lett in the expression for

w

�-1

and usi ng

we

get a shorter expression.

This

contradiction proves 1.8. 1 . 10 .

It

w

c

W

keeps all positive roots positive . or.

equivalently. all s imple roots pos1t1ve. then ,

By 1 . 8

Lew)

-

0,

whence

1�1O.

w - 1.

ROBERT STEINBERG 1.11.

Corollary.

1.12.

The following conditions are eqUivalent.

The restri ction ot

to the subspace

V generated by IT is faithful.

2I.

t is finite.

(a)

i s finite .

W

(b)

'bIT • -IT,

such that

Assume (b) holds .

by 1.9

Further

t i s finite, then

If

W

"'0

Choose

L(wo)

1 . lJ .

P

i n 1.4.

E

£

TT.

hence for every

If ( cl holds, then the number

by 1.6.

a finite number, whence (a).

( .)

is positive sem4def1nite and

wIT.

The first as sumption and 1.1 imply

and then the other assumptions yield an easy proof

wIT,

by induction on the sum of the coeffi cients o f

" � IT

we

wr1te

,,) .

-c

1·14. t,

E

Then

whence 1.lJ.

For

w
t.

' C t.

Assume

that

N.

t is f'1nite or that all coefficients in 1.4 are

l'!!.!!l

(P. p) > O.

!n ( c ) is unique .

'b

follows from 1.10.

Assume that

either that i ntegers .

'b

W

maps all posUi ve

to max1m1se the function

pos1t1ve root. whence (cl holds . of positive roots is

Wo

E

-b

There ex1sts

i s fin1te by 1., and 1.11.

is negative for every

The Uniqueness of

(c)

,2£. eqU1 valentlx.

roots onto negative o nes .

TT,

W

�

-

W

TT

W"

for the group generated by all

the co nditions 1.1 to 1.6 hold with

replaced by

-

".

the part of

t

support ed by

P

ENDOMORPHISMS OF UNEAR ALGEBRAIC GROUPS

9

This 1s easily verified. 1.15·

1!

"�"IT !!!S! w

e

Wn

'

.!m.!!!

w permutes the

posit1ve roots with support not in ". The proof is like that of

1.7.

"�TT � Wn be as above and let W, !U!. n the set of w £ W such that wn > o. (a ) Bach • £ W can be wr1tten uniquely • • w'v" w1th w ' e: W,n and .ft £ W" (b) N(w) N ( w') + N(wr) in (a). 1.16.

!.2!:

_

•

-

Assume w £ W. Let w' m1n1m1ze the function N over the elements of wW". Then W'CI > 0 for every CI £ " by 1.9. Thus ' ft " I , ft W E: W and w £ W;' ". Now assume W. w w - v v are two " express10ns as' in (a). Then w'w"v-l and Wi are 1n W�. Since .ava-l" is supported by n, 1t must be positive by what has just been said., hence equal to n '!hus w" - va by 1.10 with W" in place of W, and un1queness holds in (a) . Part (b) is an easy consequence of 1.15. •

be the cone of all v E: V such that (V,CI) � 0 for all CI £ TT, and let Vo be the cone of all v e V such that (V,CI) < 0 for only a f1n1te number of gos1t1ve roots CI. Then C 1s a fundamental domain for W on 1.17.

V • o

�

C

.

We obserY'e first that W r� lly acts on Vo because of 1. 7. Assume v £ V . Choose • e: W to m1n1m1ze the number of O posit1ve roots CI such that (wv ,CI) < o. Then wv £ c: 1f I

10

ROBERT STEINBERG

n, then because of 1.7 the elaent w(lw contradicts the choice of w. Thus each T C Vo is congruent to some u C C. That u is unique is part (b ) of the following result . 1. 18. Lemma. Assume u , u' £ C, W C w, .IB5! WI! • u ' . � (wv , (I ) < 0 for some (I

£

n be the subset of TI orthogonal to u. .!!!!:!.

(b ) u . u '

'!ben

-

•

Write w in the form w'w" of 1.l6. 'lben w"u . u because n is orthogonal to u , whence w'u . u ' . To proTe both parts of 1 . 18 we need only show that w' . 1. Asslla ' W oJ. 1. Then W ' (I < 0 for some a £ TT, whence ' 0 > (u , wm ) • (u , (I) � 0, (u , (I ) .0 , (I e n , and W ' (I> 0 because w' £ W�. This contradiction proTes w' . 1 , Bence

1. 18.

C IT !!1 Cn be the part of C defined by (u ,(I ) - 0 m (I C " , (u ,� ) > 0 m 15 £ IT - • . II c" !.!. nonapty. then the following are equal. (a ) W.. ' (b) l!!! stabilizer of C .. !! w. (c ) The stab1U!8r of MY point of 1. 19 .

l2!:

n

C n'

The first croup aboTe contains the third by 1. 18 with u ' • u, the third contains the second because the stab1Uzer of Cn fixes it pointwise by 1 . 17, and the second contains the first because C" is orthogonal to n . Hence the three groups are equal. 1.2fJ.

l!

S is a finite subset of VO ' than the sub group of W nx1¥ S pointwise is a reflection group. lD

ENDOJI>RPHISMS OF LINEAR AlGEBRAIC GROUPS W C

other words. eY!Fl

duct ot retlect10ns ot Assume

u

belongs to some

c

S.

wh1,ch fixes

W

�1ch also do .

By 1 . 17

we

po1ntw1se 1s a pro

S

W

may assume

u

and 1ts stab1l1zer 1s

C"

W

S and

obv10us 1nduct10n w1th

11

c

W"

C.

'!hen

by 1.19. S

replaced by

-

{u}

u An

and

now yields 1 . I!) •

W"

Remark.

1.21.

This result as 1t applies to real f1n1te

reflect10n groups 1s orten referred to as Cheva11ey's theorem , It 1s also true

but 1t was already mown by Cartan and Wey1 . for complex groups [27] . Henceforth 1n this sect10n, CJ"

1 . 22. and

t.

hence normalizes

It

1 . 23 .

all elements ot

W.

W c

"P

V(J"

and

If f1xed by

V and

subsets ot

V

denotes an automorphism of (f"

and

which fixes W

".

IT

denote the

•

and p !L! tr- orb1t ot roots. then W ". have the same s1gn .

S1nce ". pres erve s s1gns. this follows !'rom the equat10n n n wrr a. . ". wu (a. c pl. 1s n.n1te and ,, ' IT, L( ) R( ) denotes the formal poln'r ser1es t t w • t t w

1·24· (a) W

Rotat10ns.

,ft,J t)

summe d oYer all

w

c

Assume

W,," W� ,

Tf.

rr

w1th

W (t)

11 .

It �

the number of

tr

the same 1n case ( ,,) • (_l)n 1d.th n

not. then

It

".

(11)

•

o.

R

and

(b,) If 0;-

(J"

L

as 1n l.g;

fixes

orb1ts 1n

", 11;

then 1t

12

RO BERT STEINBERG

1.25.

1T

A.sswae

Thear..

1 . 24. Then E £ (.)W (t)/W._Ct), ' '"Q CT

IT,

.2!

tm

eguals

t!

1s Anite. egals 0 Asswae

•

C

sUJllllled over the subsets

i

(m the number of pos1t1ve roo1;s) !! W

!!

TT

Anite and the I19ta1;10n as 1n

W

1s 1nt1D1te.

1s Axed by

tr .

If

w

1s 1n

Wt! then

1n 1.16 also are, because of the UD1queness there. Thus by 1 . 16 (b) we have W (t)/W",,' O. Therefore for a t1Jted wsW CT t! the coeff1c1ent of tN(w) 1n the sum of 1.25 1s E £ ( ")

w'

wit

and

summed over all " ( 1 - l)n • on, 1f pos1t1ve by unless then 1f

n

n

" .

S1nce

-

w

fixed by n

'b

'b

Wcr

1 . 12) , 1 . 25 follows. 1. 26

W

1s f1n1te then

1s 1nt1n1te n

-

0

The senes

to

(f'

then b� 1.15 the funct10n

W,,' ftlus 1f also

same relat10nship to E St!(.) /Wna(t),

(tm

_

W"

cr

f1xes

N on

W.

computed

cr

H on

" then

as

W. (1;) bears the o W. By 1.25 then

W (t) to (1' sUDDed over the proptlr subsets of

S (l1h/w (t) or the same cr

only

aboye'z:epres8Ats a

w (t)

1n the con'text of 1.14 1s the same a s the restrict10n of W

0

1s the number of pos1t1ve roots. -1 by the UD1queness 1n (1t. equals crwO<7'

Coro llar,y. " � IT,

W

leept

N(w)

rat1ona1 funct10n. If

IT

orb1ts of

rr-

How by 1.12, 1f

wb11e 1f

1n which case 1s 1n

1.

1.e.,

Hence the coeffic1ent 1s

(see 1.24(b».

1s never 0 t

such that W1C> 0 ,

1s the number of

when 1t 1s

0

CT

,

tr

thing W1th

tm

TI,

equals

IId.ss1ng.

ftlus

I)

ENOOMOR PHISfi5 OF LINEAR ALGEBRAIC Gil) UPS 1 . 26 follows by induction 1. 2 7 .

C orollary. w

summed o ver all

we

IT

.u

£ W

1s fin1te, then Wet) -

tT.

1

of

1. 26.

deduce from 1. 25 a more general formula.

Our proof

d1ffers from that given by Solomon [22 ] for the case W tT·

and

{ E tL w )

represents a rational function.

This is the case

Next

on Illrl.

f1n1te

and can be used 1n s1m11ar s1tuations whi ch artse

1,

later.

1 . 2g. plementary

1T

Assume tT-

f1n1te and that a �

L EfT{n - a)/w1lcTet)

tm/W

tT

are com

l!!!!l

L

-

n�a

F1rst

TT.

1n variant subsets or

p

�p

Wn f1n1te

we

observe that 1f W

e t) - l/w (t-l) . tT

is f1n1te, then

For the element .0

or

1.12

W ,

1s 1n

tT

hence there 1s a correspondence w <-+J. wOw such that New) + N(w') • m. Thus 1 t among the elements ot W as noted above;

nk

IT

yen,

we

get t rom

IT

1.25;

l

e*) E EcrCY ) /W;yJt),

summed o ver

l/WnfTCt- ) 1 t Wn 1s fin1te, 0 1t Wn 1s 1nt1n1te. We m ul t1ply th1s relat10n by EfT C n - p) and sum over equals

all n � p. lett 1t 1f

we

p UY •

On the right

first sum

IT,

i.e.,

is a CT-orb1t out side

we

get the right s1de ot

c(T'(n - p)

it � Vy

'V

�

a,

on n;;2 p U and 0

Y

we

1 .2g.

On the

get £ Ca)

(T'

ot herwi se since i£ P

then the terms tor which

2E

conta:lns

ROBERT STEINBERG

p

cancel those for which it doesn't.

Thus the lett becomes the

lett side of 1. 2 6 as reqUired . m

If

is empty 1. 26 becomes 1 . 25 .

�

If

is empty it

becomes:

1 . 29 .

&

l SCT(,, ) /lI1fcr(t- )

,,

Corollary.

lI" finite

In the next sect10n

W

in case

we

1s f1n1te, 2 . 1;

that ".

W"

1s f1nite.

V" n V(1

Then

W.

Let

we

will show how

Let V"

W CT (t)

thus 1. 29 g1ves one 1n general.

To close this sect10n the same structure as

give an explic1t formula for

"

be a

(T-

WCT

can be g1ven

orb1t of

be the subspace of

V

IT

such

generated by

cons 1sts of the real multiples of the sum of

the elements of ", elaents of E t1 V"'

hence 1s one-d1mens1onal. a finite set by 1. 12 ,

we

Among the select those for

wh1ch the sum of the coeffic1ents in 1 . 4 1s maximal, then take the1r average. The result, an element of noted

m".

1. J O . finite let

For each CT- �

"

2l

m"

be as above and w" such that . " . -" (see 1. 12 ) . ,,

( a)

m"

(b )

." !!...!!!

1s a non1sotrop1c element of

WCT

rr

v" n Va-I wlll such that lI"

be d e

!!

the unique elaant of lI"

VO' .

and 1ts restriction.to

V(T

1s the reflect10n

ENOOMJR PHISMS OF LINEAR AlGEBRAIC GROUPS to

correspond1.ng

ca

"

lS

•

The group generated by

W.

cr

and

on

1.s.t1.n1te,

V"

hence f 1.xes a positive-defin1te symmetriC b1.linear form 1.s eaS1.ly seen the action 1.s 1.rreduc1.ble. 1.s a mult1.ple of

1.f

v

s

hence also 1.s def1.n1te.

F,

As just before 1.26

non1sotrop1.c. Vcr'

then

...v - v

for some real number •

(w.v,ca.) - (v,ca.)

c.

we

Vct () V.'

1.s 1.n

(a)

!!

Cb)

futhful.

• •

by Vcr'

TIcr

1.lD,

whence

the orb1.t of WCI"

�

ca. .

wp

ca,

Thus

N(ww,,) • N(w) -

WeT

ITCT, Ecr,

Wcr I Vcr

w s Wcr' w" 1.

Assume

hence equals

CQ.

we

have c(ca., ca.) •

by 1.23.

w;' 1 N ew,,)

V" '" Vcr

of the

Then

on by

By

hold

with

Tf, E , W V,

•

WCI

for every root

< °

whence (b).

V

to 1.6 E cr • WcrTIv... , then the cond1.t1.ons 1. 1 replaced

No..

and 1.ts restrict1.on to VCT !!. cr 1.s the set of ca" !!! 1.)0 !DS&

f1.xes

Wcr

1.s

may also be described

as the most positive of the project1.ons onto

1 . )2 .

V.

Wcr.

(v,w"".) - (v,ca.) • -2(v,ca,,),

var1.ous roots w1th support 1.n

on

ca.

.." s

w"" •• -�,

S1.nce

( ,)

Thus

see that

R emark . Because of Ca) ca.

1. )1.

Thus

As

F.

<° p

for some

n

ca s

with support 1.n

.

by ,

1. 12 W. 1.s finite,and clearly

VCT'

1. 16.

is generated by the various

which proves ea).

We have

'!bus by an obvious induction w..

In the context of (b ) the

condit10ns 1. 1 and 1.) to 1.6 and �he f1.rst part of 1.2 are easily ver1f1ed. Cc > 0 , ca

•

ca. , ca'

It rema1ns to prove that 1f •

ca""

.

and

.'

WCI.

as 1n 1.30)

cca' then

c

-1 .

ROBERT STEINBERG

16 1. 23

By

w

maps the pos1t1ve roots With support 1n

n' ,

those With support 1n

n' c

Thus the map 1s sur.1ect1ve,_

1nterchanged. •

.-1

and s1m1larly for

•

"

1nto

With

n

CI',

and

and

1. Corollary.

1.33.

Assume the condit10ns of 1.13 hold and tj-

W n 1s finite tor every

n 2! IT.

orb1t

lb!!

geT

!!! 1.32

1s the set obta1ned by averag1ng the elements ot the various 0"-

orb1ts ot

E

and then d1scard1ng those ot the result1ng

vectors that are smaller mult1ples ot others.

2.

The ser1es 1n the f1nite case.

sect10n 1s to prove 2.1 below.

Our a1m 1n this

This will be used 1n §ll to

derive a un1form tormula tor the orders ot the finite Chevalley groups and the1r twisted analogues.

It

will be assumed that

W

1s a f1nite group generated by retlect10ns act1ng on a real Euclidean space ot t1nite d1mens10n been chosen so that 1.1 to

1.6

n,

TT

that

hold (e.g.

E

and

E

have

can be chosen as

the set ot un1t vectors ort hogonal to the retlect1ng hyperplanes), and that

eT

1s as 1n 1.22 and ot f1n1te order.

ot §l will be used. me tric algebra

S

Chevalley [9] has shown that 1n the sym

on

treely generated by

The notat10ns

V n

the subalgebra

I

ot

W-1nvar1ants 1s

homogeneous (bas1c) elements otun1quely

determined degrees.

2.1.

Theorem. 1=!t

the bas1c 1nvariants

I.1

W �

tr

be as above.

have degrees

d(.1)

Assume that

and are so chosen

ENDOl«>RPHISMS

OF

LINEAR AlGEBRAIC GBOUPS

17

that each 1s a qMruter1st1c vector of (f': (1Ij • 8f ' iD5l j let £OJ be the characterist1c values of cr act1ng on V (3 • 1 , 2 ,••• ,n). l!l P (t) denote the rat10nal fOrm cr IT (l- 8 td (j»/Cl- � t), and as 1n 1. 24 let V (t) denote CT j j ) HCw the series E t (w 8 VCT). Then Vcr (t ) • PcrCt). Remarks. Ca) S1nce cr 1s of f1n1te order, 1t acts completely reduc1bly on the 1mariants of a fixed degree. '!hus a ch01ce of the I as above 1s always poss1ble. (b) It W1ll j be proved later that the 8j form a permutatlon of the 803• (c) If 1+ denotes the 1deal of 1nvar1ants w1thout constant 2 term and J. 1+/1+ , then PCT (t) may also be written det(lJ - CtCT )J)/detCl - tcr) . Here the denom1nator bears the same relat10n to the group of one element as the numerator does to V. Because of the tensor product decompos1t10n S. I4D H, Pcr (t) also equals the trace of tcr on the space H of har monic el.ents of S. (d) The proof to follow closely parallels a proof by Solomon [22 ] of the case CT. 1, when 2 . 1 becomes E tH Cw) (w c V) • TT (l- td (3»/ (1 - t). However, we have given a more el.entary proof of the key lemma (2. 3 below) which requ1res DO results from algebraic topology. 2. 2

The proof of 2 . 1 will be g1ven 1n several steps. S1nce the group generated by V and CT 1s f1nite, we may assume our bas1c 1nner product 1s 1nvariant under 1t. As 1s easlly seen C1" preserves the decompos1t10n of V 1nto the subspace generated by IT and 1ts orthogonal complement. It follows

ROBERT STEINBERG that on restr1ct10n to th1s subspace ne1ther P (t)

TT

Assume

Cn �

W

w

£

W.

!.2!: n QT !.!1

Cn be as 1n 1.19 d enote the number of (s1mp11c1al) cones cOngruent

Nn

and let

and f1xed by

wcr.

1l!!!!

E Ca( n)Nn

a-1nvar1ant parts of

�

(1 )

C

'

1h!s

er n .. n

Cn .!!!!1&.!!:

be congruent to

W and f1xed by

�

(resp. K ' )

(resp. V' )

2!

n

K'

V'

be the f1xed -po1nt-space for

(J"C. Cn'

and whence

by the reflect1ng hyperplanes for W.

are the 1ntersect10ns with V '

sects

V' .

whence

C'

WCT.

1s also.

1ti

c'

be the number of we

.2!!

V. � V

Let

C'

Then the ce lls K

of those cells of

be a cell of

K

which 1nterweT,

Then any po1nt 1n this 1ntersect10n 1s f1xed by C'

1s f1xed by

weT,

be as 1n (1) .!m! (2) , and let orb1ts of IT - n , � d1m C' n V'

awl

er-

may assume

Conversely , 1f

which thus 1s a p01nt common to C'

so 1s 1ts centroid, ()

WIT

be the [con1cal s1mplic1al] complex cut on

whiCh are f1xed by

Aga1n

normalizes W ,

•

(2 ) K

det w.

IT.

S1nce 0' 1s 1nvar1ant under (J" . we may assume C' . Cn' whence waC n • Cn. S1nce C n are parts of C , we get from 1 . 17 that a cnWO".

-

n runs over

Here a nd 1n similar s1tuat10ns which follow the

cr

V.

generates

2 . ).

l2.

W (t)

Thus we may assume throughout the rest of the proof

changes. that

nor

cr

then

and 1.

V'

Then

wa

acts on

en

as

V'.

-

cr

1• by

ENDOK>RPHISMS 1.19; as

OF LINEAR ALGEBRAIC

hence lt permutes the "vertlces" ot

a- .

C

19

GROUPS "

ln the

'lhese vertlces correspond to the elements of

SaM

TI

way

- "

C,,("W' , whence (3).

and each orblt contr1butes one dlmenslon to

In the complex cut on a real k-dimenslonal vector

(4 )

!i

space by a flnite number of hxperplanes let number ot cells ot dimenslon

1.

�

E (-1)

lN

denote the k • (_l) • Thls

1 ' applled follows trom Euler s tormula to the complex cut by the

hyperplanes on the unit sphere centered at the or:l.gln, but lt also follows dlrectly by lnductlon. plane, say

H,

In tact, lt an extra hyper

ls added to the conflguratlon, then each l-ce11

ot the orlglnal complex wh1ch ls cut lnto tlfO parts by corresponding to lt an (1 - l) -ce11 ln

H,

H

has

the one that

separates these parts trom each other, so that

E(-1) i N 1

re

malns unchanged. We turn nov to the proof of 2 . 3. I'

flguratlon

of (2 ) .

(_1)1 .

obse"e that

on the lett and

(_l)

k

If

we

Co-fIT -

We apply (4) to the con

comblne (1) , ,,)

we

ln (3) ,

(k - dlm V ' )

(2)

and

get E

on the right.

(3 )

and

so-(IT- JdN" Slnce

WCT

ls orthogonal, its characterlstlc values other than t 1 come k n ln conjugate complex palrs. Thus (-1) - (-1) det WO', whlch n equals S C TI) det v because det (T - (-l) ScrCIT) . '!bus lt the cr orlglnal equation ls divlded by

2.4.

RemArk.

trace tormula (cf. complex

I

of

(2).

S ( IT) , cr

the result ls 2.3.

One can also prove 2.3 by applylng the Hopf

[22] ,

[16, p.

266] )

to

¥T

actlng on the

ROBERT STEINBERG 2.5.

aDl

For

generated by

!!!! er)

V

"

and any

2! < V,er.>

"

complex character

91,

� 'X.

denote the ')(

character lnduced on < W,er> by the restrictlon ot

llwl

E eO"( "h.Cw)

0.

stab1l1zer ot (T

character

W,

W

1s

V" .

1n

IT.

By

1';

0,

lnduces on <W,et>

we

•

Thus 1t

x.

see that the

Thus the UD1t c�

It

'X 'X

under < W, d>,

0.

1s the UD1t 1s arbltrary,

2.5.

M be a (real fin1te-d1mens10nal) <W,er> module.

I (M)

m e M

such that ot

under V"'

tor the subspace ot skew-1nvariants under

n

WIll. ....

det w . m ) ,

I,,(M)

write

These subspaces

speak ot the trace ot

0"

are

and tor each

X

0"-

W

We (all

invarlant

tor the subspace ot 1nvar1ants

aU fixed by

er;

thus we may

on them.

For any <W,O"> - module

2.6.

Let

the

the character whose value

from the def1n1t10ns, whence

write

"

1.19

Us1ng as above (see (1»

< W, tr.> 1s < W ",a>

and fixed by

� <W",o:>.

W).

then 2.5 reduces to 2.3.

1,

'Yo . "

Let

part

e

1s the number ot s1mplexes congruent to

1.e., under

then

In

0"

acter on x

(wa-) det w (w

t1xes the tundamental chamber

stabll1zer ot

at

')(

" 1s a �1nvar1ant part ot

Assume

that

•

(the FOuP

be the character ot

M.

M,

First .... average

'X.(wa-)

ENDOMORPHISMS

LINEAR

rec1proc1ty theor_rU, po 271] Wn0

over

GROUPS

21

P"

on

In(M),

we

get the average of w

over

tr(a,I(M».

X(WO")

acts on

Wn

M

as

get "(P,p), 1.e. J

Similarly the average of X(lIIcr)det

tr(a, In(M» 0 we

we

Now slnce the average of

the projectlon

equals

AIDEBRA IC

By the ca1culat10n of the proof of the I'roben1us

w E Wo

over

OF

w

over

W

Subst1tutlng lnto 2.5 averaged over

w £ W,

get 2.6.

Enj'

P ,.,.(t)

for

W.

lnvariant part

be deAned for

S

Just

Wn

as

"

IT

gt

-

E S

k symmetric algebra on

let

d ("j),

d(j), E ' Part) iE! j

E Eah'Pc:r(t)fP,.,.(t) - tm , �

Then

Let

a-

For each

2.70

m

as ln 1. 25.

be the usual graded decomposltlon of the

V.

The ldent1ty to be proved may be

written.

Because of the deflD1tlons of the d's and e's the coeff1c1ent k of t on the left ls E £a Ca)tr(c:r, In(Sk». Now as ls weU

I (S) • pI (S)

known (s1nce

CI

nOmial

q)0

div1des Clearly

efflclent of

t

k

p

,

(1

-

belng the product of the poslt1ve roots wCl)q

dg p

•

m

for any root and

on the r1ght of

CI

O'p. p.

(* )

equals

and any poly Thus the co trCa;iCsk»,

hence also the coefficlent on the left by 2. 6, whence 2.7. 2.8.

Proof ot 2. 1.

This fo1101ll8 from 1. 25 a nd 2. 7 by

22

ROBERT STEINBERG

ITTI

induction on

2 . 9.

!h!.

C ct . the proof of 1.26) .

Corollary.

In 2 . 1 �

EJ form a permutation ot

EOJ• This is easily reduced to the case that

ally indecomposable and generates assume.

Since

PaCt)

V,

11

is orthogon

whlch we hencetorth

takes on a t1n1te nonzero value at

t . 1,

IWa I by 2.1 , the number 1 has the same mult1pl1city among the E,1 as among the £0 ,1. It 0'2 . 1, then all the value

E'S

are ! 1;

thus 2.9

holds in this case.

Because ot the

class1t1cat1on ot indecomposable retlect10n groups [10, Ch. 11] the only other possib1l1ty is the tr1al1ty case ot two ot the

EOj

are 1;

hence so are two ot the

D4•

& . ,1

'!hen The other

1

two terms are, in both cases, the nontrivial cube roots ot since

0'3 - 1 and a is a real transtormat1on, whence 2.9. 2 . 10 .

Remark.

A case by case check shows that the degrees

of the basic invariants of

WalVa' viewed as a retlect10n group

as in 1 . 32 , are just those

d (j )

general explanation ot this tact.

tor which

& . j

1.

We have no

Finally, we present an extension ot 2.5 in the sp1r1t ot

2 . 11 .

.2!:

TI,

then

!! � � p are complementary O-InYarlant parts En:>�&a(n - �)'l(n(WCT) • det w En:> pEa( n - phn(wcr). -

We apply 2 . 5 with

n,Wn and OX IWn in place ot

IT,

W

and

ENOOKlRPHISMS OF LINEAR ALGEBRAIC GBO UPS X,

and then induce from

1: EO"Cy hy Cwer)

•

det

1f

<W, CT>.

<Wn' a> to

·"C-..at, summed on C*)

follows from this just as 1.2g trom

�

y

2)

The result is n .

Then 2 . 11

in the proof of that

result.

Corollary.

2 . 12 .

notations a re as in

!! a !!ll!

P are as above and the

2. 6 , �

2: EO"Cn - a)tr(O",InCM»

=

n;?a ).

The affine case.

Our main aim is to prove a tw1sted

version of a theorem of Bott(see ). g and ).10 below),

the

analogue of 2 . 1 for discrete groups generated by reflections in affine hyperplanes.

In addition to the assumptions there,

we

Will assume: ).1.

1T

) . 2.

2 ( a,P) / (P,p)

(2£

1:)

generates

v.

is an integer for all

The first condition may be stated: fixes no line of ).).

V.

W

E

1:

•

is effective. i.e.

Because of 1 . 1) the second implies:

The coefficients in 1. 4 are all integers.

Our main object of study is the group reflections in the affine hyperplanes

a E 1:).

a,p

The collection

same conditions as

1:,

{�a

=

k

+

a

Wi •

2a/Ca,a)la E 1:}

generated by the 0 (k

integral,

satisfies the

hence generates a complete (n-dbnensionau

24

ROBERT STEINBERG

latt1ce, wh1ch w.lll be denoted

'!he semisUrect decompos1t10n

3 .4.

W f1xes

S1nce

k + �

reflect10n 1n through

L.

-k��

W,

•

LV

0

and the reflect10n 1n

£; LW

3. 5.

v/w'

DI

and

' W ,

�

Remark.

the

1s the product of a translat10n � .

0

and s1nce the

translat10n 1s a product of the two reflect10ns,

W'

(* )

L, LW 1s a group. S1nce •

holds.

whence

get

3 . 4. W,

It follows that

has f1nite volume.

we

1s d1screte and that

Conversely, any group

Wi

generated

by aff1ne reflect10ns and haVing these propert1es can be realized 1n th1s way.

First

we

-

IJiI

w.l th

L a complete latt1ce of

W a f1n1te reflect10n group fixing a po1nt

translat10ns and wh1ch

Wi

Our assert10n then

talce to be the orig1n [10, p. 191] .

follows from:

3.6.

�

W be an effect1ve f1n1te reflect10n sroup and

L a complete latt1ce fixed by W. to a reflect1ng brperplane for so that

�� . 2u/(eI, eI)

d1rect10n and let cond1t10ns 1 . 1 �

S!! (b)

E

W,

For each direct10n orthogonal let the vector

1s the shortest translat10n of

be the set of all such

1 . 6 , 3 . 1 J!!5l ,3 · 2 bold for

eI.

'!he following cond1t10ns are equ1valent.

L 1s generated by the

�� .

Ca)

be chosen

L 1n t¥ l1!!

an appropriate ch01ce

IT·

(1)

�

25

ENOOJllRPHISMS OF LINEAR AIDEBRAIC GROUPS (2)

LW

is a reflect10n group.

(3)

LW

1s the reflect10n group generated by the re

k

flections in the hYperplanes

n

If

+

CI

-

0

(k 1ntegral,

CI"

1s chosen as a s1mple system relat1ve to some (cf. the remark after 1. 6 ) ,

E

ordering of

then aU parts of

(a) but 3.2 are ver1fied at once, and 3.2 holds because

L: �

f1xes

is a IlUlt1ple 1s 3.2.

�W

�

Let

impl1es (3 ) .

It

WCl� of � CI

L

1s 1n

-

and in the l1ne of

(�J'CI)

whence

'

be the latt1ce generated by the

LW:

CI

+

•

But

�W

(It

real,

k 1s an 1nteger, Henceforth W,

We convert

and

W,

CI,

aence

3.4.

,

� CI .

Then

1hus

(1 )

1s also tae largest reflect10n subgroup

any reflect10n

0

W

1s 1ntegral by 1 . 5 wh1ch

1s the group desCribed in (3) by (a) and

of

E) .

CI "

w"

and

LW

1n

w

E) ,

then

�W.

L

is 1n some hyperplane

-k�CI" L

by

(.)

above,

Thus (2) 1mpl1es (1) .

are as defined Just before

3 . 5.

1nto a l1near group so that our earl1er results

may be appl1ed by mak1ng 1t act contraaredielltq on the space of 1nhomogeneous l1near funct10ns on space of homogeneous funct10ns with elements of

V'

extend the 1nner ponent. and

I 1

-

Thus

1n the form

r

product of

V

Finally, let

IT'

E'

+

v

We 1dent1ty the sub

V.

V 1n the usual way, wr1te (r

real, v " V) ,

and

to be tr1V1al on the f1rst co�

denote the set

(It

the subset obta1ned by adJ01n1ng to

+

Cllk

n

real,

CI £ E 1

the set

616 • h1ghest root for some 1rreduc1ble component of 6

- CI

V'

1s a pos1t1ve comb1nat10n 1n 1.4 for every root

E}. CI

ROBERT STEINBERG in the component of

6; the ex1.stence

[ 7, p. 256 J. 3.7.

2!

V

k + a• 0

The reflection in the affine hyperplane

V'

acts on

as the reflection 1n the hYperplane orthogon

k + a.

al to

In fact, the 1.mage of

r + v

under the first reflect10n

r - k(V'� ) + v - (v'� ) a (recall � -2a/(ata», a a a which may be wr1tten r + v - (r + v'� (k + a). k+a)

works out to

3 . 8.

The cond1.t1ons 1.1 to

V, t,

1n place of

TT, Wj

1 . 6 hold w:Lth V ', t't TI', w,

hence so do the other results of §l.

1. 1 to 1 . 3 hold. Assume k

Clearly

a.

the h1ghest root in the component of

+ a s

If

t'.

k >

0,

+

clearly holds.

at

k < 0,

and similarly 1£

F1.nally

e:

11')

then

+

w"

TT'

-

WIt

is equal to the group generated by all

i.e. to

W,

by 3 . 7 , and

we

fined for (J - 1 , 2 , •

acting on

W

1n

is the by

w(I(a

1. 13 , s

t' ) ,

1 . 6 holds.

can state Bott ' s theorem Theorem.

"

t'

so that

Now

be

1.4,

then

1 . 5 holds by 3.2, wh1.le 1.f W wa(a

group generated by all

6

a • k ( 1 - 6)+ _1le if k - 0 it

w:Lth integral coeff1.c1ents in fact, follows from k (k - 1)6 + 6

Let

Def1ne

W, (t)

[4 J.

i'or

W,

1. 2 7 (see alsQ 3 . 7 � 3 . 8) .

u.

Wet)

�

is de

d(J)

be the degrees of the basiC invariants of W d (J) -l d( » ). V. Then W ' (t) .. TI (l - t J / ( l _ t) (l - t

• •

, n)

ENDUl«>RPHISMS OF LINEAR AIDEBRAI C GROUPS 3 . 9.

(a)

RemarkS .

no prob lems because each

The denom1nator on the right c aus es

1

de, ) -

The reflect1ng hyperplanes cut

1 s pos 1t1ve by 3 . 1.

V

and one of wh1ch , denoted we

F,

s e e by apply1ng 1. 17 to

V.

result on

spac e by 3 . 7 . )

(b )

1 nto a number of s 1mpli c1al

chambers ea ch of whi ch 1s a fundamental doma1n for

as

27

1 s def1ned by

V'

on

W,

(In the pres ent cas e

Vo

�

W,

V

on

� 0 tor

a

£

TT',

and 1nt erpret1ng the ot 1 . 17

Because ot 3 . 6 an element ot

t'

1 s the �o le 1 s pos1t1ve

(resp . negat1ve ) 1t as a funct10n on V 1t 1s pos 1t1ve (resp . d( » negat1ve ) on F . Now n (l- t j / ( l - t ) • Wet ) by 2 . 2 ( c ) .

replaced by IT' and IT tormula d ( ) - l - l. t t N ( w ) ) , summ e d over 3 . 6 1 s equivalent to llr( l - t j Thus by 1 . 15 with

thos e

w- 1F and

w

£

runs

N (w )

W,

IT

and

such that

n

1 wIT> 0 , 1 . e . , IT( w- F ) > O.

through the chambers conta1ned 1n the c one

•

l N (� )

Here

TT> 0,

1 s , by the above remarks , just the number ot

retlect1ng hyperplanes s eparat1ng

w- 1F trom F.

torm 1n wh1 ch Bott or1g1nally gave h1s tormula .

This 1s the H1s proot

cons1sts 1n 1nterpret1ng both s 1des ot the equat10n with plac ed by

t2

t

re

a s the P01ncare s er1es ot the loop space ot the

s1mple c ompact Lte group cOITespond1ng to

t,

the 1nf1nite

s eri es a r1s 1ng trom the Mors e theory and the product from a c ompar1s on ot the group with a produ�t ot spheres ot d1mens10ns 2d ( j ) - 1 ( l � j S n ) .

(c )

The above tormulas aris e not only .

1n Bott's work as just descr1b ed but also 1n connect10n with c ert�n cellular decompos1t10ns of 11near algebraic groups de f1ned over local tields [14].

Thus 1t c an b e expected that a

28

ROBERT

STEINBERG

twisted form of 3 . S W1U have appUcat10ns to tW1sted vers10ns of these groups, and we shall prove such a formula . The reader primarily 1nterested 1n Botti s formula may take CT to be the 1dent1ty 1n what follows . Cons1der an atf1ne-EucUdean automorphism CT of V which permutes the elements of IT'. It follows from 1. 13 that eT f1xes 1:', and then 1f 00 denotes the Unear part of eT that eTO f1xes 1:, hence norma11zes W. Theorem. � W, Wi , d(j ) be as 1n 3.S S CT,CTO as above. .!dm Cj , SO J be def1ned as 1n 2 . 1 but � 0"0 1n place of CT . � � (t ) denote the form TIt! - S t d(j »/(l - £j td(3J-l ) U- SOjt ) .iWl W�(t ) .1ia J i l. Then Wi (t ) Q (t ) . sertes 1: t N ( w) (w C Wa' CT CT •

•

-

3 . ll. Remarks. (a) The Sj need not form a permutat10n of the SO ' but they do so 1f CT 1s Unear , 1 . e . , fixes IT J (see 2 . 2 (b» . In this case (cf. 3 . 9(b » 3 . 10 1mp11es that TI ( l - Sjt d(j )-l)-l • 1: t N (w ) , summe d over those c ells w- � which are 1n the cone C and f1xed by CT. (b) 'or each type of group W, (type An ,Bn , ) the d (j ) and the poss1b1Ut1es for eT are knOwn. O nce CT 1s g1ven the C j and 80 J are easy to work out , hence so 1s � (t ) . Cons1der Wi of type Ano Here the d(j ) are 2 ,) , , n + 1. If n + 1 • pq 1s a factorizat10n and the elements of TT' are represented as usual 1n the form of an (n + l)-cycle. . let CT, for example , • • •

• • •

ENDOMORPHISMS OF LINEAR

ALGEBRAIC

GROUPS

29

be the automorphism which II10ves each element p steps forward. Then the SO.1 cons 1st of the qth roots of 1, with 1 counted p - 1 times and the others p t1mes, 1IJh11e the e.1 are all 1 s1nce CJQ 8 W. Thus �(t) . (l_tn+ l ) / (l_t q ) P In case p . 1 , 1. e . , IT' 1tself 1s a c:r- orb1t , this reduces to 1, as 1t should s1nce w�. {l } by 1. )2. 'lh1s 1s the only t1me that W!- 1s fin1te. In case p - n + 1, 1 . e . , cr 1s the 1dent1ty , we get (1 - tn+ l )1 (1 - t) n+ l as the value of the funct10n of ). 6. •

The proof of ) 10 also proceeds 1n several steps . Reca ll the decompos1t10n Wt • WL. Let T denote the torus VI L , and K the complex cut on T by the pro.1ect10ns of the reflect1ng hyperplanes. The elements of W and cr act as s1mplic1al mapp1ngs of K. For each proper part " of TIt let F " be the "face" of F def1ned by' «I. 0 , ,> 0 , a 8 ", , 8 IT' - ,,) and T" 1ts pro.1ect10n on T. S1nce F 1s a fundamental doma1n for Wt • WL act1ng on V, 1ts pro.1ect10n, 1.e . , the Union of the T,, ' s , 1s fa1thful and y1elds a fundamental doma1n for W a ct1ng on T, and 1t further follows that each cell of K 1s congruent under W to a un1que T". .

Assume W 8 W. For each proper part " !ill. "fTl 1!! N" denote the number of cells of K cOngruent under W l2 T n and f1Ud bv wer. .Idt! � denote the 1th exterior power of V. Then the following are equal: (a) t SCT( ,,) Nn ( n proper, cr- 1nvariant ) , (b ) det(w - CT0 1 ), (c) ) .12

)0

ROBERT STEINBERG

We observe f'irst that the last sum ls det(l - WOO) and that (b ) and (c ) are equal because -sCT(TT'). (-l) ndetoo • det(-CTo� ) . ' Let T be the f'lxed-polnt set of' weT on T. Assume that T' ls not empty. Then W' ls conjugate to WOO under a transla tlo n of' T. Thus T' ls a translate of' the f'ixed-polnt set of' WC1'0 ' 1 . e . , of' the kernel of 1 - WOO on T, and so conslsts of' a f'lnlte number of' translates , say d , of' some subtorus of' T. By golng to V, we see that det (1 - ¥TO ) .;. 0 exactly when the dlmenslon of' T ' , call lt k, ls O . 'lb1s last conditlon holds even lf' T ' ls empty (In wh1ch case k · -1) J lf' det(l- wc.n ) rio 0 , then ( 1 - ¥TO )V V so that w6 ls conjugate to WOO by a translatlon and T' ls not empty. Now we will use the f'ollowing result . •

Assume that the complex cut on a k-dimenslonal torus by a f'lnite number of' translates of' (k - l)-dimenslonal subtor1 ls �ellular. and that Nl denotes the number of' l-dlmenslonal cells . l!!!!! E(-1 ) l N1 • 60 k• ) . 13.

Th1s f'ollows f'rom Euler ' s f'ormula . A direct proof' by lnductlon also exlsts (cf'. 2 . 3 (4» but will not be glven here. We apply 3. 13 to the complex cut by K on each component of' T ' and then add the results . As ln the proof' of' 2.3 we get on the left -SCT(IT ' ) E scr(1I)N1I, s umme d on the proper �lnvar1ant parts of' IT'; on the rlght we get d60 k• If' Ie (. 0 , this ls

ENDOK>RPHISMS OF LINEAR ALGEBRAIC GROUPS

)1

det el - WUo ) det (-�o l ) - det ( w - �O l ) by the above , so that (a ) and (b ) of ) . 12 a re equal in this case . 1£ k - 0 ,

o

and so is

then

T ' is congruent to the kernel of 1

- WO'"O ' which consists of

I det ( l - �0 ) 1 pOints , represented in V by the lattice (1 - WO'o ) - lL taken mod L. Now det ( l - �O ) i s positive : det (t - �O ) is pos1tive for large positive values of t , hence

for all t > 1 because wao is orthogonal. �us the r1ght side of our relat10n d6 0 k becomes det (1 - waO ) ' which proves the equa11ty of (a) and (c) 1n this case and completes the proof of ) . 12 . ) . 14.

Remark . ) . 12 . also follows from the Hopf trace

formula [16 , p. 266] applied to i ng

SUD

of the traces of

WC1

act1ng on

K.

�e alternat

wa

y1e lds on the cha1n groups -e� ( TT ' ) t e�( n) Nn and on the homology groups t (-1 ) i tr (WCTo , E1 ) because of the canon1cal 1dent1f1cat10n of E1 with the 1th

T (us1ng the fact that T 1s a product of

homology group of n

circles ) . If

n 1s a proper part of p01nt of V (the e quations � .

0

IT' ,

we

reduces to the notat10n of § 2 ) . identically on T.

� E

for

hence in View of the decomposit10n W ,

with a subgroup of W whi ch

the group

denote

•

w�

fixes a

n are cons1stent ) �

WL it may be 1dent�d

Wn (if

The groups

W�

n

� TI this

and Wn act

Because of 1 . 19 and our present construction ,

this implies :

) . 15 .

It

n 1s a proper part of TT' , then Wn , the

ROBER T

32 stab1l1 zer ot

1n W,

Tn

STEINBERG

and the stab1l1 zer ot any p01nt ot T n

are all equal. n

Hencetorth

T

and the sub s c ript

3 · 16 .

It

�T

restr1 ct1on to

normal1 zes

.!!!

')(

T.

From the det1n1t1ons ,

and each

WT

tollows that the group

3 · 17·

<WT ' �r

W 1ET • i s t1n1te .

be a complex character on <WT ' �T> !!!S! 'X n

the character on < WT ' O'T > 1nduced by the restrict10n ot < Wn

>• T ' �T

Then tor

Because ot we

Next on < W , �O > For 1 t

we

w

&

W

we

to

3 . 1S th1s tollows £rom 3 . 12 just as 2 . S trom 2 . 3.

obserYe that there 1s a homomorph1sm

such that

,wT · w

tor all

w

&

W

these two groups , the first by restri ct10n to

,

and

ot

T,

c learly conta1ned 1n that ot the s e cond . sh1rt trom a group on c omplete the proot ot

T

<. WT ' O'

'aT ·

0'0 •

onto

L,

Th1s enables us to

to a l1near group on

V,

and thus to

3 . 10 .

!! M !!...! < W ' �O > -module and

I, I

Y

the s econd by

extract10n ot l1near parts , then the leernel ot the first ,

2 . 6 , then

'X

have

cons1der the natural homomorphisms ot < W ' , a::>

3 . 16.

TT' ,

denotes a �-1nvariant proper part o t

n

are as 1n

1s

ENOOX>RPHISMS

In 3 . 17

we

OF

choose for

33

LINEAR AlGEBRAIC GROUPS

�

the charact er of

M converted

into a < wT , aT >-module w1th the aid of the homomorphism tp above , write dat (w - crO l ) in the form 3 . l2 ( C ) , and then

3 · 19. �C t )

ftle result is 3 . 1 8.

w c W.

average over

!2£

"

as above let

a s in 3 . 10. If the

d's

!!!!a and

P 1V Ct )

1: EaC ") /P 1rcTCt )

•

are a s in 3 . lD

E'S

be as in 2· 7 .!!!S! ' - s ( rr )j � (t ) .

o-

the i dentity to b e

proved may b e writt en

Here the products are over has s40wn that if nomials on wi.

th

V

and

S

,1

from

I CS )

1:

t

n.

I ( S ® E)

i s an ater10r

freely generated by the dlfferent1als of a

!his leads to the formal d (.1 » • ,1 ) and u : TT( l + s,1t d e - lu ) / e l - c t .1 dD Ei ) ) t ku1 • Setting u · - 1 , we see that the k t on the right side of C*) 1 s

tr Co-O, I ( S k ,i k c oetticient of -so-cTI ' ) 1: C- l ) i tr (0-0 , 1 CSk da

I CS ) .

�)) •

As in 2.7 the coefficient on

1: co-C ") tr (o-o , I ,, C S » . By 3 . 18 With k two cpAntities are equal , whenc e C *) and 3 . 19.

the lett is

3 . 20.

N ow Solomon [2 1 ]

S tJD E Wlth the algeb ra of ditterential forms

b asic s et of generators of identity in

to

is identif1ed Wl th the algebra of poly

polynomial coefficients that

algebra over

1

froQf or 3 . 10.

We a pply 1. 25 With

M . S

w'

k

and

these

11 '

34

RO BERT STEINBERG

1n plac e ot

IT.

W and

Isolat1ng the term

" .

IT'

",

we

and b ear1ng 1n m1nd the pres ent convent10n on

(* ) t I:a ( ,, ) /W' .,.,. Ct ) a common p01nt ot

•

-

l:a c

IT' )/W� ( t ) .

V, e . g . ,

trom W� constructed t ro m

a

and

ob ject

W" and

P ,.,.Ct ) .

1s equal to

�

W (t )

•

1n 1 . 3 2 . cr

we

cla1m that

+

ter

- 2 ( � ,el )/ (lI , el)

2 (� , el ) / Cel, el )

ta- .

w:ro..
F " . Thus

which by 2 . 1 appl1ed to

C*)

we

and 3 . 19

get

a- 1 n 3. 10 1 s 11near , so

cd

•

a nd

S

•

may torm the two twiSted - ( Va, '

TI'a- ' to-, '

Wa, } as

s at1st1es the cond1t10ns 1. 1 to 1s 1ts att1ne ext ens10n.

n + a el

by 1 . 33 (Z.

we

St

and

S'

and that

a(n +

1s 11near ,

then

IT t

and

{ VCT ' TTCT ' t(T' Wa- }

Then

C* ) t� . Z

erO '

Assume that

n

3 . 1 and 3. 2,

1.6 ,

fix

1 s equal to the corresponding

Compar1ng

Cb )

that 1 t t1xes both S .

(T

(a) One can extend 3 . 17 and 3 . 18 along

Remer,s .

the 11nes ot 1 . 28.

systems

get

as requ1red .

�(t )

3 . 21 .

W� and

the centro1d ot

as constructed

W"

No"

there

tor

+

n

el

t'

8

, whenc e II, � I: ta- '

1 nt egers �. Now 1 t

"el � - "1+11 � I: t� by 1 s an 1nteger aga1n by (* ) . +

(* ) ,

S1nce

whence

Thi s 3 . 2 holds tor

The other p01nts nec essa ry to estab 11sh our cla1m are no"

eas1ly var1t1ed.

There 1s a rather curious corollary to

Co)

3 . 12 or 3 . 17 .

3 . 22 .

It

"

Ass""

1 s a proper

1 t and only 1t k

•

dim Va- .

-1

then

that a-I:

a- 1s chos en so that

1nvar1ant part ot

w,.,... 1

t 'W/V" I

Vcr . •

(b ) n 2

.Y:

!!!Sl

1T' ,

�

n - dim

t Iwal

8

-1

Wa. -1

I:

(a)

W"tj

V !!!S

w,.,. I

•

2k , both sums

ENOO )l)RPlnSMS OP LINEAR AlDEBRAIC GROUPS

"

over those

tor yh1 ch ( a ) holds .

Assume, for example, that

2

of order d1agrams .

DS '

and

It

�.

1s ot type

fT

Then

1s

W 1s of type F4 as we see trom Dyn.\d.n cr The groups W " of 3 . 22 are ot types E6 , A X A l S and

1 +

'!he equat10ns read

1s ot type

W

W

and the correspond1ng groups

B4 •

and

35

Cn ,

c01nc1de and express

ot types F4 , C3

W1I(7

36 + 27

2

•

6

and 1

+

X

Al

1 2 + 3 - �.

on the other hand , then the equat10ns

�

as a sum of b1nomial coett1 c1ents 1n

the usual way . For the proot ot nec essar11y 11near.

we

3 . 22

fT2 . 1

We have

�

observe t1rst that

on the orthogonal complement of

VfT .

so that

cr

S1nce also

1s

acts as

-1

It

�

W,[T 1 t

W fT because -1 commutes With a' , we have 1I(7 (a ) . To prove (b ) we cons1der -1 act1ng on T . The s et ot n f1xed p01nts 1s t1n1te and cons1sts of det ( l - (-1 » • 2 p01nts . and only 1f

It

-

-1

Thes e are 3ust the centro1ds of the f1xed c ells 1n the K,

complex

for these c entro1ds are clearly fixed and

no

cell

c ould conta1n two t1xed po1nts s1nce then the s egment 301n1ng them would cons1st of an 1nt1n1 te numb er of fixed p01nts .

Now

the collect10n ot t1xed c ells cons1sts ot complete W-orb1ts and the orb1t o t J .1S .

'!bus

T

2n

iT ' such that f1xes

T"

1.e. ,

"

(as 1n

"

.

E

-1

Iw/w" I fixes

3. 1S )

Iw/w. 1

elements by

s umme d over the proper parts T

"

•

1 f and only 1f both 1s as 1n ( a ) .

conta1ns

Now 1t w

and

WCT,

then

"

-1

-

a'

do ( ct. proof ot

Thus the first equat10n of

of

-1

2.3) ,

(b ) holds .

36

ROBERT STEINBERG

Then so does the second s1nce tr

by

1

we

W by Wa l Vcr and

may replace

and use 3 . 21 (b ) .

4.

'lbe l emma 3 . 15

Retlect10n groups , latt1ces and tor1.

has the tollowing consequence ,

4 . 1. The stabili zer 1n W at each po1nt at

T 1s a

retlect10n group. S1nce each point a t at some

T 1s congruent under

W to a p01nt

T ' this is clear. n

The purpose at this sect10n 1s to extract the essence at th1s result and to prov1de a converse. retlect10n groups thereon.

The tocus 1s on tOri and

(Each retlect10n 1s relat1ve to some

subtori at codimens10n one, a Euclidean metric being assumed. )

4. 2.

.L!l

.I!!Si

T be a L

torus ,

p : V

-+

T

the WIer"l covel'T - V/ L.

1ng at

T,

� W

be a t1nite retlect10n group act1ng ettect1vely on

!!!!l

10

the (latt1ce) kernel at

(b )

T

L generated by the translat10ns

the sublatt1ce at

orthogonal to the retlectlng hYperplanes .

ll!e

p, so that

the stabilizer ln

Assume (a)

W at each polnt t g£.

L - 10 .

T !L!.

retlectlon subgroup. This tallows tram 3 . 6 (see (1) and (3 »

4.3.

Remarks. Ca)

and 4 . 1.

It f'ollows tram 3 . 6 that (a )

equlvalent to , the lnverse lmage of' V, Vi. . LW, 1s a ref'lect10n group.

1s

W ln the af'f'lne group of' (b ) If' W 1s at type

OF LINEAR AlGEBRAIC GROUPS

ENOOMORPHISMS A • ES ' F l 4

37

G , the condition Ca ) always holds . (c ) It W 2 i s not ettect1ve . (a ) has to b e replac ed by ( . ) 11 10 i s torsion tree .

or

we

Even in tb1.s torm 4.2 tails it in (b )

tak e more than

one point (ct . 1 . 20 ) [although it does hold in some cases , e . g • (see 3 . 6 )

�

if

(*)

is ot type

may tai l to hold when

or

An W

Cn ] .

•

�e reason is that

is replaced by a retlect1 0n sub

group .

!! 4 · 2 drop the as§umpt1on that (a ) holds .

4· 4·

b e the group in (b ) flections in

Lllo .

Wt

•

WtO

� (a)

�t/wtO

Every §ubgrouP ot

(b )

11 10

Ob s erve that

the subgroup generated by the re is isomorphic to a subgroup ot

LIto

can b e reali zed in

i s Abelian and 1somorpb1. c to

the quotient o t the complete inverse image ot reflection subgroup (3. 6 again ) .

2! A ; ..!!:2! C.

Here to

1!!! A

� A

Then

�

b e an abstract group , AI C

!!!

cr

have an exact sequence

A CT

...!.>B . cr � « 1

P

0

( 1 - 0"') 0

C/ (l - a ) C

C

b y its largest

!!!!

a c entral subgroup "

the natural map

b e an automorphism ot

we

6 .

Dil loW ,

Clearly (a ) impli e s 4.2 and

denote the group

B B.

W

(a ) .

The proot depends on the tollowing result .

(b ) i ts converse . 4.5.

l:!!1 Wt

- CT)A " c)1 (1

-1

"

"

-

CT) C

A

--+

whi ch t1xes

1

•

b eing the natural map trom

C

•

The relevant points , that

6

is well det1necl, that

6

is

38

BOBERT

STEINBERG

a homomorphism, and that the sequenc e l s exact , are easlly verlfl ed .

,p . n : A of A

a ctlng on B

'l'W ,

•

morphism O m • t.

We apply �

B

,p . ViLa and wr1te tP 4. 5 with A . ,p.,p (sem1d1rect

we

4.4 ,

Returnlng to

let

to

effect ed by an element

(1 C

natural ldentlfi cation of

-

,p

of

w " .'w"

L k La

li La . If of 10 ; I f

with each of

and

wt

product ) ,

such that

C

of

n ls

because o f the

and

thls follows from the defin1tlon wrlte

w)

W

the l nner auto

To do th1s we must show that the kernel

c entral , whlch amounts to :

we

CT

the natural map , and

for

w"

w

l s a reflectlon

•

ls arb ltrary ,

a product of a

smaller number of reflectlons and then us e the equatlon

ACT - ,pwDo • B O" - TWt , dl' 0 - T , t and 0" acts trlV1ally on C . Thus 4 . 5 yi elds : Wt/ J ls t0 l somorphic to a subgroup of liLa . Bow .,po i s a reflectlon

1

1

-

w

-

group by

-

+

w'

4 . 2;

.' ( 1

hence

reflectlon subgroup t,

(1

ls ln

w) v

whenc e

..,0

wit ) . Bow

.woo t

WtO

so that U

1 t flxes -

-

L,

flxes

t l s also , ls l n fact the largest

of

Wt :

v

The subgroup ln ( a )

..,

ls chosen ln

henc e ln to

If

and

w

l s just

ls a reflectlon ln

V

above

t

Wt ,

then

10 by the defiD1tlon of 10 , e riPo • From th1s (a ) f ollows . t (1 - CT).,p fl C . Because of the

defln1 tlons , (b ) follows from :

4.6.

grOUp of flxed polnts of

C.

L . 10 .

Assume as ln 4 . 2 that

Then there '" sts

t

W e

T

.2!!

T,

�

so that

�

C o

C denot e th e

any subgroup of

ENOOMORPHISMS OF LINEAR ALGEBRAIC GRO UPS

W}

{ Cl - wl t I w E:

For c E:

C

an integer for every a

C • Co

V c E: P-1c ·

choose

The condIt10n

4 . 7.

II

E:

c E: t

C

39

•

As is well known :

is equivalent to :

(vc, a )

�

(� 3. 6 ) .

(1

- wa ) vc is is as in 3. 9 (b ) then

For the last condItion is equivalent to : a multiple of � a by 1. 5 . Now i f F WVc E: F for some w e: W , ( see 3· g ) . Since

we

may assume

Vc

E

a vertex of

F,

C is finite . GoIng baCk to

pwvc • wpvc • wc = c, F by 4. 7. I n parti cular ,

T we see that the elements of

Co

3 . 15 . Let t b e i t s centroid . For a fixed c E: C the equation (1 - w) t • c has a solution in W exact ly when t = wt + c does , i . e . , when Tn wTn + c does . SInce CO , the group of verti ces of Tn ' i s central , this last condition can hold only if CO - Co + c , i . e . , c E: CO . Conversely , if c E: Co then translation by c are the vertI ces of one of the cells

Tn of

=

is a sImplicial mapping of the complex

K

of Ib ,

so that

{; TI' , then n' • n as Tn - c = wTn t for some w E: W and we see by checking vertices , and the above condItIon holds . Thus n'

4 . 6 follows .

Now for t 4· g.

W,

a-

!f!! mal:

we

have

choose t

I Co 1 2t .

o.

Thus :

to be of finite order.

W,

et c . are as in 4 . 2 ind thit (a ) �. be an automo£ehism of T fixing a fundiment al cell for

4 · 9.

�

In 4 . 6

as above

As sume T ,

hence normal1 zi!!Bj

W.

.'!'l!!!!

Wa-

acts

i§

an �!f�ct�v�

ROBERT STEINBERG

40 retle ctlon group on 4 . 2 (a ) �. (1"

Liftlng we

see that

TI (1

and on

Tao

- at T,

and In both c as es

to a l1near automorphlsm ot

Wa 1 v(1" ls

an

V and uslng 1.32,

ettectlve retlectlon group.

Row

pYa

ls a c losed connected subgroup ot

T(1" tor whlch the quotlent ls A . V, B - T, C • L, � . p ; hence lt ls p : V � Tao l s the W11v ersal covering . (1"

dlscrete , by 4 . 5 with equal to Th e kerne l

(u

llr,

c

�

T� .

La

Thus

as ln 3· 6, n

u

u

n - .nu) , hence ls ou over the various �orblts on

lntegral ,

generated by the sums ot the

TT,

� . E nu�u

conslsts o t all translatlons

� 's u

1 . e ., by mult�ples ot the elements ot

1T�

Thus

( se e 1. 3 2 ) .

4 . 9 holds 1n this case . Row cons 1der the composlte natural map

T � T/ (l - (T ) T .

(1 - a)V

q :

V-+

V (1"

ls 1ts orthogonal c omplement ( slnce

t1n1te group q :

V � �

we

may assume

S1nc e

W and

a ls orthogonal) ,

1s onto .

T/ (l - a ) T

l s ln the kernel and

The kernel ,

1 s Just the orthogonal proJect10n ot

L

on

(1" generat e a the restrl ct10n

Va n ( l - O") V

Vrr '

+

L) ,

hence 1s a

latt1ce g enerated by the pro Ject10ns ot the basic trans lat10ns

�

(u

u

c

11r ),

4 . 10 . w

c

so that (c t . 1.30) 4. 9 holds 1n th1s cas e als o . Let eVeryth1ng be as 1n 4 . 9 (1 - w) t

W tor wh1ch (T

£2!m.

It becomes

we

-

proje ct -

wt . t ,

t

onto

c

(1 - a )T -

t

c

T/ ( l

.!U t

c

T.

'!'ben those

torm a retlection sub

- rr) T ,

the condit10n

so that 4 . 10 tollows tram 4. 2 and 4. 9.

In the same way

we

can extend 4 . 6 .

41

ENOOMORPHISMS OF LINEAR ALGEBRAIC GROUPS �

4 . 11 .

- La , and

�O a subgroup of

and

4.6,

T,W, L

there exists

t

E:

T

Thus

by 4 . 9.

be as in 4 . 2 ,

(1

conta1n1ng

such that { Ci - w) t / w

Theorem 4 . 6 holds with and W ,

C

IT

E:

as in

C

- a) C .

.l'!um

Wo- } ( 1 - c d Ttl C

CO .

=

T/ (1 - a) T and Wa in place of T

there exists

side of the desired equat10n equals

t

E:

T

Co « 1

such that the left - a) T n C).

'!hus 4 . 11

follows from : 4 . 12 . l!' ( 1 - 0"' ) c

T, C ..i!lSJ.

a

( l - a) T n C

are as above , �

=

•

The character group X of

T may be identifi ed with the As such

s et of l1near functions on V wh1 ch are integral on L. 1t 1s generat ed by the dual basis to

{2�/ (� ,�) I �

E:

TT } , i . e .

{cv� I (2U1a , 13 ) / ( 13 , 13 ) - 6 � ; � , P E: n }, and c ontains E and "IT (see ) . 6 ) , via the usual 1dent1fi cation of V and its dual. If

by

E:

T,

all

�

E:

� (t )

=

0

t

we

have the equivalent conditiona : t

on

TT .

C� w�t

•

t

for

IT, by 1. 6� c.&> p « l - w� ) t ) - 0 for all � , P s TT; for all � s TT, by 1. 5 and the defin1tion of uU . p

Now as sume for all

E:

�

t s

s

T

such that

( 1 - cr )t

s

C.

Then � « l - alt)

TT, i . e . � ( t ) i s constant on each orbit of

=

0

a

Now orbit representatives project onto l1nearly in

dependent elements of Va (the pro jections have disjo1nt supports in TT) , hence restrict to linearly 1ndependent characters on ToO (see also the proof of 4 . 9 ) . By the elementary div1sor theorem [)l , II , p. 107] app11ed to the latt1c e g enerated by these characters and the latt1ce of all characters on

Ta O '

42

STEINBERG

ROBERT

t here ex1st s

to

r epre sentat1ve tto l

Thus

5.

s

e

s

T such that a Ct O ) • a ft ) £or every orb1t � a , 1 . e . for every a s TT s1nc e t o £ T . � and (1 - e d t s ( 1 - a) e , whenc e 4. 12 £o llows . Our purpos e 1 s to

The trans 1t1on to algeb ra1 c tor1 .

c arry over the results of the prec ed1ng s ect10n of algebra1 c tor1 .

K b e an alg eb ra1cally c los ed fi eld and

Let

charact er1st 1 c exponent (see §6 ) .

p

1ts

An algeb ra1 c torus shall mean

an algebra1c group 1somorphi c to the produ ct of a finite numb er

K* .

of cop1 e s of the mult1p11 c at1ve group

L

w1 11 b e an algeb ruc torus , paramet er subgroups , the extens10n of

VILe tl

of

X K* ,

1 nto

K*

o£ charact ers of 1s a latt1c e 1n

�,

1R/1l - duality Wlth

�

both

.!!!!2

X

(a ) E:

t £ T

X

T

T

a

V

the torus

1 . e . homomorphi sms

Z - duality Wlth

X

Obs erve that

(algebra1cally) and

5 . 1. that 1f

T.

rJ!l ,

1nto

to a real vecto r spa ce , and

[19 , p . 40 5 ] ) , by extens10n 1n la- duality Wlth 1n

T

1ts ( latt1 c e ) group o£ one

1 . e . homomorphisms of

L

'!'he group

In what £ollows

V,

L

( s ee

and finally

1s 1n duality Wlth

(topolog1cally ) .

a For each t £ � there exi sts t £ T � X ( t a ) = 0 1f and only 1£ l ( t ) •

o£ £1n1t e order prime to

p

�

o.

(b )

c an b e reali zed 1n (a).

This bas1 c lemma and 1ts proof are due to T. A. Spr1nger. Let

Xc

be the annihilator of

h1 lator -u-t-

Xc

1n

T.

t

a

The group

1n

XlXc

X

and

TO

the anni

1 s a £1nitely

g enerat ed Ab eli an group , henc e 1 s 1somorph1 c to a d1rect product

ENDOMORPHISMS OF LINEAR A LGEBRAIC GRO UPS

4)

Si nc e it is also isomorphi c to (to the group of values I (t a » , the finit e

o f a latti c e and a finit e group . a subgroup of

K*

group i s cyc li c [) l , I , p . 112 ] .

TO '

Thus

the dual of

X/XQ ,

i s the product of a torus and a cyclic group , henc e it has a g enerator , say

t,

T he annihi lator o f t , i . e . of dua11ty , so that (a ) holds . the pro cedure and def1ne in

rfi

on

t

lQ

that

t

of the annihilator imply that

X/lQ

If a

R emarks .

To ' t

in

I

( s e e [ 1) , p . 1)6] ) .

Xc>

is

i s as in (b ) ,

b y topologi cal

we

can reverse

a s a generator of the ann1hi lator

Xc

of

t

in

X.

The assumptions

1 s finite of order prime to

i s the a nnihi lator of

5 . 2.

T

as a c lo s ed subgroup of

Ca )

ta

p,

so

and (b ) follows .

'!he various ass ertions about anni

hilators a nd duality b ecome transparent o nc e c ompatib le bas1 s of

I

and

lQ

are chos en [)l , I I , p . 10 7 ] .

(b )

K

In case

i s not the algeb raic c losure o f a finite fi e ld , one need only assume 1 n 5 . l (b ) that the clos ed subgroup generat ed by modulo its ident1ty c omponent has order pr1me to Each automorphis m , i n fact endomorphism , o f on

V

and on

T

-

V/L.

A group

t

taken

p.

ra

a cts also

W of automorphisms of

on

L,

ra

will be c alled a reflection group 1f it 1s so on

V,

and

s1m1 larly for an automorph1sm fix1ng a fundamental c ell for 5.). hold 1f

�

V

T

Theorem.

W.

The results 4 . 2 , 4 . 6 , 4 . 10, 4 . 11, 4 . 12

1 s replaced b y a n algeb ra1 c torus

ra

and 1 f

L

are defined as above . Let

the reqUired results b e lab eled 4 . 2 ', 4. 6 ' , etc .

The

ROBERT STEINBERG

44

p roof depends on the following ext ens10n of 5 . 1 . 5.4.

It

r,

morPhism of only if

� ( C1 (t »

t

a

•

a re as 1n 5 . 1 , U

oW t

and 3,f

now 4 . 2 ' w s W, the

(w

-

l)t

4 . 2'

of.

0,

•

•

T

w1t h

X

Assume

t

(see 4 . 6 ) , and let

cO

be the ann1h1 lator of

Xo

1n

T.

a

cO

S1nce

wt

a

�

rf!

(1

- w) Co

e W,

-

as 1 n 4 . 6 .

By 4 . 6

we

«.

wt . t by 5 . 1.

b e def1ned tor

'!hus

rf1

X

and

Co

X/Xo ,

so 1s

'b

may as sume

t

Let

p

(1t

and then also

Co

(l - w) C � •

S1nce

1 s a subgroup of

C

the ann1h1 lator

- w- l ) X k Xo

(1

as

f!1.

b e any subgroup ot

1.n

1.t tol10ws that so that

0,

0

Choos e t as 1.n a • t may b e written

(In tact the three groups are 1somorph1 C ) . w

1.f and

1s ot f1n1.te order pr1me to

I(*n ) ,

1s a subgroup ot some

tor all

0

r.

e

cond1.t1 on

Now let

X

replaced b y

henc e 1s e qU1. valent to

follows trom 4 . 2 .

1 s for

Xo

For

•

O.

Cons1der a

1.s any endo

.1cJWl x (<« t a ) )

l e X,

' Th1.s tollows trom 5 . 1

5 . 1 (a ) .

«

'

0

and then that

C.

Choos e

t

e

T

1 s o t t1n1te order , and

s 1.nc e the order ot 'b 1.s pr1me to p we may replac e t by an k appropr1ate tP and assume further that the o rder ot t 1 s a pr1me to p . Now choos e t a s 1 n 5 . 1 (b ) . For w c W the cond1t10n tor all

w'

by 5 . 4 , 1t

- w)t c C s W, hence

(1

we

wr1t e

C1

'

1s equ1valent to ( 1 - w ) ( l - w )t . 0 a a by 5 . 4 . '!hus , aga1n to (1 - w) t c C for

the correspond1ng group 1n annih1 1ator 1n

X,

{ C l - w) t l w

rf1 , then

wh1 ch 1s 1n fact

Cl

lQ

e W} and s1nc e

n

C

�

and

�

tor

have the same

� .

CO '

the

ENDQl«>RPHISMS

OF

LINEAR ALGEBRAIC GROUPS

45

annih1lator ot :to. Thus Xo 1s the ann1h1lator 1n X ot both C ao and C al • Be1ng £1n1te , each ot the latter groups 1s conversely the annihilator ot :to in rae Hence they are equal , whlch ylelds 4. 6 ' . In the same way we can deduce 4. 10 ' , 4. 11 ' , and 4 . 12 ' trom 4 . 10 , 4. 11 and 4. 12 . Alternatively, we can parallel the proots given In i4 . We omit the details . 5 . 5 . Exercise. Assume ra and W as above , that L - La , and that E in 3 . 6 is identitled with a subset ot X in the natural way. It (f is an automorphism of ra which flxes E and some t a £ ra such that Cl (ta ) + 1 tor every Cl £ E , then c:r tlxes a fundamental chamber tor W. (Hint : use 5.4. ) Structure ot algebraic groups . In this section we recall the main propertles ot 11near algebraiC groups and introduce some notatlons which will be used henceforth. The basic source of thls material is [19 ] . K will be an algebraically clos ed tield , over which all algebraic groups Will be taken. K* w.J.ll be the multlpllcative group ot K and p Its characteristic exponent (whlch Is the chara cteristl c of K or 1 according as the characterlstlc ditfers from 0 or not ) . Let G be a connected (linear algebraic ) group. Then G has a unlque maximal connected solvable normal subgroup R , called the radical of G. It G/R is glven the structure of an algebralc group , it has a trlvla l radlcal , 1 . e . , is semislmple . The semislmple groups turn out to be products , with some amalgamatlon ot (tlni tel centers , ot 6.

46

ROBERT STEINBERG

s i mple groups , Which have b een c las s1f1 ed by Chevalley [ 19 ] i n t h e Kl l11 ng- Cartan trad1t10n (types t hat

1s a s em1s imple group

G

b een chos en . group of E

X

in

A ,B , • n n

• •

, G2 ) .

Assume now

whi c h a maximal torus

T

has

will always d enote the (di s crete ) charact e r

T , written add1t1v e ly most (but not all) of the t1me ,

�

the sub s et of roots , and

group correspond1ng to the root

the one- paramet er Un1pot ent sub �.

We hav e :

X 1 s unipotent and 1s omo rphi c ( as an alg ebra1 c � group ) to the additive group ( ot K) . 1l x � 1s an 1somorphism !!:.2!!! K to X , � tx ( k ) t- l ,. x (� (t ) k ) for all � � � � �

6. 1.

k.

N group .

Wi ll denot e the normali zer of A s i s known

and

acts fa ithfully on

W

85

g roup i n the s ens e of

T

With

E

T

W ,.

NI T

as a reflect10n

as a correspond1ng root

system , so that the notati ons now that an ordering of ( res p . TJ ) wh1 ch

�

w , et c . may b e us ed. � has also b een g1ven. Then

wi ll denot e the group generat ed by tho s e 1 s pos itive (resp . negat1v e ) , and

g enerat ed by

T

and

B

Assume

U

X�

for

the group

U.

i s a maximal un1pot ent subgroup of

G,

�

1s a Borel (maximal c onnect ed solvab le ) subgroup of

G.

(b )

(a )

6.2. B

E

the Weyl

U

(� > � and from T x U

The natural maps from the Cart esian product arb 1trary order of the factors ) to are i somorphisms o f var1 et1 es . U- x T >( U

12

G

CC )

U

TI x

0;

�

B

'lb e natural maP from

1s an 1somorph1sm onto an open subVariety of

G.

ENDOMORPHISMS OF LINEAR A LGEBRAIC GROUPS

W,

{� I w

l!!

6.).

1 . e . £or

e

!e

NIT ,

Wl

b e a system o£ representat1ves £or {nw l

(a)

N.

47

1s also a system o£ re

presentat1ves £or the doub le c os ets o£ G relat1ve to B . (b ) - 1 ..... 1£ we s et Uw - U " Dw u n w ' then each element o£ BnwB can be written un1quely ulnu2 � ul e U , n e �T , u2 e Uw • In terms o£ a

X

�1nvar1ant 1nner product on

extens10n to a real vector spac e

we

V,

and 1ts

have the c learly equ1-

valent cond1t10ns , each o£ Wh1 ch may b e taken as a de£i n1t1 0n :

6.4.

The s em1s1mple group

o£ the £ollow1ng holds . subgroups o£

T e

{2aI Ca , a ) l a

s1mple root ]

valentl, ! so that p.

6. 5

X

L 1s the group o£ one-parameter 1s the subgroup generated by

10

There eXists a bas1s {w� l � so that ( 2 w � , P ) / ( P , P ) .. 6 aP or. eQUi

El, �

,g!:

1s s1mply c onnected 1 £ one

II

(a)

In

i!!SL

G

L ·

(b )

wawp . w p - 6ap a

Reduct1ye groups .

£or ' all s1mple roots

A connected group

�

!DS

1s called

C

reduct1ve 1£ 1ts rad1 cal 1s a torus , or , equ1valently (see [) , p.

63 ] )

1£ it can be decomposed

s1mple group and T'

T'

G - G ' T'

a central torus .

G'

with

a s em1-

In th1s decompos1t10n

1s un1que ly determined as the 1dent1ty component o£ the

c enter o£

G, and

G'

as the der1ved group o£

G

or e lse as

the largest sem1s1mple subgroup . 7.

Endomorphisms o£ algebra1c groups .

prove 7 . 2 and

7. 5

below.

Let

Our purpose 1s to

G . be a l1near algebraic group

ROBERT STEINBERG

48 and

a an endomorphism of

G , aG, aZ G , . . .

Now

is a decreasing sequence o f clos ed subgroups

[ 19 , p. 304 ] ,

eIlG

hence must level off :

Since all these groups conta1n

G by aP G ,

rep lac1ng

Ga '

We want to study

G.

0"

that

we

GO" ,

=

aP +lG

n.

may as well assume , after G.

G o nto

maps

for s ome

Here are some

elementary results which hold in this case .

7· 1-

�

G � G.

omorph1sm of

GJ

the c enter of

II GO

(a )

0"

Ker

If the c enter of

is sem1simple , then

ker

0" ...

is finite . in particular

GO

all

C H,

(c)

�

!! R 0"

Ker Now

cr

ker

aH . H .

Il

0

by [ 19 , p . 304 ] ,

has dimension i s containe d in

GJ ,

the map

Go '

aH .

erG

would have

i s a di s crete

it must b e c entral .

0"

which proves ( a ) .

Since

ker

0-

is finite

crH C H

Thi s implies that i f also H,

n

ker ell � ker ell- I is onto I so that I ker ell I • n Since ker a is contained i n the c enter of GO ' it

follows that if the latter group is finit e then

index in

�

is onto , it follows that for every pos itive integer

I ker a l n •

cit . ) .

H

hence i s finite .

ker a

Since

Il

then so is

since otherwi.se

G.

is an auto

is conne c t ed and

G,

G l .3:.!!!m O"R . R.

normal subgroup of the c onne c t ed group 0"

H

!!

is the radical of

fewer connected components than

Since

(b )

G.

is normal in

H

CT

so that

1I

:y!.!!! dim O' H ... dim H.

G,

an end- ·

is finite and contained in

morph1 sm of the abstract group underlying subgrouE of

CT

G b e a linear a�eb rai c grouE and

hence is equal to

H

in case

I ker

cr

I

...

dim cr H • dim H then H

1, (lo c .

O"H 1 s o f finit e i s conne ct ed .

If

49

ENDOMORPHISMS OF LINEAR A LGEBRAIC GBO UPS is normal and. x is arb 1 trary in G , we may wr1 te x - ay and x (aH) x-l • (ay) (aH) (ay- l ) • a l yHy- l ) - all , so that o-H H

is also normal.

Since

which is normal in

is a solvab le connected subgroup

o-R

G

by a part of (b ) ,

we

have

o-R � R ,

and

equality holds by another part of (b ) . Our main aim is the proof of the following result .

�

Theorem.

7. 2.

an endomorphism of

0"

be a l1near algebraic group and

G

G �

�

G,

0"

fixes a Borel subgroup

2! G . Th1s follows from : Lemma .

7. ) . cr

�

be a connected l1near algeb rai c rsroup ,

G �

an endolllOrph1 sm of 0" .

G

� II

ihen the map

:

G,

a Borel subgroup fixed defined by lI (x , b ) - xbCTx- l

and B

G X B .-. G

i s surjective . 0"

If element of

is the identity , this reduces to the fact that every

G

is con3ugate to some element of

lemma for the moment , let be a Borel subgroup of

G.

by 7 . 1 , so that

( *) l' �

Applying 7. ) to

1yCT

and

b £ B,

see that

0"

G

CT

and Then

1'- 1 .

B

as

B.

be as in 7.2

Assum1ng the and let

B

is also a Borel subgroup

l' in G [16 , p. 60 9 ] l Xby . ax- . y- l for some x £ G

for some

get 1'-1 . whence l' - b- lx- lox . Substituting i nto ( * ), we fixes the Borel subgroup :xBx- l , wh1ch proves 7. 2, we

mod 7. ). Since the rad1cal of B,

we

need only prove 7. )

G

is a normal subgroup conta1ned in

when G

is s em1s1mple , wb1c h

we

50

ROBERT STEINBERG

The subset of x- lyox

e

B

G )( G

!' ,

X

B

1s c losed .

proJe ct10n of

S

ls also c losed. dCl

cons1st1ng o f all

G/B

we

dCl

maximal torus

T

(l ,b )

We c a n choose

1n

B

G

and

G

at

b

the CI ,

that of B.

-

A

1.

slmple

followed by translatlon to

maps the tangent space to

(1 - d C1b O" ) ) .l .

c all

show that at some pOlnt the dlfferentlal

w1th the tangent space to

c alculatlon shows that +

ls complete [19 , p . 609 ] ,

be the Li e algebra of

-

-

G/B )( G ,

on the second factor , 1 . e . the 1mage of

Next

We 1dentlfy g

e.

such that

hence lts proJectlon 1n

1;

S1nce

Let g

1 s onto .

G

(x , y )

1 s closed.

ls closed and conslsts of complete cosets relat lve

to the subgroup 1t

CI

We f1rst show that the 1mage o f

henceforth assume .

1n b

G >< B so that

1

1n

onto 1bO" f1xes a

(by the conjugacy theorem [19 , p . 609 ] ) .

The negat1ve roots are then permuted ln orb 1ts , so that by modify1ng that

§6 ) .

b

by a suff1 c1ent ly general element o f

we

T

can assume

1b CT fixes no element ln the Lie algebra '2.- of Thus the above map covers .2 + � CI

f1rst part the 1mage of a nonempty open subs et of connected;

Corollary .

y:

G,

G

conne cted solvable subgroup of group of

G

f1xed by

0"

" ,

as reqUired .

(see By the

1s closed , by the second 1t conta1ns

hence 1t 1s all of

7. 4

•

tJ

whi ch 1s dens e b ecause G, and

which 1s
G

1s

7. 3 .

are as 1n

7. 2 ,

then allY

GeT' 1s conta1ned 1n a Borel sub

The subgroup acts by conJugat10n on the var1ety of Borel subgroups f1xed by

[ 19 , p . 609 ] .

CT ,

7 . 2 and complete by [19 , p . 514] there 1s a

wh1ch ls rionempty by

Thus by Borel ' s theorem

ENDOMORPHISMS OF LINEAR A LGEBRAIC GRO UPS

51

f1xed po 1nt , whence 7 . 4 . Next

we

present a re lated result wh1 ch will be us ed lat er .

G

An automorph1 sm of an algeb ra1 c group

Wi ll be called s em1-

s1mple 1f 1 t can b e ach1eved by conjugat10n by element of some algebra1 c group c onta1n1ng

a

G.

s em1s 1mple

If

G

1s sem1 -

s1mple , then 1ts group of out er automorph1 Sms 1 s f1n1t e s o that ev ery automorph1 sm group ,

a

can b e rea11 zed by conjugat10n 1n a larger

and the cond1t10n that (1'

e ach of :

k

s ome 1 nt eg er 7. 5.

�

1 s a s em1s1mple 1nner automorph1 sm for

prime to

G

p.

Every s em1 s1mple automorph1sm

Theorem.

algebra1 c group

1s s em1s 1mple 1s equ�valent to

acts sem1s1mply ( 1 . e . d1agonally ) on the Li e

(2)

G',

algebra of

a

(T .2!...J!!:!

f1xes a Borel subgroup and a maximal torus

thereof . Be caus e ot 7 . 2 this follows trom : 7.6.

If

G !n 7 . 5 1 s so lvable , then

a

f1xes a max1mal

�.

G . TU

We may assume

G

c onne ct ed , wri t e

a maximal torus and

U

1 s the normal s ubgroup c ons 1 st 1ng o f all

un1potent e lements of

G

[19 , p.

606 ] ,

s o that T

and extend

G

1s

so that

�

1 s rea11 zed by conjugat1 0n by the s em1 s 1mple element s . S1 nc e l s Tsand T are maximal tort 1n G , we have us Tsu- l - T for

s ome

u £ U

s1mple part

(by c onjugacy), and on replac1ng we

may as sume 1t to be s emis 1mple .

f1xes a maximal torus , 1 . e . that

is

us

by 1ts s em1 To prove that

CT

does , 1 t 1 s thus enough to

52

RO BERT STEINBERG

(*)

s how

if

us

i s s em1s imple it i s c onjugate to

s.

By an U,

obvious induction on the length of the derived s eries of we

U

c an as sume that =

u

-

that we

l

1U

2 SU lS

us

-1

i s commutative . I n thi s c a s�

U

with

u

l

i s conjugate t o u • 1, 2

must have 7.7.

Remark.

e U

and

u2 s .

u

2 e Us

[19 , p . 412 ] ,

so

The last element b e1ng s emis i�

whi ch proves

(*)

and 7 . 6 .

This result may b e vi ewed as an ext ension

to disconne cted groups of the fact that every s emisimple element of a connected algebrai c group is contained in a maximal torus . It has also b e en proved by Wint er [)) ] by a d1ff erent method .

8.

A basic theorem on connect ednes s . This s ection is de-

voted to the proof of the following result . 8. 1 .

Theorem.

algebrai c group and G�

� cr

G

b e a simply connected s em1simple

i s em1s1mple automorphism of

G.

l!!!m

i s a conne ct ed reduct1ve group. Because of 7 . 5 this is a consequenc e of tbe following

result , whic h is s lightly more general becaus e

�

need not b e

�

an auto

s em1simple . 8. 2 .

Theorem.

morphi sm of �

T

£!

Let

G

b e as i n 8 . 1 �

which fixes a Borel subgroup

G B.

!h!!!

G �

B

and a maximal

is a connect ed reductive group.

Sinc e the s imple components of

G

are all s1mply connected

( s e e , e . g . , 6 . 4 and the d1s cus sion at the start o f

§6)

and

�

ENOO)l)RPHISMS OF we

permutes them ,

LI NEAR

ALGEBRAIC GROUPS

G

may as sume that

the notat10ns o f §6 a s soc1ated with

53

1ts elf 1 s s1mple . B

and

T.

We us e �

S1nce

f1xes

on

X,

and on

mut1 ng the positive roots s1nc e 1t f1xes

B,

and for each root

T

CI

1t also acts on we

CF'lltb (k)

have

Ger

(1 )

NIT, 1 . e . on W , X ( C k) CI CfCl

for all

k £ K

1s generated by

Ucr ..IDQ.

Ncr .

•

E,

per

and some

c

Ci £ {".

'l'h1s follows from

the uniqueness 1n 6 . 3 . (2 )

U

the s et

cr

( s e e 1. 22 ) are proport10nal to that of

cr

CI .

we

of roots no two of wh1 ch add up to a root ,

A

2n

( 2" )

CI , CfCl , P

wClwerCi

• • •

In

a

E

1s

The correspond1ng reflect10n 1n 1n the first cas e and

wp

1t follows that 1f

S 's CI Thus ( 2 ) 1'ollows from : The part 01'

1s part of a s1mple

� .

supported by the various

(2"" )

whi ch o ccurs only 1f

{CI , OU , • • • }

both cas es above

From 6 . 2 (b )

U • cr

ClIJ ,

CI +

(b )

.

syst em of roots fixed by (2"')

•

conc lude :

( a ) an orb1t

1 s of one of the following types :

CI

s et of the form of type

V�

pres erves the s et of pos 1t1ve roots , and the

S

{a , erCl , • • • }

form

Us1ng 1 . 32 ( b) , the

c las s1f1 cat 10n of simple root systems [ 20 , p . 13 0S] , (2' )

we

of all ( pos1t1ve ) roots whos e project10ns on

S CI

fact that

CI

1 s connect ed . For each pos1t1ve root

Uer

W cr

( see 1. 30 )

1s

1n the s econd . we

form the parts of

and take their product

support ed by each

SCI

we

Ucr get

1 s c onnected.

54

ROBERT STEINBERG

a nd 1n c as e ( 2 ' b )

' 1£ and only 1£ 1n cas e ( 2 a ) c c • « �«

1

It 1s d1££erent £rom

1,

c c . ! " u"

x" (x")x� (xa")

� ,. CCyXay £or X" , koa, . . .

are det ermined by the equat10ns wh1 ch have a nonzero solut10n c c " oa

• • •

•

x p (lU. )

xp

x"

1 n wh1ch cas e

x' s

the other morphism

1,

B ecaus e o£ the equat10n

from th1s relat 10n , the cond1t10ns £or

- o.

u

u

(y . " ,a ", . . ) , .

only 1 £

we adjust the homo

ex" (k) 'Xotl (t » ,. c c . -c p whi ch £ollows " � x" (k")Xa "(xa")xp (kp ) to C "k« , k" k� + (1 + c"ccm)xp Jus t be o . 1£ c c - 1 " CT«

c a otI ' Xoa " then all x ' s

•

';' ! 1 , c c " cm p � 2 then k may b e chos en arb1trar1 ly and the other «

x's

solved for . kp

then

T a

1 s connect ed .

w

li:

respond1ng s et ,

�

(here

w

S

"

Q1: ( 2 ' )

there , all components repres ent ed by generat ed by a s requ1red .

n

U;UaU; . ul , u

with

u 'n,u l 2 • • •

,

CT

permut es the

such that the cor u

B y 6 . 3 with 2

e

UCT -

e

t'a ,

are 1n

1,

rr

and

(.!!.!!

{l }

N by an element o£

1 s d1££erent £rom

w" ,w a" ,

Wa

supports some

1s repres ented 1n

u - u nu l 2

k's

p . 2 1s allowed ) , 0 , whence ( 2" " ) .

X.

1 s a re£lect10n of

£act by an element o £ changed ,

- -1

Th1s 1s b ec aus e

{lU" I o f

elements o£ the bas1s

(4)

c c " oa

F1nally , 1f

b e comes arb1trary and the other

(3 )

2

x

x" ,.

are :

I£

and

un )

x

fi ed by

so that the commutator relat10n

ho lds .

b e fixed by

• • •

may b e chos en arb1trar1ly and

In cas e (b )

solved £or .

l

and then 1t 1 s a one- paramet er

In case (a ) the elements

group.

• • •

111

G

U

ao '

1nt er

and by the uni queness Ga .

The element o£

W

1s 1n the group

and 1s fixed by

a ,

henc e 1 s

w,

ENDOMORPHISMS OF LINEAR (5) root

N n

Choose

l!!!n

(I .

of thos e £

NO' p

T

(I (t ) . c a (l

s o that

1 s generated by

Wcr

w £ Wcr .

c 's £

(1 - 0' ) T we

S econd 1 f

(I , a (l, •

of ( 2 ' )

• •

.

First ea ch p

define

are all s1mple . 0"

. as s ert10n follows from ( 2 ) and (4 ) WJ.th

t n l

£

0' .

by

by

1.

Now if

NO"

for some

(6 )

Even

t

l

(1

whi ch works out to

£

w

£

£

n

T

N

replaced by

repres ents

p

£

(1 - O') T

w

e

W,

(O"- l ) n

if and only if

- w) t

and proves

whi ch is repres ent ed in

Wcr

But then our p

a nd

then e

(1 (5 ) .

Ncr

O' ) T ,

is a pro

The

duct of refle ctions ( i n the s ens e of 1 . 3 2 ) whi ch also are .

T. If the latti c e

§5 are applicab le to the torus

results of

T

of one-parameter subgroups of

and its dual

X

dual to the basis

Io ·

(7)

L.

{w l (l

L

for the corresponding s et

containing

We use

and

«I +

w £ WO'

It

(5)

S

(see 6 . 4 ) , ' ( s ee 5 . 3 ) . and 4 . 10

(5)

Ncr '

is repres ented in

�

2! (2 ' ) equation (a ) 2£ (b ) 2!

(I

WJ.th

B

replac ed by the Borel subgroup

T and corresponding to a s imple system of roots

" chos en as 1n ( 2 ) .

(1 - w ) t

(2cz/ «1,(1) 1

is just

ihus (6 ) follows from

If a reflection

( 2 "" ) holds .

X

of

L

are imbedded

in a real vector space in the usual way , then the b asi s of

so that

�P,

W , then each w £ WO' 1s represent ed by an element p By 1 . 3 2 1t 1s enough to prove this when w has the form

1 . e . , when

the

for every s1mple

TO' and representat1ves 1n

(1 - w)t

such that

55

=

WO" •

N

wn '

e

Na

repres ents a

so that of

w

e

t

GROUPS

ALGEBRAI C

(1

We apply "'

- O' ) T.

am ) ( t2 )

. 1

(I +

GJOtI +

By 6 . 4 and ( 2 "' )

• • •

we

get

to the cond1tion

«I + atI+

in cas es (a ) and (b ) o f ( 2 ' ) ,

b ec aus e of the definition of

t

• ••

)( t)

• 1

whi ch

are equivalent to ( a ) and (b ) of

ROBERT STEINBERG

56 ( 2 "" ) , as requ1red. Row we

can prove 8. 2 . Let Gl be the group generated by TCT' Uo- and U; . By (6) , (7 ) , (2" " ) and (4 ) we get RO" S Gl , so that GO" . Gl by (1) . Then GO" 1s connected by (2) and (3) . x.t R be the unipotent radical ot GO" . By 7.4 there � sts a Borel subgroup ot G conta1n1ng R and t1xed by 0- wh1ch we may take to be the group B above . Thus R C UO" . It tollows trom (2"" ) and (4 ) that the only subgroup ot UO" normal1zed by RO" 1s 1. Thus R . 1 , GO" 1s reduct1ve , and 8. 2 1s proved. 8. 3 . Remarks . ( a ) It should be clear that UO" and Rcr torm the mak1ngs ot a Bruhat decompos1t10n 6 . 3 ot Go- ' and that the correspond1ng root system may be 1dent1t1.ed with a subsystem ot the system tcr ot 1 . 3 2 . (b ) It every w c WeT' 1s represented. 1n RO' , tor example 1t 0" - P wi th P as 1n ( 5 ) , then GCT 1s even sem1s1mple s1nce T(f' and to' have the same rank. ( c ) In case CT 1s 1nner the s1tuat10n becomes transparent , even when G 1s not s1mply connected. It 0" - 1 x with x c T, then 1 t tollows trom 6 . 3 (b ) that Gx 1s generated by T, those � tor which CI (X ) - 1 , and those n. tor wb1ch wx - x , so that GxO 1s generated by T and the � alone , hence 1s reduct1ves ( see [19 , p. 17-02] ) . It .x (resp . WxO ) �s the subgroup retlect10n subgroup ) ot W generated by all w ( resp. such that wx - x (resp . CI (X ) - 1) 1t tollows that 1s 1somorphic to WxlWxC . The s1tuat10n 1s s1m1lar to that ot 4 . 4 , but not exactly so s1nce 1t can happen that wClx - x even J

ENDOMORPHISMS � (x ) �

though

1

OF

LI NEAR A LGEBRAIC GROUPS

(see 6 . 7 (b ) below) .

If

G

57

is simply connect ed

this pos s1b 1 l1ty i s obviated as in ( 7 ) above which becomes very simple when

(j"

i s i nner , so that

.x

-

.xC)

by 4 . 2

•

and

Gx

is

c onne ct ed . 6. 4 .

Corollary

•

.

Ev ery e lement of

contai ned in a Borel subgroup fixed by

G

fixed by

a'

!!

a'.

Since

G is connected , each of its elements i s contained a' in a Borel subgroup , so that 6. 4 follows .from 7 . 4 . 6. 5 .

2t G

The c entrali zer o f a s em1s1mple element

Corollary.

is c onnected. This i s the special cas e o f 6. 2 dis cus sed in 6. ��

2t

G

Ev ery two commuting s em1s1mple elements

Corollary .

6. 6 .

are contained in a max1mal torus . If

x

and

manmal torus of

are the elements ,

�

whi ch contains

Remarks

6. 7 . i n 6 . 6 or

y

"a "

by

(a ) "two"

we

need only choos e a

y.

" Two" can not b e replac ed b y "three"

i n 6. 5 since the s em1s1mple part of

the c entrali zer i n 6. 5 need not b e simply c onnect ed (cf. 9. 1 b elow) •

(b )

The extent to whi ch s 1mpleconnectednes s is nec es sary

for the above results is studi ed in the next s ection;

that it

c an not be entirely dropped is shown by the following example : G

PS� (K ) ,

x - d1ag (1 , - 1 ) , 1 2 .. -1,

0'

GCT

1

con x . Here s i sts o f the monomial matri c es , so that 6. 2 , 6. 5 and 6. 6 all -

-

fai l , and s inc e the only Borel subgroups fixed by

0'

are the

58

ROBERT STEI NBERG

s uperd1agonal and subdlagonal subgroups 6. 4 does als o . Finally , t o clos e th1s sect10n we sketch

an

alternat e proof

of the connect1V1ty In 6. 1 which 1n part parallels a proof of Borel for compact Li e groups [ 2 , p. 22 5 ] , other po1nts of 1 nt erest . 6 . 6. X £ G

Assume

G

Clearly 1t will b e enough to prove :

!.!!!!

CT

1 n c as e either (a )

aO

and which brings up

are as 1n 6. 1 .

x

!! x

£

1 s s em1s 1mple of (b )

x

G , O'

�

!§.

un1pot ent . Part ( a ) follows at onc e from : 6. 9. group

B For

that torus CI

CT

x T,

x

1! Ca ) ho lds . then and a map-mal torus

T

1 s conta1 ned 1 n S2l

B,

a

Borel sub

both fixed by

0' .

1 s connected ( s e e (3 ) above ) . In 6. 9 as sume first O' 1s regular . Thus x 1s conta1ned 1n a uni que maximal T

c learly fixed by

( s e e , e . g . , [ 2 6 , p. 53 ] ) .

0',

and

Then

CI (x)

L . �

� 1

for every root

as In 6 . 4 ,

whenc e

f1xes a pos1t1ve system of roots by 5 . 5 , hence also the

correspond1ng Borel subgroup ( s e e 6 . 2 ( a » . duct1ve ( even If

G

S1nce

�

1s re

1s not s 1mply connected; s ee 6 . 3 (c » , and

the s et of f1xed points of an automorphism of a no ntriV1al s em1s1mple group 1s always 1nf1n1te ( s e e 10 . 5 below) , the g eneral c as e of 6 . 9 may be reduced to the first cas e as 1n [2, p . 225 ] . We om1t the details . For the proof of 6. 6 (b ) the as sumpt110ns can b e cons1der ab ly weakened .

ENDOMGRPHISMS OF LINEAR ALGEBRAIC 8. 10 .

GRO UPS

The conclusion of 8 . 8 (b ) ho lds if

c onnected linear algebraic group (and

�

G

59

1s any

is sem1simple ) .

Since this i s proved for s olvab le groups in [19 , p. 602 ] , it follows from :

!!! 8. 10 every unipotent element u fixed by

8 . 11 .

c ontained in a Borel subgroup fixed by By 7 . 2 the automorphi sm G.

i urT

rT !.§.

rT .

fixes a Borel subgroup

B

and

0"

do als o ,

The case of 8. 10 i n whi ch

�

is i nner is

Its unipotent and s em1simple parts

iu

of

whenc e 8. 11 . 8. 12 .

Remark.

proved in [19 , p . 615 ] , but the proof Just given is simpler.

9.

Extensions of 8. 2 .

It wi ll b e convenient to formulate

our results as in 8. 2 in t erms of automorphisms (of s em1simple groups ) whi ch fix a Borel subgroup and a

maxi mal

torus thereof ,

which wi ll b e c alled quasis em1simp1e (abbreVi ated quas s ) .

We

obs erve that every s em1simple automorphism is quass , by 7 . 5 , that a quas s automorphism i s s em1simple i f and only i f its c las s in Aut GlInt G Aut GlInt G

has order prime t o

p,

contains a quas s element .

and that every c lass in The main result i s as

follows . 9 . 1. rT

Theorem.

�

G

a quass a utomomorphism of

sal COVeri ng ,

F

b e a s em1s imple algeb rai c group and G.

the kernel of

�

n :

G' � G

b e the univer

n , i . e . , the fundamental group

ROBERT STEINBERG

60

G , !!!!! C ' the center ot G ' " (a) Gr/Gao is isomorph1c to a subgroup ot F/ (l - cr ) F wh1ch contains «l- cr)C ' f\F) /(l-crlF. (b ) !! cr varies over its class in Aut GlInt G , every such subgroup is ach1eved.

2!

In the statement and proot ot th1s theorem, the tollow1.ng tacts about the un1versal cover1ng are used. G ' is a simply connected selll1 s 1mple group and " is an isogeny (1 . e . a sur j ect1ve homomorphism with a t1n1te kernel , a part ot the center ot G ' (ct. 7 . 1» . Every (quass ) automorphism a ot G can be lilted uniquely to one ot G ' , also denoted cr (this 1s proved in 9. 16 be lo w) . The act10n ot t:T on F and C ' depends o nly on the class ot Aut G/Int G to wh1ch t:T belongs . (a ) 710:" . Gao " ' ( ( 1 - a) G I'\F)/ (l - cr ) F. 9. 2.

�

Lemma .

(b )

Gc/Gao is isomorphic

We apply 4. 5 with A , B , and C replaced by G ' , G, and F. By the exactness 1n the th1rd pos1t10n 1t 1s enough to prove (a ) . Now G:" is connected by 8 . 2 . Hence so is nG� . Thus nG� � Gao ' Also nG� . ker d 1s a closed subgroup ot Ga , ot t1nite 1ndex since F 1s t1n1te . Thus nG� � Gao and 9 . 2 tollows . Part

(a ) ot 9. 1 tollows at once trom 9. 2 (b ) .

9. 3 . Corollary. Gcr 1s connected 1t and only it (1 - a )G' n F • (1 - cr )F •

9 . 4 . Corollary .

Gao

1s a reduct1ve grouP . and G"IGao

ENOOJI)RPHISMS OF

LINEAR ALGEBRAIC GROUPS

61

1 s Abellan and cons1sts of sem1s1mple elements . The f1rst result follows from 9. 2 (b ) , the second from 6 . 2 and 9. 2 (b ) . ' ' for G ' Tb prove 9. �) we use the notat10ns B , T , • • •

J ust as we used B , T, . . . 1n the proof of 6. 2 . 9. 5 . lemma . (1 - 0-)<1 n F - {l - wh ' Iw £ W!--} Cl - 0") T ' n F. Assume f . (1 - o-)x w1th f £ F , x £ G' . Then fa x - x . If x . ulnu2 as in 6 . ) w1th n represent1ng w £ w' , then the uniqueness implies that w, ul and u2 are all f1xed by 0-, so that f . (1 - o- ) n. Proceed1ng as 1n ( 5) of the proof of 6. 2 , we get (1 - 0-) 1) £ (1 - w) t ' ( l - cr) T, whence 9 . 5 . Now we can prove 9. 1 (b ) . let T . nT' , a maximal torus of G. The elements of 1� are all quass . O n lift1ng them to G ' and comb1n1ng 9 . 2 , 9 . 5 , and part 4. 11 of 5 . ) (which 1s applicable because W , acts trivially on the center of G' , 1n part1cular on F) , we get 9 . 1 (b ) . Remark. It follows from 4 . 6 and 5. 1 (b ) that 1n 9. l (b ) can get by Wlth automorphisms of f1n1te order. 9. 6.

we

9. 7. Corollary. � G be a sem1s1mple algebrai c FOUP and F 1ts £'undUiental group. Then the foll0!!!l.M con41t10ns on a class S � Aut GlInt G are eqUivalent . (a ) the act10n on F of some (hence of every) 0- £ S 1s f1xed-p01nt-tree. (b ) For every guass 0- S S the group Go- 1s connected. S1nce F 1�

tin1 te ,

(a ) 1s equ1valent to (1 - 0-) F • F ,

62

ROBERT STEINBERG

hence to (b ) by 9 . l (a ) . As an 1l1ustrat10n ot how (a ) we

1t c learly general1 zes ,

a

(b ) go es b eyond 6. 2 wh1 ch

have :

1s the adJ01nt group ot type

, E6 2[ D4 2n 1 s any qUiss outer automorph1 sm 1n the t1rst two c as es

9 . 6.

ASS

!!

-.;>

G

A

or guas s tr1al1ty automorph1sm 1 n the th1rd cas e . then connected . 9. 9.

are

tollowing cond1t10ns F

1s tr1v1al..

2!

G 2! G

For a s em1s 1mple algebraic group

Corollary.

(b )

eqUivalent .

(a)

G

lh!

The fundamental group

The c entrali zer ot every s em1s1mple element

1 s c onnected .

(c)

Every two c ommut1ng s em1s1mple elements

are conta1ned 1n a maximal torus . Here ( a ) and (b ) are equ1valent by 9 . 7 w1th

and (b ) 1mplies (e ) as 1 n 6 . 6 .

It ( c ) holds and

s1mple , then every s emts1mple element ot

Gx

S . x

Int

G,

1 s semi

1 s conta1ned 1n

by ( c ) and every un1potent one by 9 . 4 (or by 6 . 10 ) , whenc e

GxO (b )

!§.

Ga

•

9 . 10 .

caut10n .

It

F

1s tr1 Vial

c onnected , although 1t 1s so 1n cas e

G

p . 1.

need not b e s 1mply In the g eneral c as e

1nseparab1 l1ty can o c cur [ 19 , Exp . 16] . 9 . 11 .

The 1mpli c at10n ( a ) � ( c )

Remark .

1 zed as tollows . elements such that

It

A

AIAo

1 s an Ab e11an subgroup ot s emts1mple 1 s generated by two element s at least

one ot whi ch bas o rder prime to that ot 1n a max1mal torus ot

G.

can b e general-

F,

then

A

This eas1ly tollows trom :

1s c onta1ned 1t

a

1s a

ENOOMORPHISMS OF LINEAR AIGEBRAIC GROUPS

6)

quass automorphism of G of t1n1 te order q pr1me to that of F, then Gcr is connected. To see this assume y £ Ga and choose -1 Y, £ n y , so that a y " .. fy with f £ F , whence f q • 1 , f 1, y ' £ G0"' , and y £ Ga O b y 9. 2 (cf. [2 , p. 227] ) . Next we observe that the lower limit in 9. 1 comes directly from the center of G ; more precisely : =

9. 12 . Corollary. 1£ C 1s the center of the following groups are naturally isomorphic .

G !a

9 . 1,

�

Here (a) and (b ) are isomorphic by 4 . 5 applied with C t , C , F in place of A , B , C , wh11e (b ) and ( c ) are because CCT () Ga o .. nc� by 9. 2 (a) . We observe that the group in (b ) measures the amount by which n : C� -+ Co- fa11s to be surj ect1ve. 'lbe equality of (b ) and (c ) may be stated : the natural map Ca/nC� -+ Ga /Ga o 1s 1n j ect1ve. MOre prec1sely one can show that 1f B and T are as usual then the maps Ca / nc� � To-ITao �Ba/Ba o are b1ject1ve . 9 . I) Corollary. .!!. G !m! S are as in 9. 7 .!!!S! C II the center of G , then the following cond1t1ons on S are egu1va !!!!t . (a) 1 - a) C t () F ( 1 - a) F for some , hence for every, (b ) The natural map n : c;'- � Co- is surject1ve. ( c ) CT £ S . Go- 1 s connected for some guass 0- £ S . •

=

Here (a) and (c ) are equivalent by 9. 1 , while ( a) and (b )

ROBERT STEINBERG

64 are by 9 . 12 .

G

We observe that the cond1t10ns o f 9 . 13 always hold When 1 s an adJ01nt group since then Corollary.

9 . 14 .

S

�

are equ1valent . �

for some . hence for every '

£ S.

(b )

(d) x

For all quass G

f1xed by

(e) .

(e)

(1 g.4

CT) x ' •

x' (1

-

c ,,

CI

as sume ( e ) . x ' £ G'

G'

x'

Choose •

and

CI

x b e a s in

0" £

(F

S.

Assume

G.

c we

and sinc e (e) holds

Then x'

and con-

is 1n the c enter get ( e ) .

quass . so that £ £ F.

cr

f1xed by

s1nc e

Projecting th1s group to

(see 9 . l (b ) . 9. 2 (b » . some

follows from 9. 1

Then (1 - CI)x ' c F. whence by (a) . , - 1 £ G ' £or some c c C ' • By CT) c . i . e . , x c � hence conta1n1ng

[19 , p. 609 ] .

0" £ S .

1s contained in a Borel

lx .

there exists a Borel subgroup of

ta1n1ng

S,

is isomorph1c

For every quass

Assume that (a ) holds , and let

Choose

£

�

For evert auass

The e qUivalence of (a ) , (b ) . (c ) and (d) and 9. 12 .

�) C' � F

0".

fixed by

G

subgroup of

2!

-

ao � £ S . Ga/G aO has the same

value . 1n the s ense of 1somorph1sm. every element

(a ) (1

� £ S , Grr/G

(c ) ' For every quass

F/ (l - �) F.

•

If eVeryth1ng 1s as 1n 9. 13 , then the

following conditions on G • C� GaO O"

F . C'

£ .

(1

_

(F) G

Now '

�

(1 - �) x '

11'

for

1s contained 1n a Borel

subgroup B ' o£ G' fixed by a . Applying '.3 and 4 . 12 to B ' modulo its normal subgroup o f unipotent elements , we conclude that £ £ (1

-

�)C ' .

Thus

(a ) holds , and 9. 14 1s pro.ved.

have used the fact that 1£

c £ T'

1s fixed by every

Here w £

we

W,

•

ENOOJ()RPHISMS

then

c

&

C ' . 'l'h1 s follows from the slmple cOl1ll8c tedness o f G' : (I

I f for every simple root •

w(lc

we

c,

65

OF LINEAR ALGEBRAIC GROUPS

(I ( c )

get

•

1,

we

apply W(I

to the equatlon

whence our as sertlon.

As an example to lllustrate 9. 12 and 9. 14 9.15 ·

!! p oj. 2 ,

G•

S0 2n , !m!

tr

we

have :

ls any automorphism

( necessarily outer) effected by a s em1s1mple element of not ln

l:!!!!!

S02n '

GCT has two components and

1

0 2n

a4 - 1

always lle ln dlfferent components . As a flnal example , typlcal of how some of the above results can b e extended to groups whlcb are not s em1s1mple ,

G

we

have :

If

ls a connected l1near algebralc group whose Sem1s1mple part ls

slmply conne cted , then the conc luslon of 6. 5 holds .

!be reader

may 14sh to prove this for b1mselt . Fina lly morphlsm of

we

G

prove the fact used l n 9. 1 , that ev ery auto can b e l1fted to

G' .

Tb1s follows from :

� G b e a sem1s1amle algebta1c group and n : G' � G lts universal coverlng. Let cr : G � G be an ' lsoge&. !ben there exists a unl que ls0ge& at : G -+ G' 9.16.

-

such that

'IT ' .

an

By modifying

•

C1

by an lnner automorph1sm,

a fixes a maximal torus X,X

'

-

T.

Set

T ' . n- 1 T.

we

may

assume

Let E , E '

and

be the correspond1ng root systems and character groups .

Let Y be the automorphlsm of the real vector spac e generated by * x ' defined by y n* • n CT* (here CT* denotes th e transpose of the restrl ction of

CT

to T ,

and s1m1larly for

'Ic'.)

•

By

ROBERT STEINBERG

66

[ 19 , p . � 9 ]

cr

the existenc e of

following tw propert1es of Y .

'

1n 9. 16 1s equ1valent to the

we

yx'

(2)

onto an 1ntegral mult1ple of another. 1s the un1veral cover1ng ,

maps every root 1n � ,

Y

(1)

( * ) n*�

have

an 1sogeny , for each � £ � there exists 1nteger

n such that a*�

and use

(* )

bas1s

B

X

'

and � £ � , •

II e � ' X'

X

X

'

we

We have (3 ) normal1 zes

S1nce

P

e

and a pos1t1ve

�

g et

If (1) .

we

1s

apply

X' ,

this and (3 ) ,

into peYB

i . e. , 1f

1s an integer (see 6 . 4 ) .

p . yllJ

We

W' ,

and

qll . y p with W e B and q a posit1ve 1nt eger by ( 1 ) .

the funct10n x , y � (Yx , Vy )

we

> (x ,y )

-

on

on each simple component .

also holds with cr

'

G'

-

henc e Using

get 2 (p , � ) / (� , II ) . q2 (YtO , y p ) / (y p , y p ) . q2 (u), p ) / ( P , p ),

an integer , wh1ch proves ( 2 ) and the existence of so that

n*

P1ck a

is a symmetric bi l1near form invariant under W ' ,

i s a multiple of x ,y

Since

cr

�' .

and � in the same component s1nc e otherwise

in the same component and

Since Y

Y

n

S1nce

•

p. 1004 ] .

We must show yB C

then 2 (P , � ) / (� , � ) p

o.

[19 ,

X' .

compatible with the decomposition of � ,

its simple components .

(p , � )

np

and the def1n1t10n of

of

may assume

-

�

cr

"

in place of

cr

'

,

then

cr ' .

If 9. 16 on G ' ,

cr " 1 s a homomorph1sm of G ' into 1ts c enter.

is connected , or els e sinc e G '

equals 1ts derived

group , th1s homomorphism is tr1v1al , wh1ch proves the Uniqueness 1n 9 . 16 . 9 . 17 .

Remarks .

(a)

From the above d1s cussion 1t should

b e c lear that for each G there 1s a Unique covering (1sogeny ) x : G ' -> G whi ch 1 s Universal sub ject to (* ) above. The fact

67

ENOOMORPHISMS OF LIIJ-!AR A LGEBRAI C GROUPS that

G'

1s s1mply connected 1s proved 1n [ 19 , p. 2)01] .

(b )

Theorem g. l holds ror complex s em1s1mple Lie groups with connected ness taken 1n the usual s ens e as

we

see by taking

K to be the

complex fi eld there , and also ror compact s em1.s1mple L1 e groups as has b een shown by Borel and others [2 ] .

(Or

cours e all

elements and automorphisms are sem1s1mple 1n th1s case. )

S1nce

the results of th1s s e ct10n are deduced from g . 2 and rormal propert1es of root systems , they rema1n equally val1d 1n thes e " 1n 9. 14 ( e ) c ases . (In the compact cas e " Borel subgroup has t o be replac ed b y "max1mal torus ror wh1 ch there 1s a funda • • •

mental chamber fixed by

10 .

cr.

"

)

Automorphisms versus finiteness .

Henceforth

we

are

concerned wi th the case 1n wh1ch G� 1n §7 1s f1nite . In th1s s ectlon we study the general s1tuat10n and develop the ma1n tools . O ur

1mmed1at e aim is : 10 . 1

group and �

f!l G

Theorem.

b e a connected l1near algebraic

� an endomorph1sm or G 2!!l2. G.

( 1 - �) G

•

G.

In the various steps of the proof be r1n1te unless

U Gcr 1s finit e ,

we

we

do not assume

Gcr to

exp11c1tly say so .

10 . 2

G� 1 � fin1te 1f and only 1f dense open part of G. This 1s b ecause

(1

-

CT) G

contains a

d1m ( 1 - CT) G - d1m G - dim Gcr .

66

ROBERT STEINBERG

It

10. 3 .

H

1s a nOrmal subgroup of

It

G

1s solvab le . then 10 . 1 holds .

G � 1,

We assume derived s er1es of

G,

let

H b e the second last t erm 1n the

and use 1nduct10n on the length .

By the 1nduct10n assumpt10n applied to G/ H there exists y e G so that y- �cry e: H . S1nce H

x

1

G.

- cr

Gcr .!:!.

10. 2.

10 . 4 .

e

and

( G/ H) cr '

f1nit e . then so 1s By

G

acts as a homomorphism there;

thus

Assume

(see 10. 3 ) 1s Abelian ,

(1 - cr) H

1s c los ed

[19 , p. 304 ] hence equal to H by 10. 2 . Thus there exists II e: H so that y- � . IIcr ll l , 1 . e . , x - (1 - cr) (YII) , whenc e 10. 4 .

!!:

10. 5 .

1 s s em1s1mple . then

G

1f the d1fferent1al

Gcr 1 s f'1nite 1f and only (at the 1dent1ty) 1s nilpotent .

dcr

Assume

cr

Gcr f'1nite . Let (see 7 . 2 ) . For each b e: B

10 . 4 . B.

Thus CT

Then

we

may

as sUllle that

B

be a Borel subgroup f'1xed by

we

have

CT

f'1xes a maximal torus

permutes the pos1t1ve roots up to factors

occur , s1nce otherwise the b e f'1xed by

CT,

and the map

1

T

the map -

cr : T

1n

must

Sulll

of the roots of that orb1t would * 1 - cr : X -+ X would not b e 1nject1ve ,

�

T

would not be surject1ve 1n contra

It follows that some oil (n > 0 ) mult1plies each root by a factor pm (m > O� whence ( e . g . by 6 . 2 ) daP • 0 d1 ct10n to 10 . 4 .

and

dcr 1s nilpotent . ,

finite only 1t

p > 1.

by

pm ( m �O)

and on each orb1t of roots a factor > 1

[19 , p . 1604 ] ;

cr

1b cr conjugate to

We see 1nc1dentally that Assume

dcr n11potent .

GCT can be Then 1 - do- ,

ENOOMORPHISMS OF LINEAR ALGEBRAIC GROUPS

69

the d1fferent1al of 1 - a', 1s surj ect1ve at the 1dent1ty , so that (1 - CT)G covers an open part of G and GO"' 1s fin1te by 10. 2. 10 . 6 . Y: G 1s sem1s1mple , ,. - 1 xa' for some f1n1te.

x !!!

G .2!:

,.

1s f1n1te. and e1ther an (n > 0) , � G,. 1s

G CT =

For 1n both cases the n1lpotence of dCT 1mp11es that of d,.

•

10. 7. ![ G 1s sem1S1mple . then 10. 1 holds . Assume x It G. Let ,. - 1XCT . By 10. 6 and 10. 2 appl1ed twic e , (1 - CT) G and (1 - ,. ) G . x overlap : (1 - a} y • (l - ,. ) z . x for some y , z It G. Then X . (1 - a) (z- ly ) , Whence 10. 7. Now we can prove 10. 1. Assume x It G. Let R be the radical of G. By 10. 3 and 10 . 7 there exists y It G so that y- lXay E R , and then by 10. 4 Z E R so that y-�ay . (1 - CTh , whence x E (1 - al G, 10. S . Remark : '!he above proof s1mpl1f1es cons1derably 1n an 1mportant spec1al case , 1n wh1ch G 1s def1ned over a fin1te f1eld of q elements and CT 1s the Froben1us map (which replaces each matr1c entry of G by 1ts qth power) . Then dCT . 0 even 1f G 1s not sem1s1mple so that we need only comb1ne the argument of 10. 7 and the last few l1nes of 10. 5 . This proof 1s a var1ant of one due to Lang [15 ] . A shorter proof of 10. 1, Which does not bring up the 1nterest1ng po1nts 10. 5 and 10. 6 , 1s as follows .

70

ROBERT STEINBERG Then as sume

First prove lO . 4 as b efore . i s a Bore l subgroup b

of

G

c £ B

and

,.

b .

'lben X .

1£ 0' ,

is con.lugate to

By 7 . 2 there

and by 7 . 3 elements

X . ybay- l .

C O' C- l

Corollary.

(1

�

fixed by

such that

such that lO . 9 .

.thaD

B

B

x £ G.

y

and

By lO . 4 there eXists

(1

- �) (yc )

as reqUired.

G O" is finite , x e G , � ,. • 1xO' , � G,. is 1somorph1,c to G . a

- O') y ,

For if

X .

1O . lO .

Corollary .

l!

GO' i s finite . then

Borel subgroup and a maximal torus thereof.

fT

fixes a

Further anY two

such couples are conjugate under an element of

G . O"

The first sta tement is by lO . 9 and the usual conjugacy theorems . Assume B , B' are the Borel subgroups i n the second statement . 'lben xBx- l - B ' for some x e G , and X- 1O' X normali zes

B

since

0'

fixes

hence has the form bO'b- l B'

i s conjugat e to tori can

now

for some

by the element

and

B' ,

b e B xb

of

hence belongs to by

10. 1 ,

GO' .

whenc e

B, B

'lbe maximal

b e treated s imilarly . Corollary .

lO . ll .

a re subgroups of c onnected .

B

(a)

GO' is finite and A !nsl B � such that A :J B !!lS! B !!

Assume

G

fixed by

(1

- O') B - B .

(b )

The natural map

AO"-> (A/B) a

i s surjective . Here (a ) follows from lO . l. Assume aB e (A/B ) O" (with a £ A ) . Then a- lO'a £ B. 'lbus a- laa . bO'b- l with b £ B by (a ) .

Since

ab

e

A

O'

we

have (b) .

ENDOMORPHISMS OF LINEAR ALGEBRAIC GROUPS 10 . 12 .

G

Corollary.

� is finite , then G Let

If

� is an automorphism in 10 . 1

and

is solvable .

R b e the radical of

placed by

71

G.

We use 10 . 1 with

G/R , whi ch is permis sib le by 10 . ) .

a n1 lpotent automorphism ,

we

conclude that

Since

G/ R

G

re

d� is then

is triv1al , as

requ1red . Remark . A s1milar but d1fferent proof has been found , 1ndependent ly , by Winter [)) ] . 10 . 1) .

Corollary.

For an endomorph1sm

group there ex1sts the d1chotomy : (b )

(a)

�

of a simple

� is an automorph1sm,

G� 1s f1nite . We are excluding the tr1v1al group from the list o f s1mple

groups .

We may assume

trivial.

�

is an automorphism , then

Assume now that

10 . 12 .

place

If

� is sUrjective since otherwise

� by any ix� '

G� is inf1nite . hence assume that

G� is infin1te by By

10 . 9

we

may re

� fixes a Borel aP

i s now chos en , as in the proof of 10 . 5 , to multiply each root by a fa ctor pm , the

subgroup and a maximal torus thereof .

If

� 1s

factors are all e qual since the root system 1s irreduCible , all e qual to

1

s ince otherwise

d1ction to 10 . 5 .

d� would be nilpotent in contra

It follows [ 19 , p. 1809]

that

� is an auto

morph1sm, whence 10 .1) . 10 . 14 . Remark.

Th1s result justifies to some ext ent the

dichotomy of this paper 1n wh1ch only endomorphisms which s atisfy ( a ) or (b ) are studi ed.

72

ROBERT STEINBERG 10 . 15 .

s1mple group G . a

If

l:!!

Corollary. g

0-

and .,.

be endomorphisms of the

GO' 1s fin1te , then so are

GO'... !!!!! G.,.CT . (1'f

1s not an automorph1sm , then ne1ther are

or

'1' 0- ;

thus 10 . 5 follows trom 10 . 1) . 10 . 16 . G .,.

even 1f

Caut10n.

Th1s result 1s false for s em1s1mple groups

1s also f1n1te (cf. 10 . 6 ) .

The c lass1cal f1n1te s1mple groups . Our ob ject1ve 1s

11.

to study Go- 1 n case G 1s sem1s1mple and CT as usual 1s an endomorphism of G onto G such that GCT 1 s f1nite . In this sect10n

we

1dent1ty the poss1b111t1es for Go- ' obtain a cellular decompos1t10n , and obta1n a new formula for the order (see 11. 6 , 1 1 . 1 , and 11. 16 below) .

The development 1s to a c erta1n extent

expos1tory , provid1ng a un1f1cat10n and s1mpl1f1 cat10n of known results , s1nc e each group turns out to be ess ent1ally a product of Chevalley groups are as above ,

we

and

the1r twisted analogues .

know that

ker

CT

10 . 5 .

As 1 s permiss1ble by 10 . 10

1n §6 to be f1xed by also are .

a,

then U ,

tI ,

we

N and W

The transpose of the restri ct10n of

w

e:

WO"

Theorem.

�

CT

G !!!!!

1s represented by some

n"

e:

�

be as above .

NeT" -

(b )

B a nd

(1 . e.

0"* .

11 . 1.

and that

G,

choos e

extended to the real vector spac e V generated b y denoted

�

and

1s tr1v1al by 7 . 1 so that CT

1s an automorph1sm of the abstract group underlying p > 1 by

G

If

to

NI T) T,

X,

wi ll b e

(a ) �

For each w

e:

Wo-

T

ENDOK>RPHISMS OF LINEAR AIGEBRAIC GROUPS

Uw .2! 6 . 3 1s fixed by a . ( c ) !! I1w 1s as 1n ( a ) , then 6 . 3 hOldS with Ga , Wa , Na , etc . 1n place of G , W , N, etc .

the group

A and

Here (a ) follows from 10 . 11 with a nd

a

replac ed by N

whi le (b ) follows trom the def1n1t10n of

Uw. a s 1 n 6 . 3 , so that also

T,

x

e Gu 1s wr1tten x . ulnu2 x • ou · on . au2 , we have w e Wcr by 6 . 3 (a ) l by 6 . 3 (b ) , whence 11. 1.

each root

a power

CI

the pos1t1ve roots . (CI » ox (k) • x CI p Cl ( c CIkq

and then u l, n , u 2 eGa of the roots and for

There ex1sts a permutat10n p

11. 2 .

Now 1 f

q (CI) 2! p such that : (a ) p permutes (c) (b ) a*p Cl • q (CI ) CI for every root CI. c�... e K*

for some

k

and all

e K.

The groups

XCI ( CI > 0 ) are the m1n1mal subgroups of U that are normali zed by T [19 , p . 1305 ] , hence they are permuted by

a ,

and s1m1larly for negat1ve roots .

If

we

def1ne

p

by

uXCI • X, CI ' then (a ) holds , and then by [ 19 , p. 1�4] so do (b ) and ( c ) (wh1 ch are proved by s ett1ng ax ( l ) • X ( c ) and then Cl P Cl Cl applying u to the equat10n 6 . 1 with k - 1 ) . We

observe that a d1fferent ch01ce of

morph1sms

xCI

11 . 3 .

will not change

Remark.

p

or

q

a,

T

or the 1so

1n any essent1al way .

It follows from 11 . 2 (a )

that

Wu

1n 11. 1

can be g1ven the strqcture of a reflect10n group as 1n 1 . 3 2 . The fact that

*

u

permutes the roots only up to pos1t1ve mult1ples

1s of no 1mportance there. The cond1t10n for

Ga to b� f1n1te d1scussed 1n the proof

ROBERT STEINBERG

74

of

10 . 5

reads :

11 . 4 .

Y

CI

runs

over any p - orb1t of roots . then th e

produ ct 1T q(CI) > 1 . For the identifi cation of

the

groups

G� we need :

11. 5 . In any irreducible component E l of E , q !! constant on roots of a given length . and i f q is not cOnstant on all of El � CI !!!S! P are long and short roots . re s pec t iv e ly . then (CI , CI) / (P , P ) . p and q (P )/q (CI) . p . [19 , p . lS06 ] . Th e proof i s as follows . in E l • Assume first they, have the same

This i s a variant o f

Let CI and P length .

Then

b e roots

p . WCI

for some

w £ W.

Since

(1'*

clearly

lie ""lfO"' normal1 zes the action of W on V , �* -1 - WI for some wI £ W. This equation and 11 . 2( b) imply that q (CI)U*pWCI . q (CI) q (wm)wm q (WCI )W�,.*p Cl q (wm) O"*WlP CI. Sinc e p WCI and WIPCI are roots and q (CI) a nd q (WCI) are positive numb ers , q (CI) - q (wu) - q(p) . Now assume CI long , P short , and q (CI) � q (p) . We may also as sume (CI , P ) > O . In this case ( * ) <-P ,CI> . 1 and (CI , CI) / (P , P ) , in the not at ion

. 2 (P ,CI)/ (CI ,CI) : for the product of the positive int egers

and < CI , P> is less than * Applying 4 by Schwarz ' s 1nequal1ty . By 11. 2 (b ) , 0" W, CI - wClO"* both sides t o p P and using 1. 5 and 11. 2 (b ) , we get

q (CI)

q { P ) . Sinc e < P,CI> '" 1 , . 1 , 2 , or 3 , and q (p ) /q (CI) i s a power o f p different form 1 , we conclude that q (p ) /q(CI) ... p and < p P,p Cl > p. Then . 1 so that the equation just us ed Wlth CI and P interchanged yields < CI , P > - p •

•

-

,

ENDOMORPHISMS OF LINEAR ALGEBRAIC GRO UPS

75

i . e . , (« ,<<)/ (P , P) . p by ( * ) . We see incidentally that in this situation p maps long roots onto short ones and vic e versa. Identitication o t GO"' . We consider the posSibili ties when G is s1mple , i . e . , E is irreducible . We know in any case that p permutes the positive roots , hence the simple ones , and preserves orthogonality ot roots since 0"'* normali zes W and orthogonal roots correspond to commuting retlect10ns . (a) Assume q (<< ) 1c constant. Then p preserves root lengths , e . g. , by the remark tollowing the proof of 11 . 5 , hence 1nduces an automorphism ot the Dynk1n d1agram. The cases are : ( 1 ) p 1s the 1dent1ty , 1 and G 1s of one ( 2 ) p 1s not the 1dent1ty and e1ther p 2 of the types An (n � 2 ) , Dn (n � 4 ) , E6 , or else p 3 . 1 and G 1s of type D4 • Cons1der case ( 1 ) . The 1somorphisms x« can be adjusted so that each c« . 1 1n 11 . 2 (.rite c« c«, l-q and then replace k by c�k) , and then X« (k) is fixed by u exactly -nen k e F q , the field of q elements . It is clear that Gu 1s just a Chevalley group parametri zed by Fq ( 6 ] (in Cheva lley t s treatment G 15 an adjoint group , but this makes little d1fference ( ct. 12 . 6 » . Sim1lary in case ( 2 ) Ga is one of the twisted analogues of the Cheva lley groups considered in [24 ] . (b ) Assume q (<<) is not constant . Then p interchanges long and short roots , so that p 2, 1 and G is of type C2 , F4 or G2 • By 11. 5 the corresponding values of p are 2 , 2 or 3 , and if q (<<) - pa for a long root , then q (p) pa+l , so that GO'" can be parametrized by a field of p2a+l elements . '!bus we 11 . 6 .

•

•

•

_

76

ROBERT STEINBERG

G� 1n the f1rst case as a SuzUki group [29] and 1n each of the other cases as a so called Ree group [16 ] . We thus see that vers10ns (cf. 12 . g below) of most of the known .f1.nite

re cogni ze

s1mple groups can b e rea11 zed 1n the form

G . The s1mpl1c1ty , a 1nc1dentally , 1s most eas1ly proved from 11. 1 and 11 . ) by the method of T1.ts [30 ] .

We obs erve that 1n c as e (a)

rea11 zed as the gro up of rat10nal po1nts o f def1nit10n of the Frobenius

G

G

G�

can be

for an appro�

G over a f1nite f1 eld (of q elements , � b e1ng qth power map) , that 1n case (b ) 1t can not . If

G� 1s essent1al ly a product of groups of the type just d1scussed ( see §6 ) . Comb1ning th1s With 10 . 11 (b )• we

1s not s1mple , then

can therefore assert : 11. 7 .

1!

G 1 s a connected l1near algebra1c group and

1s an endomorphism of

G

Gcr 1 s finite . then each 1s either cycl1c or els e one of the

such that

compos1t10n factor of

G a groups discussed 1n 11. 6 .

Next we will develop a formula for the order o f 11. g.

T

and

.!!

f1xed by

U1

�

taken over those roots Let

1: 1

CI

for

•

1 1: 1 1 .

Let

CI

for whi ch

1:2

G� .

wh1 ch 1s normal1 zed by

nq(cd , the wh1 ch XCI <: U1 •

I U1 � I

b e the s et of roots

proceed by induct10n on

��

U

1s a subgrOUp of

�,

product b eing

�

C UI '

We

b e the s et of roots such

U2 , the derived group of Up and 1: . 1:1 - 1:2 • ) Then [19 , p . 1)0 5 ] U1 (resp . U2 ) 1s generat ed by the � 1t conta1ns and U1/U2 1s canonically isomorphic to the direct

that

u

ENOOMORPHISMS OF

tI NEAR ALGEBRAIC

GROUPS

77

product of the XU for m e t3 ; and all o f these groups are fixed by CT . By 10. 11 with Ul ' U2 in place of A , B we have I U ler I I U2O" l i (Ul/ U2 ) 0- I · Now I U2cr l .. ITci (m) (m e t2) : if t l is empty this is clear, and if not it comes from the induction assumption. Thus it remains to show that I (U1" U2 ) 0" 1 TTq (m) (m e E3 ) . Now in the representation of Ul/U2 as a direct pro duct of the Xm (m e E3 ) , 0" acts on the factors according to the formula 11. 2 . Breaking up t into orbits under the action of p and changing the notation slightly , we are reduced to shOwing that the number of solutions of the system Clk! (1) k 2 ' q (2 ) k3 , , c q (n) .. k is TT I l q ( j ) , given t he c j e K * • • • nkri l C 2 k2 and the q (j ) whi ch are integral powers of p and have a pro duct > 1 (by 11. 4 ) . Now we may use the first n - 1 equations to express k2 , • • • , kU in terms of kl and then reduce the last equation to the form Ck! .. kl with c e K* and q . llq (j ) . The last equation has q solutions . Hence so does the system. •

•

•

•

11. 9. � Q denote the product of all q (m) for which m > 0 , and for each w e W let Q., denote the product of all q (m) for which m > 0 � wm < O. ( a ) I Uer l · Q. (b) l2£ each w e Wer ' I Uwer l .. Q., . Because of 11. 1 (b ) this follows from 11. 6. 11. ID . For each p-orbit n of simple roots let � .. TIq ( m) (m e n ) . .l!w! I Ter l · I det (0"* - 1) I .. IT ( Q n - 1) , the product over the various orbits . By 1l . 2 (b ) , I det (O"* - 1) I

=

TT ( Q

n

- 1) , a nurrber prime to

ROBERT STEINBERG p.

X

By the dual1 ty b etween

o rder of

ker l""

-

1)

on

T,

and

T this number 1s also the

whi ch 1s

!! Q, o" !!!!! Q n are Q TT(Qn - 1 ) E 0" (w £ \'/0' ) .

as 1 n 11 . 9 � 11. 10 , !!!!!!

11 . 11 . -

I Ga I

Thls follows from 11. 1 , 11 . 9

UO'

11 . 12 . group ) of

by

p

by

and

11. 10 .

i s a maximal unipotent (l . e •

•

a p- Sylow sub

Ga '

Q,

For dividing

I Ta l .

the order ot

/ Ger l ,

U"" ,

because each

11 . 4 .

11. 1) .

Coro llary .

II

orders of p- Sylow subgroups of

I s the largest power of

Qn

'I'

�

and

.. an

GT

�

!ill!

p

(w + 1)

is divisible

0.,.

�

.!!!S! �

GO' ,

Q,. :o

are the

�

•

Thi s tollows from 11. 9 and 1 1 . 12 . To

The formula 11 . 11 is only useful for crude purposes . get our final formula

we

will have to assume that

"" permutes the simple cor:tponents of

i.e.

G

G

is OI-simp1e ,

in a s1ngle orbit .

Thi s presents no serious loss in generality since

we

may recover

the general case by taking products .

11 . 14 . (a)

P

.!!

G !!

cT-Slmple . then the following are true .

pe rmutes the irredUCible components of

a* -

E

in a Single

q""

l � q a posit1ve number and transformation o f (f1nite) order equal to that of p orbit .

(b )

0"1

!

•

S1nce the irreduc1ble components of simple components of

G [19 , p. 1713 ] ,

we

E

correspond to the have ( a ) . Let r be

ENDOMORPHISM3 OF

LINEAR AIDEBRAIC

GRO UPS

79

tF the order of p . By 11. 2 (b ) , CTi a. - f (a. ) a. with f (a.) pos1t1ve for each root a.. To prove (b ) we must show f 1s constant . I f a. and P are roots which are 11nearly 1ndependent and not orthogonal , there 1s a root of the form a. + cP (c • .:t 1) . Then f (a. + cp ) (a. + c p ) f (a.) a. + cf (p ) P s1nc e a* 1s linear , and f (a.) - f (a. + cp) - f (p ) s1nc� a. and P are linearly 1ndepend ent . Thus f is constant on each component of t. Now evaluat1ng a-.r+l p a. 1n two dlfferent ways ( r + 1 1 + r) , we get q (a.) f (a.) a. - q (a.)f (pa.) a., whence f (a.) f (pa.) . DNs f Is constant by (a) . •

•

•

11. 15 . Remark. In case (a ) of 11. 6 the number q above Is Just the common value of all q (a.) and �l - p (on the roots ) whl 1e In case (b ) q Is thelr average value (p�a+l ) l/2 a nd CTl

oJ. P

•

We are now in a posltlon to apply the consideratlons of §2 with CTl of 11. 14 in place of CT of 2 . 1 , slnce CTl flxes a pos1tlve chamber and permutes the collectlon of unit vectors 1n the dlrectlons of the roots (thls s caling down Is Immaterlal slnce 2 . 1 Is a result about W rather than t) . 11. 16 . Theorem. Assume that G !!. CT-s1Dlple and that q � a'l are as in 11. 14 . f!!i I J be baslc invariants for W actlng on V , so chosen that each Is a characteristlc vector for CTl : al I J - SJI J � E J a root of 1 , !!!!! d ( J ) !!!! degree of I J ( J 1 , 2 , ,n) . � N - t (d (J ) - 1) be the number of positlve roots . Then 1 Gc:r I qt;r (qd ( J ) - &J ) . •

• • •

•

ROBERT STEINBERG

We get the tormula tor H , which 1.s ot course well known , by compar1.ng degrees in 2. 1. The rest ot the proot wIll be g1.ven 1.n several steps . 11. 17. � a denote the number ot roots such that XCI � U1 !!! 11. S, � H denote the total number ot pos1.t1.ve roots, and tor each w & W � H (w) denote the number ot roots CI such that CI > 0 and 1IICI < o . (a ) I Ula I .. qa !!! 11. S. (b ) I UCT I qH !!!!! I UWCT I qH (w) !!! 11. 9 . ..

_

It we torm the product ITq (CI) over a p-orbit ot r e1.ents , we g et qr by 11. 2 (b ) , 11. 14 (b ) , and the tact that CT1 has the same order as p . Th1.s y1elds (a) , hence also (b ) . ' 11. lS. !! 0" 1.s as 1.n TT ( qd ( j ) - & ) / l det (CT* - 1) 1 . j We have E o" .. E qH (w) a- replaced by 0'1 and t by 11. lS 1.n n ew ot the equat1.on TI (q&O - 1) , 1.t &O� denote j It we now combine 11. 10 ,

11. 11 ,

�

E 0"

(w

& W� -

by 11. 17 (b) . We now apply 2. 1 wIth q. We get the rtght s1.de ot det (0'* - 1) .. det (qCT1 1) the character1.st1.c values ot CT1 • -

11. 11 , 11. 17 and 11. 16

we

..

get

11 .16 .

11. 19. R.erb . (a ) The t1.rs'tf tactor 1.n the tormula tor I GCT I represents the act1.on ot 0-* on the Jacob1.an ot the I ' s , s1.nce the latter 1.s a s calar mu1t1.ple ot the product ot the pos1.t1.ve roots , and the second tactor may be wr1tten I det (CT� - l; 1 wIth J as 1.n 2. 2 (c ) . In th1s torm the tormula 1.s val1.d even 1.t G 1.s not �s1.mp1e as we see by tak1.ng products . 1he same 1.s true

ERDOJI)RPHISMS OF LINEAR ALGEBRAIC GROUPS

Sl

ot the tormula written as I Ga I • " I det O'* l d8t (lJ - crt l ) w1th Q as 1n 11. 9 . (b ) It G 1s o:-s1mple and s1mply connected, then GO' modulo 1ts center 1s s1mple 1t we exclude a tew cases (ct. 11. 6 and 12. S) . Thus the order ot the s1mple group ot the taDl1ly becomes 1GO'I / I Ccr I 1t C denotes the center ot G. 11. 2<) . The 1nd1v1dual cases ot 11. 16. We Will d1scuss the tormula tor I Ga l 1n the cases ot 11. 6 , leaving the d1scuss10n ot the values ot I cO' , to the 1nterested reader. Recall that 1n 11. 6 (a) 0'1 . p . It 0'1 1s the 1dent1ty , then the & .1 are all 1 and 11. 16 becomes Chevalley ' s tormula tor the order ot his groups (see [S] where the d C .1 ) tor the various types and the orders ot the centers are a lso g1ven Cct. l2. S) . Bow assume 1n case Cal that O"l .fa 1, af • 1 and -0'1 & W. Then crl acts on I .1 .lust as - 1 does , so that £ .1 . (_ l) d (.1) . Thus we get the tormula tor I GO' , trom CheVa 11ey ' & s1mply by replac1ng the term qd Cl) - 1 by qd C .1 ) + 1 Whenever d Cl) 1& odd. For example when G 1s ot type E6 the d Cl ) are 2 , 5 , 6, S , 9,12 so that only 5 and 9 have to be treated as above. This expla1ns the tormula 1n thSa case , proved 1n [24, p. SSS] by a tric ky calculat10n, and also expla1ns the alternat10n 1n the other cases : G ot type An ( n � 2) (and Gcr the unitary group) , and G ot type Dn (n odd) (and Gcr a type ot orthogonal group) . Next a ssume 1n (a) that 0'1 " 1 , af 1 and -0"'1 I. W. Then G must be ot type Dn (n even) , but the tollow1ng d1scuss10n 1s val1d even 1t n 1& not even. Relat1ve to a sU1table bas1s vl ,v2 , ,vn ot V the I l may be taken as •

• • •

ROBERT STEI NBERG

TIv1 and the first n - 1 elementary symmetric polynomials 1n the v� , and 0'1 as the change 1n s1gn of vn [20 , p. 1308] . qd ( qn + 1) IT ( q21 - 1) , the product on 1 from 1 Thus I GeT I to n - 1 , and d . n (n - 1) , 1n agreement w.lth [1] . Finally assume 1n ( a ) that eT1 '" 1 , a? 1 , so that GeT 1s a tr1a11ty form of D4 " The d (3 ) are 2 ,4 ,4 ,6 (see the preVious case ) , and the £ 3 are 1 , £ , £2 , 1 with £ '" 1 , e3 1 1 this 1s true to w.lth1n a permutat10n by 2. 9 , and £ and £2 have to occur 1n the same dimens10n s1nce 0"1 1s a rea l transformat10n. Thus IG I q12 ( q2 - 1 ) ( q6 1) ( q4 £) (q4 £ 2 ) .. q12 ( q2 1) (q6 - 1) O" (q8 + q4 + 1) , whi ch expla1ns the factor q8 + q4 + 1 1n [24 , p. 888] . Now cons1der case (b ) . If G 1s of type C2 (resp. G2 ) , then d (3) 2 , h w.lth h - 4 ,6 (resp . ) . The quadrat1c 1nvariant 1s always f1xed by 0"1 (s1nce W and 0'1 generate a f1n1te group) , thus the other one changes s1gn by 2 . 9. Thus 1GO' I . qh (q2 - l) (qh + 1) (cf. [16 , p. 432 ] , [29 ] ) . F1nally 1f G 1s of type F4 we have d ( 3 ) 2 , 6 , 8 , 12 , and we c la1m that £ 3 . 1,-1 , 1 ,-1, respect1vely. We know by 2 . 9 that exactly two of the £ 3 are 1 and as above that £ 1 1s one of them. We cons1der the form E a.8 + 1: ( J2 p ) 8 , the first sum on the long roots , the second on the short ones ; clearly 1t 1s fixed by W and 0'1 (see 11. 2 (b ) , 11. 5 , and 11. 14 (b » . It 1s not a mult1ple of the quadrat1c 1nvar1ant : to see this choose a bas1s vI ' v 2 tV3 ' v4 for V so that the roots are obtained from VI + v2 (short ) , 2V 1 ( long ) , and VI + v 2 + v3 + v4 (long) by arb1trary permutat10ns and s1gn changes and then compute the •

•

•

•

_

_

_

_

•

•

t

ENOOMORPHISMS

OF

LINEAR

AWEBRAIC GROUPS

terms of the form that involve only vI and v2 • Thus it may be taken as 13 , whence £3 . 1. It follows that I Ga l q24 ( q2 - 1) (q6 + l) ( qg - 1) ( q12 + 1 ) (cf. [16 , p. 40 1] ) . =

Topology of Ga. The group GO" continues as in §ll. In this section we consider the "connectedness" and "Simp ly connectednesstt properties of Ga . The role of identity component is played by Gau , the group generated by the unipotent elements of Ger · 12 .

Remarks . (a) The situation is parallel to the "continuous" case in which a is an automorphism since in that case if each w £ Wa is represented in NO" ( cf. 11. 1 and g. 3 (b », then GO" O turns out to be not only reductive but semi Simple , hence is generated by its unipotent elements . (b ) In most (but not all) cases Gau turns out to be the derived group of Ga . 12 . 1.

First

we

extend 11. 1 to Gau • 12 . 2 . Theorem. (a) In 11 . 1 � nw c an be chosen 1n Gau ' in fact in the group generated by Ua !!!.9. �. (b ) !! this 1s done . then 11. 1 is true With Gau and NO" n GO" u !!! place of Ga and Ncro Let n be a a-orbit of s1mple roots , S the set of all positive roots with support in n, and Ul the group generated by all lCL for which CL £ S . By 11. 8 there ex1sts u £ Ulcr ' u � 1 . As in step (4 ) of the proof of 8. 2 this 1mplies that the

ROBERT S'l'EINBERG group generated by Uo' and 1'0- contains a representat1ve tor w" (see 1. )0 ) . Now the elements w" tor the various p - orb1ts generate Wcr (see 1. )2 and 11. ) . This proves (a) , hence also (b ) •

Corollary. (a ) Gcr 1s generated bx Uo" U; S!lSl TO' ; � Gcru !!I Ucr � U; �. (b ) Ga/Gcru 1s 1so morph1c to Tcr /(Tcr n 0cru ) ' hence 1s Abel1an. Here (a) tollows trom 12. 2 (a ) and (b ) tram the tact that TO' normal1ses Uo' and U; . 12 . ) .

'Dleorem. Y: G 1s s1mply connected, � GO' • GO'u ' 1 . e. Gcr 1s generated by 1ts unipotent elements. In other words 1n view ot our analogy , Ocr 1s "connected" (ct. S. l) . By 12 . ) we BlUst show: (* ) TO' <: Gcru . For each s1mple root CI let GCI be the group generated by � and X_Cl • As 1s known GCI 1s semis1mple and TCI • T n GCI 1s a maximal torus ot 1t [19 , p. 1702 ] , ot dimens10n 1. Let "'CI I Ie* -+ TCI be an 1somorphism. Then "'CI c L, the group ot one-parameter subgroups ot T and WCl"'CI • -"'CI . It tollows tram 6.4 that the "'CI a re up to s1gn the bas1c generators ot 1. '!hus T 1s the d1rect product ot the TCI • Break1ng the set ot s1mple roots up 1nto p - orb1ts , we see that 1t 1s enough to prove (*) wen there 1s a s1ngle orb1t. Now 1t Gl 1s a s1mple component ot G , 1t r 1s the number ot components , and 1t , • t:l', then Gcr 1s 1somorphic to (Gl ) , . Thus 1t may further be assumed that G 1s s1mple , that E 1s 1rreduc1ble. By now the oilly possibil1t1es 12 · 4 .

ENDOMJRPHISMS OF LINEAR

ALGEBRAIC

GROUPS

are : G 1s of type A l , A 2 , C2 , or G2 , and the corresponding group GO' 1s S � , SU.3 ' C� (SuzUk1 group ) , or G� (Ree group) (see 11. 6 ) . In the second case ( * ) can be proved by s1mple calculat10ns . We exclude this case henceforth. ihen the follow ing holds : ( ** ) the element w of W" other than 1 is Just -1 � in other words wa . -a for every root a. Assume now t £ T� . Choose nw as 1n 12 . 2 and t l e T so that t f . t . By -1 ..L -1 ( ** ) nwt ln-1 w • t l , whence t l�t l - tnw · If p ,. 2 , then nw 1s semisimple because n! e T. Thus nw and t � are semi s1mple elements of G". which are con jugate in G. They are thus also conj ugate in G". (the argument , taken from [26 , p .39] , 1s given 1n 12 . 5 below) . The last statement also holds if p . 2 since then I T� I is odd by 11. 10 and t l may be chosen in T" . Since nw £ G� , so is tnw ' and then also t , as required. •

I� the course of the proof

we

have used :

12 . 5 . !l x and y are semisimple elements of Gtr wh1ch are conlugate 1n G , assumed to be simply connected. then they are con lugate in G� . Assume Then �'X'�z-1 • y , so z e G. that z- lcrz e Gx ' By g. l Gx is connected. Thus b y 10.1 there ex1sts u e Gx such that z-lcrz ucru-l • Then zu £ G" , and (zU )x (zu ) - l y , as required. =

•

The reader may have noticed that 1n this last step of the proof of 12 . 4 1ts continuous analogue g. l has been used. 12 . 6 . Corollary. If the assumpt10n that G 1s simply

ROBERT STEINBERG

86

connected. is dropped , n : G t -+ G is the universal covering , � F 1.s the kernel of n , then the following are true. (a ) nO� . GtrU • (b ) GI1"G".. u is 1somorph1.c to F/ (1 - IS ) F . ( c ) Dually Gou is 1.somorphic to G� /Fa . We apply 4 . 5 with G t , G , F in place of A , B , C. By 12 . 4 G f is generated by 1.ts un1.potent elements . '!hus nO� S; GO"U Since F is sem:1 Si.mple , 50 1 5 Ga/ker d , whence Gcru k ker d. Since nO".. ker d by exa ctness we get (a ) which implies ( c ) . By 10 . 1 ( 1 - a ) Gt � F , whence (b ) . •

"..

=

Corollary. n induces a bijection of the un1.potent elements of G� onto those of Grr• Th1.s 15 clear. 12 . 7 .

RemarkS . (a ) If G 1.5 a Simple adjoint group then with a few exceptions GtrU 1.s s1.mple (cf. 11. 6) , and most of the known f1.nite s1.mple groups are obtained 1.n th1.s way. It follows from l2 . 6 (a ) that these groups may also be obta1.ned by start1.ng with G simply connected and then d1v1d1.ng G� by 1.ts center. (b ) In analogy with the cont1.nuous case n : G'
About representat1.ons . In this sect1.on we present a reformulat1.on and uniflcat1.on of the pr1.nc1.pal results of [ 26 ) . 1) .

ENDOKlRPHISMS OF LINEAR A LGEBRAIC aROUPS

87

We recall that an 1rreduc1ble rat10nal representat10n of a s1mply connected group 1s character1zed by 1ts h1ghest we1ght , a character on T which 1s a nonnegat1ve 1ntegral comb1nat10n of the � a of 6 . 4 [ 19 , Exp . 15 , 16 ] . 13 . 1. Theorem. Let a b e a s1mply connected sem1s1mple algebra1c group and � an endomorphism of a onto a such that a� 1s f1n1te. Let q (a} be as 1n 11 . 2 , and let R denote the set of 1rreduc1ble rat10nal representat10ns for which the h1ghest we1ght � t � (a) LQ a sat1sf1es 0 S �(a) S q (a) - 1 , so that It I I det �* I TTq (a} (a s1mple ) . Then the collect10n (tensor product , R1 & i , !!!2ll � triV1al) !§. a complete set of 1rreduc1ble rat10nal representat10ns of a , each counted exactly once. =

=

=

13 . 2 . Remarks . (a ) I n case all q (a} . p and p 1 1n 13 . 1 (so that a may be thought of as be1ng defined and "split" over the pr1me f1eld with � the pth power map ) , th1s reduces to [26 , Th. 1. 1] . (b ) However , 13 . 1 also conta1ns a ref1nement of (a ) proved in [26 ] , namely , that 1n case a 1s simple and two d1fferent root lengths occur and are related to p as 1n 11. 5 , then each of the bas1c representat10ns of (a ) splits as a product of two others , for one of wh1ch � vanishes on all long (s1mple ) roots , for the other on all short ones . Th1s can be deduced from 13 . 1 as follows . Let a ' be the simply connected group whose root system t ' 1s the dual of that , t, of a (abstractly t ' 1s got from t by the 1nversion a -+ a' .. 2a/ (a ,a) , hence interchanges long and short roots } . Let a- be •

ROBERT STEI NBERG

66

an endomorphism of G '" G ' (interchang1ng the factors ) such that ' and cr*a. ' • a. (resp. ,.,.*a. . a. ' and O"*a. ' • pa.) for a-* a. • pel every long ·(resp . short ) root a. of I: (thus tr2 1s the pth power map ) . 1n [19 ] .

The 8ld.stence of

IT'

by no means s1mple , is proved

If we now apply 1) . 1 with

G

G )( G'

replaced by

and

IT'

as just descr1b ed , and use the canon1cal decomposit1on of a representat10n of G X G ' as a product of representat10ns of G and G ' , we eas11y get our assert1on. ( c ) Convers ely , 1) . 1 is a formal consequence of the special case (a) as refined in (b ) . The proof , a bookkeeping job , consists in break1ng each � 0 fT i 1nto the bas1c components described in ( a ) and (b ) and then reass embl1ng the plec es .

(d)

Among the elements in R there is

one wh1ch 1s especially interest1ng , namely , the one for whi ch the h1ghest weight � takes on its highest value : � . E ( q (a.) - l � , which may be written

(cr*

-

1) to

W

with

half the sum of the pos1tive roots .

•

E aJa. ' the famous In this case the representa

t 10n ls very much the same a s 1n characterist1c

O � 1n parti cular

TTq (a.)

(a. > 0 ) ( cf. [26 ,

the degree 1s , as ln chara cteristic Cor. 6 . ) ] ) .

Every

as we see from

a

0,

other representat10n 1n If. has a smaller degree

compar1son with the characteristic

Weyl ' s formula [) 2 , p . ) 69] may be used . s1mply connected , then of

G

(e)

If

case where

0 G

is not

1) . 1 holds for project1ve repres entations

(which correspond to 11near representat10ns of the

un1versal coverlng ) .

1) . ) .

Theorem.

The elements of R.

Let

G,

cr

�

i

b e as 1n 1) . 1.

remain d1stinct and irreduc1ble on

(a)

BNDOJI)RPHISMS

OF LINEAR

AlGBBRAIC

GROUPS

restr1ct1on to G� . (b ) A complete set ot 1rreduc1ble re presentat10ns ot Ga (.2!!£ K) 1s obtained 1n this BY . (a) It G 1s not s1mply connected , then 13 . 3 holds tor proJect1ve representat10ns ot Ga u (ct. 12. 6 (b ». (b ) By wr1t1ng G as a product ot 1ts (1'- s1mple components we may reduce 13 . 3 to the case 1n which G 1s s1mple which 1s proved 1n [26] . Once (a ) has been proved , (b ) tolLows , by a theorem ot Brauer and Nesbitt [ 5 , p. 14 ] , trom : 13 . 4 .

Remarks .

Theorem. It G , ss& ft. ate as number ot sem1s1mple con.lugacy classes ot G� tr

13 . 5 .

This is proved 1n [26 , more general result , will be

1n

!!.

13 . 1 ,

then the

I ft. 1 .

Another proot, wh1ch yields a g1ven 1n the next sect1on. 3 . 9] .

As a t1nal related result we state without proot : Theorem. � 13 . 3 the representation algebra ot Gtr 2!!! K has a defin1tion 1n terms ot generators and relations CI {YCI ( CI s 1 mple ) : y� ( ) Yp Cl ) � p !!...!!! 11 . 2 . 13 . 6 .

•

Sem1.s1mple classes and maxi mal tOri . In this section we determ1ne the number ot sem1s1mple c lasses and the number ot maximal tori ot G fixed by tT (with G and tr as 1n Ill) (see 14 . 6 14 . 14 below) , and obta1n 13 . 5 as a consequence . A lthough the two numbers depend eventually on the evaluat10n ot averages over the Weyl group ot rec1procal sets ot numbers (see 14 . 4 and 14 . 6 ) , both turn out to be powers ot p (1n tact products ot q (U) IS (see 11. 2 ) , and the1r product 1s Just the 14 .

,

ROBERT STEINBERG

90

princ1pa1 term 1n the formula 11. 16 for the order of I Gtr I . We start our deve10pment with some pre 11m1nary material about re f1ect10n groups . 14 . 1. Lemma . Let Y be a real f1n1te-dimens10na1 Eucli dean space and E1 the component of degree 1 of 1ts exter10r algebra E . If W 1s a (not nec essar11y f1n1te ) ref1ect10n group wh1ch acts effect1ve1y on Y , then 1t also acts effect1ve1y on � E1 • We proceed by 1nduct10n on n , the d1mens10n of V . Let v 1 ,v 2 , • • • , vn be a bas1s of Y so chosen that the ref1ect10ns w1 ,w2 , • • • ,wn 1n the correspond1ng (orthogonal) hyperplanes be long to W. Let V ' be the subspace generated by v1 ,v2 , , vn_ 1 , let v be a nonzero vector Orthogo':l81 to Y ' , and let W , be the restrict10n to V ' of the group generated by w1 ' w2 , • • • ,wn_ 1 • We have the decompos1t10n � . F1 + F2 with Fl · E1 CY ' ) and F2 • E1_ 1 CY ' ) " v . Now let x - X l + x2 (X l e F1 , x2 s F2 ) be f1xed by W. Then Xl 1s f1xed by W' , hence 1s 0 by the 1nduct10n hypothes1s app11ed to W' . S1m11ar1y x2 • 0 unless 1 • 1, 1n wh1ch case x . cv for some number c . Then wnx . x 1mplies that e1 ther c - 0 or wnv • v. The last equat10n 1s not poss1b1e because vn 1s not orthogonal to v. Thus x . 0 as requ1red. • • •

Wi th1n

the framework of the preceding proof one can also prove the two fo11owing results , wh1ch will not be used here. 14. 2. If W

!:!!.

14. 1 1s not effect1ve on V , then

ENDOK>RPHISMS OF LINEAR AIGEBRAIC GROUPS

!!

14 . 3 .

w

� 14 . 1 is irreducib le on

irreducib le on each � (1

-

1,2,

• • •

V,

91

then it is

and the corres pondi ng

,n)

repres entations are all 1ne gu1valent . 14 . 4, .

Theorem.

b eing effective . � I w l -l �WEW det (l - YW)

Assume in 14 . 1 � 'Y -

W

is finite b esides

be any endomorphi sm of V.

.Ih!!:!

1.

v l , v 2 ' • • • , vn b e a basis of V . Then d et( l-yw� A • • • A vn

Let

( 1 - 'Vw) v �

( 1 - 'VW) v n ' whi ch c an b e written as a sum of .± yl A • • • It Y1 1\ "/ lfY1 +1 A • • • " YlfYn ' forming a permutation of the v ' s . If the part of this • • •

1\

2n t erms of the form the

Y's

t erm involving W is averaged over is

0

unless

1 . n,

i . e . , unless the term involves

Thus the average of det ( 1 - 'YW) 14 . 5 .

Remarks .

then b y 14. 1 the result

W,

(a )

is

1

W

vacuouSly.

as required.

Similarly one can prove that if

a s econd endomorphism then the average value of det (P - "Iw) det p .

(b )

and

W

consists of the matrices

The preceding results 14 . 1 to 14 . 4,

are also true if V is

automorph1 sm whos e fixed-point s et is a hyperplane . Parallel to 14 . 4 , but involving reCiprocals , Theorem.

nec essarily effective .

Assume that � 'V

formation which norma 11 z es

W

W

"I

diag (�l , ±.l , • • J.

a complex unitary space and a reflection is defined to b e any

14 . 6 .

is

The reader may Wl sh to consider the cas e in which

is in matrt c form (c )

is

P

we

have :

is f1n1te but not

b e an invertible l1 near trans and is such that

every

1 - 'Yw,

92 a nd also IJ

ROBERT - 'If

STEINBERG

J ( J as in 2. *» , is invertible.

I w r l 2: det (l - 'V w) - l - det (lJ - l' J ) - l we W

l'!l!!!

•

Since l' normal1zes W , it acts on J, hence the right side ot 14 . 6 makes sense. We introduce a small parameter t and prove 14 . 6 with #If replaced by 'by . Because ot the identity det (1 - 'Vwt ) -l - E tr (-vw'Sk ) t k (S . E Sk is the usual grading ot the symmetric algebra on V ) , which becomes clear once 'VW has been put in superd1agonal torm, the coett1c1ent ot t k on the lett ot 14 . 6 is the average ot tr fvs ,Sk ) , which is tr (y ,I (Sk » since the average ot w on Sk is the pro j ection on I (Sk ) . Let the homogeneous basic �1nvar1ants 1 1 ,1 2 , . . . ot degrees d (1) ,d( 2) , be so chosen that -V acts superd1agonallr relative to them: '1Ij e i J + lower terms. Then since the I ' s are basic , tr ('V ,I (�» is E ei ( 1 ) e� (2) summed over all sequences p el) ,P(2) , . . . such that p (l)d (l) + P (2) d (2 ) + - k, l j k d ( thus i s also the coettic1ent ot t in TT(l - e jt ) ; , i . e. , in det (lJ - 'V J ) - l , which proves 14 . 6 . •

•

•

•

• • •

•

•

•

We will also use the tollowing simple combinatorial principle. 14 . 7 . � S be a set , W a t1nite group act1D8j on S , � a a map ot S � S yh1ch respects the equ1valence relation S/W. '!ben I (S/W)«r I - IW r l E.w lke r ( <
ENOOl()RPHISMS OF LINEAR

AWEBRAIC

93

GROUPS

Let So be the set ot all x £ S such that C7'X . 1fX w £ w. S1nce So 1s clearly �stable , each x £ So Iw/wx I conjugates al� 1n SO . Thus I ( S/W) cr ' e quals summed on x c SO ' hence also I w r l X (the number ot x , w such that ax . 1fX) , which 1s the right s1de ot

tor some has I W r l l: IWx I couples 14 . 7 .

Th1s concludes the pre11m1naries . 14. 6. Theorem. Assume as betore that G 1s a sem1s1mple algebra1 c group , cr an endomorphism ot G � G such that Gcr 1s t1ni te . and cr* !!!2 q (cd !!....!!! 11. 2. (a) The number . ot sem1s1mple conjugacy classes ot G t1xed by C1' U I det a* I • TTq (m) (m s1mple ) . (b) It also G 1s s1mplY connected. then the number ot s em1s1mple conjugacy classes ot Gcr 1s the same.

We t1rst

prove :

14. 9. !!! 14. 6 each ot the numbers det (cr* - w) the same s1gn as det a* , !!!2 (b ) 1s prime to p. Let ., - w- � * . Because the there exists a pos1t1ve 1nteger n t (m) > a tor all roots m. S1nce n ., • wlcr* n with wl C W. Then wl t (m) 1s p to a pos1t1ve power by

(a)

.!!!!

number ot roots 1s t1nite , such that .,nm • t (m)m with cr* normal1zes W, we have . 1 by 1. 10 , so that each 11. 4 . It 1n the 1dent1ty

s1mple ) we now set c - 1 and then let c decrease to 0 , 1s prime to p and has the same sign as W8 see that det (1' - 1 ) det � , whence 14. 9. (m

ROBERT STEINBERG

94

Every s em1s1mple element of

We cons1der now 14 . S Ca ) . conjugate to an element of

T

G

1s

( chos en as at the b eg1nn1ng of §ll ) ,

and two e lements of

T

are conjugate 1n

G

1f and only 1f they

are conjugate under

W , as eas 1 ly follows £rom the uniqueness 1n

Thus the number sought 1n (a) 1s Just I( T/W) I . By 14 . 7 cr l ) th1s equals IW I - E l ker «(7' - w I . By 14 . 9 (b ) and the usual
requ1red.

To prove 14 . S eb )

we

wi ll show that the na�ural map

l�om the c lass es descr1bed 1n (b ) to thos e descri b ed 1n Ca l b 1 Jective .

It 1s 1nJective by 12 . 5 (the s1mpleconnectedness 1s It 1s also surJect1ve , s1nc e

us ed here ) . 14 . 10 .

!!

G

tr

Let some Then

C

14 . 11 .

Corollary.

(a)

defined over

n

cr

(7'

!!..!!

conta1ns an element

£

C

so that

z £ G

yaxy- l - x

so that

i·or

y . ( 1 - cr) z .

Gcr as requ1red . �

k

G

b e a sem1S1mple algeb rai c grouE

sU:

q

elements . Let

r

b e the

'!be number of s em1s1mple conjugacy classes of k !! qr . (b ) The numb er of s em1s1mEle

c onjugacy c lass es of connected.

1·1xed by

By 10 . 1 there ex1sts

defined over a f1nite fi eld G

G

C b e the c lass and x

z- � z £

G.

have more g�nerally :

•

y £ G.

rank of

we

1s any connected l1near group and

usual , then every class of i·1xed by

1s

Gk

1s the same 1n case

G

1s s1mEly

ENDOMORPHISMS OF LINEAR ALGEBRAIC GROUPS We need only take

�

95

1n 14 . 8 to be the qth power map.

We observ� that the formulae of 14 . 11 also hold 1f G 1n 14 . 8 1s �-s1mple and q 1s def1ned as 1n 11 . 14 . By essent1ally the same calculat10n as 1n the proof of 14 . 8 one can show : If G 1s a semls1mple group of rank r and 1f n 1s a mult1ple of p such that I n l > 1 , then the number of sem1s1mple classes fixed by the map x � xn 1s In I r 14 . 12 .

•

Here the group can be an algebraic group or a complex or compact Lie group (1n wh1 ch case p 1) . As 1t applies to Li e groups th1s result bears a superf1c1al resemblance to H. Hopf t s theorem that the topolog1cal degree of the map 1n 14. 12 1s even 1f G 1s not semlS1mple [ 13 ] . =

As a combinatorial corollary of the preceding considerations ( e . g . of 14 . 12) , � have : If the symmetr1c group Sr+l acts naturally on the s equences of complex numbers ( c l , c 2 ' , c r+l ) (Trc i 1) , 10in the numb er of classes fixed by the map c i -? c � ( I n I > 1 ) II I n lr . 14 . 13 .

• • •

Now

we

consider maximal tori f1xed by

•

u .

14 . 14 . Theorem. Assume that G !!!.!! (T' are as in 14. 8 ( a ) and that Q denotes the order of a maxi mal unipotent (i . e. � p-Sylow subgroup) of GC1' , so that Q TIq (cx) taken over the positive roots . Then the number of maximal tor1 of G fixed by =

ROBERT STEINBERG

96

Remark. In the next sectlon 'tit' wi ll show thls Is a lso the number of un1potent elements 1'1.xed by cr . We know of no way of relatlng these facts . 14 · 15 .

Slnce T Is flxed by � and N 1s the normall zer of T , the number sought In 14 . 14 15 1 (G/N) n- I • If we conslder G/T 1nstead with W actlng from the rlght (w. x T _ xTn-wl ) , then th1s 15 that same as I « G/T)/W) I whlch by 14 . 7 may be written ( * ) I w l - l � I kerG/T (0' - w) I . Flx W .. W , wrlte n for nw ' choose g .. G so that n-l - (1 - �)g (by 10. 1 ) , and set 1 A dlrect calculat10n shows that l eft multlp11catlon by � - 1 - �. n g maps kerG/T (cr - w) onto (G/T) " , so that the two sets have the same sl ze . Now " Is conjugate to tT , under 19 In fact , so that 1 (G/T) 1 1 a,. I I I T" 1 by 10 . 11� and I a,. I - GO' I , and ,. I T", I I ker T (w- lO' - l) l - lker T (n' - W) I - Idet n'* r ldet (l - O'*- lw) -l by 14 . 9 as In the proof of 14 . 8. Substltutlng Into ( * ) we get *-l -l l aa l l det C1'* I - I I W r l� det (l - � w) . If we use 14 . 6 with Y (1"*- 1 and then 11. 19 (a ) , we get Q2 as requlred. ff'

•

=

_

Corol lary.

�

G be a connected linear algebra1c group defIned over a flnlte field k 2l q elements . � n b e the dlmension of G !ru! s that of a Gartan subgroup. Ih!!! the number of maximal torl (or Cartan subgroups ) deflned over k n-s 1& q 14 . 16 .

The Cartan subgroups are the centrall zers of the max1mal tor1 [19 , p . 701] , hence are In 1 - 1 correspondence with them,

JSNOOIDRPHISMS

LINEAR

OF

ALGEBRAIC

97

GROUPS

and 1dent1cal with them 1n the s em1s1mple case . In th1s case 14 . 16 fol lows 1�om 14 . 14 app11ed to the qth power map � s1nc e then cf .. q2N ( s ee 11. 17 ) With 2N - tota l nurrb er of roots n - s.

Gl

•

R

In the general case let

G/R.

Let

Cl

(i . e . defined over

and

conta1ns a cartan subgroup

S

1s solvab le ,

S C

�

b e chos en to b e f1xed by S1nce

C

s .. s

S1nce

Su

Let ue

and

of

G

(because

e

�

f1xed by

[ 19 , p. 70 5 ] ,

G.

'!hen

S

and th1s may

(by the conjugacy theorem and 10. 9 ) .

1s 1ts own normal1 zer 1n Cu

and let

1ts 1nv erse 1mage 1n

Hence the nurrb er of C&rtan subgroups of

I {S/C)"..I .

Gl

b e a cartan subgroup of k)

G

b e rad1cal of

S

f1xed by

S

[19 , p. 6 04}

�

1s S

be the un1potent parts of conta1ns a max1mal torus of

and C.

S)

and

Su " e , we may 1dent1fy S/ e with SuI Cu · Thus the preced1ng numb er becomes I<Su/eu) �I , wh1ch equals 1 Suer 1/ 1 e� by 10 . 11.

eu

=

Now 1 f

I A� I

=-

A 1 s a connected un1pot ent group defined over k , then d1m A : by Rosenl1cht [17] A poss ess es a normal s ertes q

def1ned over

k

such that each quot1ent 1 s 1somorph1c to (the

add1t1ve group of ) and then

(T'

K,

so that by 10 . 11

has the form crk .. Ckq ( c

e

we

may

K* ) ,

assume

A - K,

whence our

assert10n.

It follows that the numb er of C&rtan subgroups of wh1 ch are 1n S , 1 . e . , wh1ch map onto Cl 1 s qk with k .. d1m S - dim e .. d1m S - d1m e - d1m R + d1m Cl - d1m e , u u which reduc es 14 . 16 to the -sem1s1mple case and thus proves 1t . 14 . 17 .

Remark.

The number

n - s

1n 14 . 16 1s just the

d1mens10n of the var1ety of cartan subgroups of 15 .

Un1potent elements .

G.

Th1s , our final, s ect10n 1s

G

96

ROBERT STEINBERG

d evoted to the proof of the following result .

15 . 1. ff'

group and fin1te .

Theorem .

Assume that

an endomorphism of

i s a s emiS1mple algeb ra1c

G G

�

G

such that

Then the number of unipotent elements of

G �

Gcr

!!.

1s the

square of the numb er of elements 1n a max:Lmal unipot ent subgroup. The last numb er 1s , of cours e ,

Q

i n o ur usual notat1on.

S1nce an element 1s un1pot ent 1 1' and only 1 f 1t 1s a p- element p) ,

( i ts order 1 s some power of

15 . 2.

th1s ca n b e reformulat ed :

G i s the s quare of O" the numb er of elements of a p-Sylow subgroup . The numb er of p-e lements of

A s an 1 nt erest1ng cons equenc e ,

15 . 3 .

� G

Coro llary . a

group defined over

we

have :

b e a connect ed linear algebrai C k

f1nit e f1 eld

2;(

q

elements .

Let

n

the d1mens1on of

G

that of a mAE mal torus (1 . e .

r

the rank of

Then the numb er of unipotent elements of

G

defined over

G)

•

le ,

!ns& r

1.e

•

We ob s erve that

of

•

n - r

G

GJ

1n 15 . 1 1s a-s 1mple and

n-r q .

and that the same formula holds q

1s def1ned as 1n ll . 14 .

The deduct10n o f 1 5 . 3 trom 15 . 1 1 s analogous t o that or 14 . 16 from 14 . 14 , henc e wi ll be left to the reader. The proof of 1 5 . 1 depends on the follOW1ng two results .

If

-

s1mple element of

G

and

-

GO" '

tT

.u.

1s just the d1mens1on of the var1 ety

of all un1potent e lements o f 1f

�

Gle

2!

are as 1n 15 . 1

and

-

:It

1s a s em1-

there exi st s a s em1s1mple subgroup

Gt

OF

ENIX)Jt[)RPHISMS 2!

Gx

which is 1�xed by

Gx

elements of

15. 5 . G

�

and contains all of the unipotent

G

Assume

is simply connected.

� 15 . 1 and also that

�

!!

G(J' with the following property.

.2!!

( p-Sylow)

Here

!ES

Then there exists a complex 1rreduc1b l�

s ems1mple element of Go- !ES unipot ent

99

•

Theorem.

character 'X

LINEAR ALGEBRAIC GROUPS

subgroup

Gmt " GO' ()

Q ex) is of Gox '

x

is any

the order of a maximal •

� ')( ex)

.±

Q (x) .

Gx '

Let us deduc e 15 . 1 from 15 . 4 and 15 . 5 , by induction on dim G. ed .

G

By 12 . 7 and 9. 16 , we may assume that

The degree X ( l )

of

�

in 15 . 5 is equal to

GO"

order of a p-Sylow subgroup of Nesbitt [6 ] ,

X (x)

-

0

is simply connect

i s semis1mp le .

Thus by 15 . 5

and the orthogonality relations i·or characters , 1Gcr l

ex

.sems1mple ) .

elements of

xu

with

x

Let

P ex)

denot e the number of unipotent

u.

and for a fixed x

1 Gcr l

Thus

=

1:

p ex )

ex

there are

semis1mple ) . < dim G .

by 15 . 4 to yield

p ex)

If

x

e *) 1: p ex )

is not in the c enter of

•

Gcr ,

The induction hypothesis may b e appli ed •

Q ex) 2 .

We may thus cancel in

terms that correspond to values of we

p ex)

sem1s1mp le ) .

Combined with the preceding equation this yields

If

Qex) 2

1:

'

po ssibi lities for

Q ex) 2 ex then dim Gx

•

G x Each element of Ger c an b e written uniquely cr and u commuting s em1s1mp le and unipotent elements

respectively [ 1 9 , p. 406] ,

1:

the

By a theorem of Brauer and

•

x

unless

Q (l) ,

x

we

get

all

Ger · p e l ) .. Q el) 2 ,

not in the c enter of

then divide by the order of the center ,

whi ch is 15 . 1 (mod 15 . 4 and 15 . 5 ) .

e* )

100

ROBERT STEINBERG

Next we conslder 15 . 4 . By 9. 4 the group Gxa ls reductive and contains all of the unipotent elements of Gx ' hence so does its se�s1.mple component G' (see 6 . 5 ) . Slnce x is 1"1xed by CJ" so is G' , whence 15 . 4 . •

It remalns to prove 15 . 5 which is not as Simple. First we construct the needed representation. For w e W� let s ew) denote the determinant of the restriction of w to V� (see 1 . 32 ) . In the group algebra of G� over the complex f1.eld let e - 1: e (w)nw• 1: b , the first sum over Wer t the second over B� t let· E be the left ldeal generated by e , and let R be the natural representatlon of G� on E by left multlplication. Theorem. Let the notations be as above. (a) . In! dimension of E II Q .. I UIT I . The elements ue (u e U� !:2!!!! a (linear) bas1.s of E . (b) The representation R ls l.ntegral relative to the basls ln ( a) . (c) It is absolutelx lrreducible and it remains so when reduced mod p rela�lve to the basis in (a) . 15 . 6 .

This is proved in [23 ] on the bas1.s of certain axioms (1) to ( 14 ) which, because of the properties of the decomposition 11 . 1 developed ln §l2 , are at once seen to hold 1n the present case. We omit the detailed verification. We will show, 1n several steps , that the character the representation R of 15 . 6 has the property in 15 . 5 .

�

of

Q(x) . We have xue . xux- le since xe . e. Thus x permutes the basis elements ue , and (1 ) .!!

x

s

T� ,

� x, (x )

•

ENDOl«>RPHISMS OF LINEAR ALGEBRAIC 'l

its character by 15 . 6 is

(X )

I UO'X I .

-

Now

101

OROUPS Ux

is a max1ma1

Ox (by [19 , p. 1702 ] ) , and both are By U . 12 (applied to the seDl1simple component ot

Unipotent subgroup ot f1'.

tixed by

Ox ) UXt'J" is a max1mal Unipotent subgroup I u I - Q Cx ) , whence ( 1 ) . xn-

(2 )

x

E

a positive int eger ) such that

Cn

t

� t

For each s,m1 simpJ,e

E

T.

x

0 f1'

01"

�.

Thus

there exists some

is conjUgate in

E

0 ...

... _ er n to

T . Since x is seDl1s1mple it is conjugate to some ... Since t 1s ot tinite order and T has only fin1tely

many elements ot a g1ven t1nite order , there exists a positive integer

n

Now x

and

gate

in

gate in

t

G. 0...

°

Since

tensor

0 ,.

then t

s

T, wb1ch are conju

•

is simply connected , they are also conju

, by 12 . 5 , whence (2 ) . •

ell

representation in 1 5 . 6 ...

terms ot

rJI1 . ,. ,

We set

are sem1simple elements ot

� ...

(3 )

A - t.

such that

.

power

(n

positive ) .

� R,

!£ll! I\r

tor the

tor the corresponding one in

Then the restri ction ot RO" . The proot is

ot

B.r

a bit

�

0 0"

1:s the

nth

10Dg and is postponed

to the end ot this s ection. (4 ) 0 f1"

� ...

and

Deduction ot 15 . 5 . let

(t ) - � ( t )

n, ' •

and

� (x)

•

Let

x b e a sem:l.simple element ot

b e as in ( 2 ) . We have �O"Cx ) n - �" (x) -

t

�(x) n .

The tirst equality is by ( 3),

the s econd and tourth by the conjugacy b etween

x

and

t,

the

11 . 13

in place ot rr, and the firth by applied w1th the group O f ot 1 5 . 4 in place ot O. Since

� (x )

and

third by ( 1) applied w1th

� (x)

T

are int egers , they must agree up to sign , which

102

ROBERT STEINBERG

y1elds 15 . 5 . It rema1ns to prove () ) connected.

•

Recall that

G

1s s1mply

( 5 ) l!!l "- . (cr* - l)w � 1) . 2 (d) � P,,- l!!! correspond1ng 1rreduc1ble representat10n ot G. � Ra- be the reduct10n mod P .2! I\r (see l5 . 6 (c » . � P,,- � iO'" .!!:! equivalent on GO'" By 1) . 2 (d) and I) . ) any 1rreduc1ble re presentat10n ot G0'" whose degree 1s TTq (�) (� > 0) , 1 . e. , l utT l , must be equivalent to P,,- . By l5 . 6 ( c ) , l er 1s such a representat1on , whence ( 5 ) . (6 ) It ,. . ell the restr1ct1on ot R,. to GO'" 1s the nth tensor power ot �. Let I' and PI' be det1ned as 1n ( 5 ) but With , 1 n place ot fr. Thus �-l � + 0' '" . {,. * - l)W . (er - 1 )l.&J .. "- + ('J"*"- + "- . ,

•

•

•

The terms on the r1ght are the h1ghest weights ot the n represen tat10ns p,,-. a'- (1 . O , l , ,n - 1 ) . The tensor product lfl\. ,r1 conta1ns P", as a component by the above equat10n tor 1' , hence 1s eqUivalent to 1t by 1) . 1 (or else by a compar1son ot degrees ) . On restr1ct1ng to Ger where er acts tr1v1ally and us1ng ( 5 ) tor (T and tor 'I' we get (6 ) . •

•

•

0

Proot ot () . We Will show that xa (x ) n .. � (x) tor every x e Ger . As a lready noted both numbers are zero unless x 1s sem1s1mple. Assume then that x 1s sem1S1mple , and let m be 1ts order. Let M (resp. ii) be the group ot mth roots ot 1 1 n a field conta1n1ng the character1st1c values ot (7)

ENOOH>RPHISMS OF LINEAR ALGEBRAIC GBO UPS R(J' ( X ) (resp .

�(x) ) .

an 1somorph1sm 8

of

Because

p

1 s pr1me to

M onto Ji.

1ntegral l5 . 6 (b ) 1t follows that 1f charact er1st1c values of

Becaus e also s l , s2 ' . . .

m,

103 there ex1sts

R(J" (X ) are the

1s

R� (X ) , each written accord1ng to 1ts

mult1plic1ty , �hen 8 (s l ) , 8 ( s2 ) ' • • •

are those of

i� (x) .

same r emarkS apply to the character1st1c values

The

t l , t 2 , . • of � (X ) . Cons1der the equat10n t 8 (t J ) • (t 8 (s1 » n . By ( 6 ) the terms o n the lett form a permutat10n o f the terms obta1ned on the right by formal expans10n. S1nce e 1 s we get t t J • (t s1 ) n , 1 . e . , x.,. (x ) • 'X.a (x ) n , The proof of 15 . 1 1s now complete . 15 . 7.

Remarks .

(a )

•

an

1somorph1sm ,

whenc e (3 ) .

Our proof of ( 3 ) by reduct10n mod p

1 s by no means elementary , although 1t 1s qu1te natural . Perhaps some reader can replac e 1t by a s1mple proof based d1rectly on the construct10n of the representat10n

R.

(b )

In cas e all

q ( a)

a bove are equal , say to q , then the number Q (x ) of 1 5 . 5 may be wr1tten qd (X ) , with d (x ) the d1mens10n of a maximal Gx • Observe that � . ( q - l)w 1n ( 5 ) 1n this c as e . There ex1st s an analogue 1n character1st1c O .

un1potent subgroup of Assume that that

n

G 1 s s em1s1mple and s1mply connected , that

1s a pos1t1ve 1nteger , and that

the c lasses descr1bed 1n 14 . 12 . presentat10n of character of

x

x

p . 1,

b elongs to one of

Then 1n the 1rreduc1ble re

G whos e h1ghest we1ght 1s (n - l )w the 1s t. nd (x ) • Th1s and a correspond1ng result

for compact L1e groups may be deduc ed from Weyl' s formula [3 2 ,

P . 3 89] .

(c )

As an exerc1se the reader 1s asked to prove 1n

B:>BERT STEINBERG

104

15 . 1, that the number or s em1s1mple elements or Gcr is a multiple or

det (al - lJ )

(see 11. 19 (a » .

ENOOMORPHISMS OF LINEAR ALGEBRAIC GROUPS

105

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• • •

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•

•

•

, Math Zeit , vol . 24

ROBERT STEINBERG

10 9 JJ.

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UNIVERSITY O F

CA UFO RNIA ,

IDS ANGELES