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 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"vl and Wi are 1n W�. Since .aval" 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",,'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
a1nvar1ant 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 po1ntspace 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 kdimenslonal 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 lce11
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
CofIT 
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 fin1ted1mens10nal) <W,er> module.
I (M)
m e M
such that ot
under V"'
tor the subspace ot skew1nvariants 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 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 (ndbnensionau
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 atf1neEucUdean 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(3Jl ) 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'lxedpolnt 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'ixedpolnt 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 kdimenslonal torus by a f'lnite number of' translates of' (k  l)dimenslonal subtor1 ls �ellular. and that Nl denotes the number of' ldlmenslonal 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
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 CoO, I ( S k ,i k c oetticient of socTI ' ) 1: C l ) i tr (00 , 1 CSk da
I CS ) .
�)) •
As in 2.7 the coefficient on
1: coC ") tr (oo , 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.. 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 ut
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 (< 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 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 ( * ) . 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 . , (« , 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 f1n1tedimens10na1 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( lyw� 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 fixedpoint 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 ( 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 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. � pSylow 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 _ xTnwl ) , 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 nl  (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 ns 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 pSylow subgroup . The numb er of pe 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 as 1mple and
nr 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!!
( pSylow)
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 pSylow 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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ROBERT STEINBERG
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UNIVERSITY O F
CA UFO RNIA ,
IDS ANGELES