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SIMPLE GROUPS
OF
LIE T PE


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ROGER W. CARTER



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A Wiley-Interscience Publication


D
WILEY


JOHN WILEY & SONS
London. New York. Sydney. Toronto






!




,,-
'-, -10- ""- (;I..)::)(.:__Q... 'D
__T
\1 ()t rQ
Preface
ISBN 0471 137359
Printed by Adlard & Son Ltd, Batholomew Press, Dorking
Since Chevalley showed in 1955 how to construct analogues of the
complex simrle Lie groups over arbitrary fields, these 'simple groups of
Lie type' and their twisted analogues have been the subject of detailed
investigation from a number of dilTcrent points of view. This book is
intended to serve as an introduction to the theory of ChevalJey groups.
It is not an exhaustive account or these groups, but concent rates on the
basic results in the structure theory of the Chevaliey groups and the
twisted groups.
The Chevalley groups are studied in this book as groups of auto-
morphisms of Lie algebras. This appnnlch implies a concentration on
the adjoint ChevaHey groups-indeed the universal Chevalley groups and
other isogenous groups have only been touched on in the development.
In developing the theory we have found it necessary to assume a certain
familiarity with the theory of simple Lie algebras over the complex field.
Fortunately several good accounts of this theory are now available, and
the information which we need about the simple Lie algebras has been
collected together in a survey chapter.
Considerable emphasis has been placed in the development upon the
Weyl groups associated with the groups of Lie type, as it is becoming
increasingly clear that many properties of the ChcvaJlcy groups which
are independent of the lIeld of definition can best be dealt with in terms
of the Weyl group. Thus several chapters have been devoted entirely to
the \Veyl group. Most of the development is independent of the field of
definition, although we concentrate attention from time to time on finite
fields in order to determine the orders of the finite groups of Lie type.
We also show that the classical linear and symplectic groups and some
of the orthogonal and unitary groups are examples of Chevalley groups
or their twisted analogues.
One of the most fruitful ways of thinking about the ChevaJley groups
is as split forms of simple algebraic groups, for many of the most striking
properties of the Chcvalley groups are special cases of general results
on algebraic groups. The theory of algebraic groups thus sheds much
light on the structure of the Chevalley groups. This theory, however,
requires a considerable background knowledge of algebraic geometry,
which is beyond the scope of this volume. A reader who masters the

1'1
lo,j
-1'-

r=
1-
,""'"
r...........-
J.
. f.5..,ji
- ,/
-r)
Copyright  1972, by John Wiley & Sons Ltd
All rights reserved. No part of this book my be reprouced by
any means, nor transmiUeo, nor translated II1to a mchme
languagc without the written pcrmission of the publisher.
Library of Congress Catalog Card Number: 72-39228
 "
lit
-:'.._,_!...-v"'"
\
0, __ :'-
......-"''..

IV
SIMPLE GROUPS OF LIE TYPE
material in this book is nevertheless encouraged to follow up by acquiring
a knowledge of the algebraic group approach. This approach is advan-
tageous ill that it includes in addition to the split forms (Chevalley groups)
and quasi-split forms (Steinberg groups) also the non-split forms, and so
includes for example those classical ort hogonaI and unitary groups which
are not Chevalley or Steinberg groups. Our approach is, however, sufficient
to include all the known finite groups of Lie type.
A considerable part of the theory of ChevalJey and Steinberg groups
can be developed in the more general context, introduced by Tits, of
groups with a (B, N)-pair. We have followed' this approach since it
removes the necessity for repetition of several of the arguments. Groups
with a (B, N)-pair operate naturally on geometrical systems called build-
ings, also introduced by Tits, and we have included a chapter in which
these buildings are described and their connectioll with the groups estab-
lished.
It was decided not to include any account of the representation theory
of Chevalley groups, although elegant theories have been developed by
Chevalley for groups over an algebraically closed field, and by Curtis
for p-modular representations of groups over finite fields of character-
istic p.
Finally, almost all the finite simple groups at present known may be
regarded as groups of Lie type, and we have included a chapter which
completes the contemporary picture regarding finite simple groups by
giving a brief description of the sporadic simple groups not of Lie type.
Part of this book was written while the author held visiting appoint-
ments at the University of Chicago during the Autumn of 1968 and
Simon Fraser University, British Columbia, during the Summer of 1969.
Thanks are due to both these universities for their support. The book
was based partly on lecture courses on Chevalley groups given at the
University of Warwick and the University of Chicago, and on Weyl
groups at Simon Fraser University and Warwick University. A number
of helpful comments were made. by members of the audience at these
coulses.
Thanks are due to C. W. Curtis for suggesting an improvement to
my original approach to chapter 6 and to r. G. Macdonald for permission
to include a hitherto unpublished result of his in chapter 10.
Finally I wish to thank NIl's. Susan TaJi for typing the manuscript.
University of Warwick,
Iv/oy 1971
ROGER CARTER
........''\
Contents
Preface iij
-.
1 The Classical Simple Groups 1
1.1 Introduction 1
1.2 The linear groups 2 ......
1.3 The symplectic groups 3
1.4 The orthogonal groups 4  "-
1.5 The unitary groups 7
1.6 The orthogonal groups in characteristic 2 8
2 Weyl Groups 12
2.1 Systems of roots ]2
2.2 The length function 18
2.3 A geometrical interpretation of the fundamental systems 20
2.4 Definitions by generators and relations 23
2.5 Parabolic subgroups of a Weyl group 27 \
2,.6 The Coxeter complex 30
3 Simple Lie Algebras 33
3.1 Lie algebras and subalgebras 33
3.2 The Cartan decomposition 35 II
3.3 The roots of a simple Lie algebra 36
3.4 The Dynkin diagram 39
3.5 The existence and isomorphism theorems 41
3.6 Description of the simple Lie algebras 43
4 The Chevalley Groups 51
. 4.1 Properties of the structure constants 51
4.2 The Chevallcy basis 55
4.3 The exponential map 60
4.4 Algebras and groups over an arbitrary field 62
4.5 The groups A1(K) 65 \
v

VI SIMPLE GROUPS OF LIE TYPE LOON II:NTS vii
!!1' 5 Unipotent Subgroups 11 Further Properties of the Chevalley Groups
68 170
'. 5.1 The subgroups U, V 1 I. 1 The simplicity of the Chevalley groups
68 170
'y 5.2 Chevalley's commutator formula 71 11.2 Classical Lie algebras in matrix form 176
i 5.3 The structure of U and V 77 11.3 Identifications wilh some classical groups 183
; I
-
6 The Subgroups (Xr, X-r) 81 12 Generators, Relations and A u to/J1orph isms in Chevalley Grm1ps 189
r-
I 6.1 The group SL2(K) 81 12.1 A theorem of Steinberg 189
L6
6.2 The homomorphism from SL2((:) 83 12.2 Diagonal, field and graph automorphisms 199
6.3 The homomorphism from SL2(K) 87 12.3 Graph automorphisms of B:!,(K) and F.ICK) 202
f
L 6.4 The elements hr(i\) and nr 92 ] 2.4 A graph automorphism of G2(K) 206
12.5 Automorphisms of finite Chevalley groups 210
7 The Diagonal and Monomial Subgroups 97
r' 7.1 Properties of the subgroup If 13 The Twisted Simple Groups 2]6
-_il 97
\.- 7.2 Properties of the subgroup N 101 13.1 The reflection subgroup VI 216
 - 13.2 The system <I> 1 in VI 21
i 8 The Bruhat Decomposition 104 13.3 The struclure of VI 221
I L 13.4 Definition of the twisted groups 225
8.1 Bruhat's lemma ]04 13 .5 Existence of a (B, N)-pair in the twisted groups 227
8.2 Groups with a (B, N)-pair 107 13.6 The subgroup U 1 231
8.3 Parabolic subgroups 111 13.7 The subgroup HI 237
8.4 A canonical form 114
8.5 The Levi decomposition 118
r- 8.6 The finite Chevalley groups 120 14 Further Properties of the Twisted Groups 250
L. 14.1 The finite twisted groups 250
9 Polynomial invariants of the Weyl group 123 14.2 Factorization of the polynomial Pw1(t) 254
I 9.1 The algebra of polynomial invariants 123 14.3 The orders of the finite twisted groups 259
I _ 9.2 A theorem of Chevalley 125 14.4 The simplicity of the twisted groups 262
9.3 The degrees of the basic invariants 129 14.5 ldcntification with some classical groups 268
] 9.4 A theorem of Solomon 135
l-_,,. 15 Associated Geometrical Structures 274
10 The Exponents of the Weyl Group 145 15.1 Chamber complexes
274
I 10.1 A theorem of Weyl 145 15.2 Foldings 277
I
I 10.2 A theorem of Macdonald 15.3 Abstract Coxeter complexes 281
\ - ,- ]51
10.3 The class of Coxeter elements 156 15.4 The complex L( W, n) 288
10.4 A dihedral subgroup of the Weyl group 158 15.5 Buildings 202
10.5 Eigenvalues of the Coxeter elements 161 15.6 Retractions onto an apartment 295
\......iF'i.... 10.6 A theorem of Coleman 166 15.7 Groups of type-preserving automorphisms 299
L
L.
I.Y

16 Sporadic Simple Groups 303
16.1 The Mathieu groups 303
16.2 Sporadic simple groups characterized by the centralizer of
an involution 305
16.3 Towers of permutation groups 306
16.4 The Leech lattice 307
16.5 Groups generated by a class of 3-transpositions 309
CHAPTER 1
-,
The Classical Simple Groups
1.1 Introduction
Bibliography 311
[ndex of Notation 323
The theory of finite simple groups is at present at an extremely interesting
stage of development. Until about 1955 the only known finite simple
groups were the cyclic groups of prime order, the alternating groups, the
families of classical simple groups over a finite field discovered by Jordan
[I] and investigated by Dickson [I] and Dieudonne [1, 2], some finite
analogues of the simple Lie groups of type G2 discovered by Dickson
[2, 3], and the five 'sporadic' simple groups of Mathieu [1, 2].
Since that time the families of classical simple groups have been
described in a unified way by means of the Lie theory, following the
fundamental work of Chevalley [4]. The groups obtained by Chevalley's
method include Dickson's groups of type G2 and also other groups,
previously unknown, of types F4, E6, E7 and E8.
Further families of simple groups of Lie type, which may be obtained
by modifications of ChevaHey's procedur, were later discovered by
Steinberg, Tits, Hertzig, Suzuki and Ree. In fact all the infinite families
of finite simple groups known at the time of writing, with the exception
of the cyclic and alternating groups, may be regarded as groups of Lie
type over finite fields.
In addition, the five Mathieu groups have been supplemented by the
discovery during the last few years of several new 'sporadic' simple
groups, and it seems not at all unlikely that these will be followed by others.
At the same time, classification theorems for simple groups have been
proved of increasing strength. Following the Feit- Thompson theorem,
proving that every non-cyclic simple group has even order, results have
been established by Brauer, Gorenstein, Suzuki, Walter and others
classifying simple groups in many cases where the Sylow 2-subgroup or
the centralizer of an involution is assumed to be known. These results
impose severe restrictions on the structure of simple groups as yet undis-
covered. In fact it is not at all clear whether the simple groups known at
present constitute 'almost all' the simple groups which exist, or whether
there are large numbers of such groups still to be discovered.
-
Index 329
".

..
Now the simple groups of Lie type have many structural features in
common and our purpose in this volume is t,) describe some of the pro-
perties of these Lie families in a unified way, following Chevalley, Stein-
berg, Tits and others. In order to do this some knowledge of root systems
and Euclidean reflection groups is required, and also some knowledge of
the simple Lie algebras over the complex field. The results about rellection
groups have been developed from first principles in chapter 2, and the
results we shall require about Lie algebras have been collected together
in chapter 3. With this introductory material, we begin in chapter 4 an
exposition of the properties of Chevalley groups, regarded as groups of
automorphisms of Lie algebras. The properties of the Lie families of
simple groups are to a large extent independent of the field over which
the groups are defined, and this field will usually be an arbitrary one.
However we do specialize from time to time to the complex field or a
fini te field.
In the present introductory chapter we shall describe the c1assical
groups of non-singular linear transformations of a vector space.
1.3 The Symplectic Groups
'--'
We now suppose that the vector space 'P is endowed with a non-sIngular
bilinear scalar product which associates with each pair x, y of elements
of 1) an element (x, y) of K. We assume that this scalar product is skew-
symmetric, so that
"j
(y, x)= -(x, y)
for all x, Y El'. A space endowed with a scalar prod L1ct of this type is
called a symplectic space.
Consider the non-_i ngular linear transformations of V into itself which
arc isometries, i.e. which satisfy the condition
[:
,..
J
l,.;c4
1.2 The Linear Groups
(Tx, Ty) = (x, y)
for all x, y EV. The isometrics form a subgroup of GLn(K) called the
symplectic group Spn( K). Now this group is, to within isomophjsm,
independcnt of the choice or thc scalar product. In fact any non-slIlgular
skew-symmctric scalar product can be represented, with respcct to a
suitable basis, by the matrix
t..,
L
t:
Let 1J be a vector space of dimension n over a field K. The group of all
non-singular linear transformations of V into itsdf is called the gcneral
linear group CLn(K). The transformations of determinant 1 form a
normal subgroup SLn(K), the special linear group. The factor group
CLn(K)jSLn(K) is isomorphic to the multiplicative group of non-zero
elements of K. The centre Z of CL-n(K) consists of all transformations of
form T(x)=Ax for AEK with A#O. The factor group GLn(K)jZ is the
projective general linear group PGLn(K). It operates on the projective
space of dimension 11-1 associated with V. The centre of SLn(K) is the
subgroup Z n SLn(K), and the factor group SLn(K)jZ n SLn(K) is the
projective special linear group PSLn(K).
The projective special linear groups are generally simple. In fact
PSLn(K) is a simple group for all n  2, except for the groups PSL2(2)
and PSL2(3) (cf. Huppert [I], p. 182). The finite simp'e groups in this
family are obtained by taking for K the Galois field CF(q), where q is
some power of a prime. The projective special linear group over this field
is denoted by PSLn(q) and its order is given by
1
I PSLn(q) 1- qn{n-l}/2(q2 _1)(q3_1) . . . (qn_1).
(n, q-I)
o 1
-1 0
1._.
.....----
o
o 1
-1 0
A=
{
L
o
; ': f_._- 
:j'L_
,i
o 1
-I 0
In particular, the dimension of any non-singular symplectic space is even.
With respect to the above basis the symplectic transformations are repre-
sented by matrices T satisfying the condition
T'AT=A.
Now a symplectic transformation necessarily has determinant 1. (cf.
Dieudonne [1 D. The centre Z of Spn(K) consists of the transformations
-,-
, r'
; L...,..

4
SIMPLE GROUPS OF LIE TYPE
THE CLASSICAL SIMPLE GROUPS
5
Tx= AX, where A= ::!: 1. The factor group Spn(K)/Z is called the projective
symplectic group PSpn(K).
The projective symplectic groups are generally simple. In fact they are
all simple except for PSp2(2), PSp2(3), PSp4(2). The symplectic trans-
formations of a 2-dimensional space are just those of determinant J,
therefore we have
form is its index, which is defined in the following way. A subspace
W of 1) is called isotropic if
...........,...1\
(X, y)=o
I
1
I PSp2l(q) 1=(2, q-l) ql2(q2_1)(q4_1) . . . (q2l_I).
for all x, yE ijJ1. It was proved by Witt that the maximal isotropic sub-
spaces of 1) all have the ame dimension (cf. Dieudonne [I]), and this
dimension is called the index of f We shall denote the index of f by v.
v cannot be greater than !n, and we say that a form f has maximal index
if v=-ln when n is even and v=-!(n- J) when 1l is odd.
The determinant of an orthogonal transformation is :t 1 and the
orthogonal transformations of determinant I form a subgroup SO'll(K,f),
the special orthogonal group of f The centre Z of On(K,f) consists of
the transformations Tx= AX where A= ::!: 1, provided n> 2, and
J
Sp2(K) = SL2(K),
PSp2(K)=PSLz(K).
The finite symplectic groups are denoted by Spn(q), and the orders of the
finite symplectic groups are given by the formula
Z n SOn(K,f)
1.4 The Orthogonal Groups
is the centre of SOn(K,f). Factoring by the centre we obtain the cor-
responding projective groups
We assume now that 1) is a vector space of dimension n over a field K
L__t_c_-it_i _!_equal to 2. (The orthogonal groups ove-field-cl
characteristic 2 must be defined Ii1 --l-dTft-e'rent way, which we describe in
section 1 .6.) We assume that there is defined on V a non-singular bilinear
scalar product which is symmetric, so that
POn(K,f) = On(K,f)/Z,
PSOn(K,f) = SOn(K,f)/Z n SOn(K,f).
I
I
---.j
'!
f(x) = (x, x).
Conversely, the quadratic form determines the scalar product by the
formula
PD.n(K,f) = On(K,f)/Z n On(K,f).
i r
L
t
i
I,
r
I-
I-
-
[,
f
i- fIDI.
(y, x) = (x, y)
for all x, YEV. This scalar product determines a quadratic form f given
by
One might expect, by analogy with the symplectic groups, that the groups
PS01t(K, f) are generally simple. !hiI.!lot__.?-,__?!J.._r.!.j!.is.__!1__!.L
to descend to a smaller subgroup. We denote by On(K,f) the commutator
'subgroup of On(K,f}:--'Si11C,e---On(K,f)/SOn(K,f) is abelian, D.n(K,f) is
a subgroup of SO'll(K,f). We define the corresponding projective group
"". -
(x, y) =_! (+ y) - f1=L<!2:.
(We have used here the __I___!e ch_aracteristic of K is not 2.) The
non-singular linear transformations of 1) whi-ch- ar-e isometries form a
group Ou(K, f), the orthogonal group associated with the quadratic
form f In contrast to the symplectic group, the structure of the group
OuCK, f) does depend upon which quadratic form is taken.
The number of inequivalent quadratic .LQm.__Jlih__L.LQ.n.--1>-js de--
pendent on the field K. The most important invariant of a quadratic
Then the groups POn(K,f) are generally simple. In fact POn(K,f) is
simple provided n 5 and v 1. It is not generally true that PD.n(K,f) is
simple if n=4, or if n 5 and v=O.
We now describe the families of finite simple orthogonal groups. Let
K be the finite field GF(q) and 'V be an n-dimensional vector space over K.
We distinguish between the cases when fl is odd and even. Suppose first
that n is odd, and let 11 = 2/ + 1. Then there are just two inequivalent
non-singular symmetric scalar products on 'V, which can be represented
""""\
i

............. -.,I..J '--'.I.,'-"V.t..., \J.1" LIC 1 1 ","t.:..
...
by the matrices A and EA, where E is a non-square in K and
\.-.i "\.
1 0
o
given by
1
I Pfl.i(q) 1=--) ql(1-1}(q2-1)(q4_1) . . . (q21-2-1)(ql_I),
- (4,ql_J
1
I PD..-(q) 1=-- q1(1-1}(q2_1)(q4_1). . . (q2l-2__I)(q1+ I).
21 (4, q1 + 1)
r
o 0
11
A=
1.5 The Uilitnry Groups
r:
o I1
o
We now consider a vector space 1) of dimension 11 over a field K and
slIppose A-)A is an automorphism of K of order 2. Suppose 1) is endowed
with a non-singular Hcrmitian scalar proJuet (x, y). Thus (x, y) is linear
in x, conjugate linear in y, and
[-.."
.=,
These two scalar products give rise to the same orthogonal group 021+1(q),
and the order of the associated simple group is given by
1
I PD.2l+1(q) 1- (2, q_ J) qZ2(q2_1)(q4_1) . . . (q21_1).
(y, x) = (x, y).
This scalar product determines a Hermitian form f given by
---
Note that this is the same as the order of the group PSp21(q), although the
two groups are in general not isomorphic.
Now suppose n is even and let n=2/. There are again just two in-
equivalent non-singular symmetric scalar products on 13, but this time
they give rise to distinct orthogonal groups. These scalar products can be
represented by the matrices
f(x) = (x, x).
[--<>
.;,iLl
o
11-1 0
o
The values f(x) of the form lie in the fixed field Ko of the involutary
automorphism of K. The non-singular lincar transformations of 1) which
are isometries with respect to this scalar product form a group Un(K,f),
the unitary group associated with the Hermitian form f The structure
of Un(K,f) again depends upon thc choice of the Hermitian form f
The index v offis again defincd as the dimension of the maximal isotropic
subspaces of 1).
The unitary transformations of determinant ] form a subgroup
SUll(K,f), the special unitary group off The centre Z of Un(K,f) consists
of the transformations 1\= AX where AX = 1, proviued J1 > J. The centre
of SUn(K, /) is Z n SUn(K,/). Factoring by the centre we obtain the
corresponding projective groups
I
l_ ,,'o
,;:d.
(0 I,).
II 0 11-1 0 0 0
0 0 0
0 0 0 -E
PUn(K, f) = Un(K,f)fZ,
PSU.u(K,f)=SUn(K,f)fZ nSU.u(K,f).
The groups PSUn(K,f) are usuaJly simple. In fact PSUn(K,f) is always
simple provided n2 and v I, with the exception of three finite unit3ry
groups mentioned below. Again there are examples which show that
PSU11(K,f) need not be simple if v=o.
We now describe the finite unitary groups. A finite field K admitting
an automorphism of order 2 must be a field GF(q2) for some prime-power
.' i
""-
J.
where E is a non-square in K. The first of these gives rise to a quadratic
form of maximal index I, and the second has a quadratic form of index
I-I. We shaH denote the corresponding orthogonal groups by Oiz(q)
and 02i(q) respectively. The orders of the associated simple groups are
.-'"
\.

o
SiMPLE GROUPS OF LIE TYPE
THE CLASSICAL SIMPLE GROUPS
q, and the involutary automorphism is given by
A= ';..q.
Thus (x, y) may be regarded as a symplectic scalar prouct on 1': It is
not assumed to be necessarily non-singular but by a SUitable chOice of
basis for V it can be represented by a matrix of form
By choosing a suitable basis for the vector space 19 over GF(q2), any
non-singular Hermitian scalar product can be represented by the matrix
In. Thus there is essentially only one such scalar product, and this gives
rise to a Hcrmitian form I of maximal index. The correspomling unitary
group is denoted by Ull(q2), and the order of the associated projective
special unitary group is given by
o 1
1 0
o 1
1 0
o
1
I PSUn(q2) 1= qn{n-l}/2(q2_1)(q3+ 1)(q4-1).. . (qn_( -1)n).
(n, q+ 1)
The three exceptional groups which are not simple are
PSU2(22), PSU2(32), PSU3(22).
o 1
1 0
1.6 The Orthogonal Groups in Characteristic 2
o 0
The orthogonal groups over a field of characteristic 2 have to be defined
in rather a different way from the orthogonal groups already considered.
If K is a field of characteristic not 2 and (x, y) is a symmetric scalar product
on a vector space V over K, the corresponding quadratic form I is defined
by I(x) = (x, x), and therefore satisfies the condition
o
I(AX+ fLY) = A2J(X) + fL2f(y) + 2AfL(X, y)
o
for all A, fL E K.
Now suppose K is a field of characteristic 2. A quadratic form on 11
is a function I with values in K satisfying the condition
Let 17 be the dimension of 11 and 21 be the rank of the above mat.rix.
Let 1)0 be the set of XE1) such that (x, .1')=0 for all YEV. Then 1)0 IS ,a
subspace of 1) of dimension d= n - 2/. On this subspace 1)0 the quadratic
form I satisfies the condition
f(AX+ fLY) = A2J(X) + fL2f(y)
and f is said to be non-degenerate if no non-zero vector x E110 satisfies
f(x) =0. The dimension d of 1)0 is calle.d the defec_t off:. _
The non-singular linear transformations T of V whIch satIsfy the con
dition
f(AX+ fLY) = A2J(X) + fL2J(y) + AfL(X, y)
for all A, fLEK where (x, y) is some bilinear scalar product on 19. In
particular, putting fL=O we have
I(AX) = A2/(x)
and putting A = fL = I we have
(x, x)=O
f(Tx)=f(x)
form the orthogonal group On(K,f) associated with f Since
(x, y)=f(x+ y)+f(x)+f(y),
and
(y, x) = (x, y).

........",
\
--.
i
-\
j
--... -
,.
l
I
.
..

II
""- ..
- ....
i"t'
l
1
r
r-"-"-
L
L'
. '-- -,-
1
i
1.;
,'"'-
l .'
- .. ,
.. "0 . .. -, .. .z ..
it is clear that
(Tx, Ty) = (x, y).
Thus each element of On(K,f) is an isometry of the scalar product
(x, y). ._-.---
A vector xEtl is called singular iff(x)=O, and a subspace of'}) is caJled
t?tally singular if each vector in it is singular. Any two maximal totany
smguIar subspaces of'}} have the same dimension 11 and 11 is caJIed the
index off. As before 1J is at most -1-/1. '
We shall again denote by f2.n(K,f) the commutator subgroup of
On(K,f). -----
We"snall consider only non-degenerate quadratic forms and distinguish
between the cases when d=O and d> O. Suppose first that f is a nOIl-
degenerate quadratic form of defect O. Then (x, y) is a non-singular
symplectic scalar product and so On(K,f) is a subgroup of Spn(K).
(ote that n = 2/ is even.) The commutator subgroups !2n(K,f) are usually
simple. In fact !2n(K, f) is simple provided 11  6 and v  J. f271(K, f)
need not be simple if 11=4 or jf n6 and 11=0. We now describe the
finite groups of this type. A basis el, e2, . . . , e" e-l, e-2, . . . , e-l can be
chosen for V such thatfis one of the two following forms:
f(x) = XIX-l +X2X-2+ . . . +XIX-I,
f(x) = XIX-I +X2X-2+ . . . +XI-IX-(l-l) +axf+XIX-Z+o:x-f,
where
J is non-degenerate, f(1)o) has dimension d over K2. In particular
d I K : 1(21.
It can be shown (cf. Dieudonne [I]) that On(K, f) is isomorphic to the
subgroup of Sp2l(K) of transformations T satisfying the condition
f(Tx) + f(x) Ef(1)o),
where x lies in some suitably chosen non-singular symplectic subspace
of lJ of dimension 2/.
If K is a perfect field, i.e. K2= K, it is clear from the above that d= 1
and
021+1(K,f)  Sp21(K).
In particular there is one finite family of orthogonal groups 02l+1(q)
of this kind, and 02l+1(q) is isomorphic to SP'2l(Q).
If K is not perfect it can be shown that the commuator subgrup
OII(K,J) of On(K,J) is simple provided I):, 1 and v):, I (with the possible
exception of the case 1=2, v = 2). .
The facts given above about the classical linear, symplectic, :>rthogonal
and unitary groups will sufTIce for our present purposes. Ior further
information about these classical groups the reader is referred to the
books of Dieudonne [I, 2]. We shull show in later chapters that many
of the simple classical groups, including all the finite families, can be
interpreted as simple groups of Lie type.
X= L; Xiei
1
and o:t2+t+0: is an irreducible polynomial over K=GF(q), where q is
a power of 2. The indices of these forms fare / and /-1 respectively,
and the respective orthogonal groups are denoted by 0ii(q), 02i(q). The
orders of these groups are given by the same formulae as the orders of
the groups O:}j{q), 02i(q) when q is odd.
Now suppose that f is a non-degenerate quadratic form of defect
greater than O. The defect d=n - 2/ is the dimension of the subspace
'}}O of '}}, and on 'Po the formf satisfies the relation
f(Ax+ fLY) = A2J(x) + fL2J(y).
The set of valuesf{x) for XEtJO therefore forms a subset of K which is a
subspace Over the subfield K2 of K. Since 1)0 has dimension d over K and
....,.................................:!.>:;""'!:."'Gr':UIIW1II.o....:kl-t,...". 4 "'. .....;.>...,,:.'......,.f'" -m".¥11_'_... .."'''''''_..,... .........-... r.. _, ......,...__ _

CHAPTER 2
\VeyJ Groups
In the present chapter we shall describe som .
groups of orthogonal transform'ltion h' I e proertles of certain finite
Such rdkction groups called We; Is w IC 1 arc gL:nerated by reflections.
, e; ey groups Phy , . I  .
theory of groups of Lie t . I d .' e; a crUCIa role 111 the
ype-1l1<' ee there IS" . J .
group may be regarded as the < k I t' d sense 111 w lIch the WeyJ
.'._ Le type. In order to deGne theS ;: n r of the corresponding group of
.".' . " a ro t . . y g oups we 1I1tIoouce the concept
.r-!?!:i ...., 0 systcm. We do this aXIOmatically but the w,' .
,.... ...: -r -IJj(jt. systems arise naturaIly I'n tl tl '. ay 111 whIch such
'_ _ "" e; le leory of Lit.> aJ b . I .
,. . chapter 3. '" ge ras IS exp amed in
.,.

..  I
2.1 Systems of Roots
Let)) be a Euclidean space of finite dim .
vector r of V we denote by H-' th 11 . :nsIOn l. For each non-zero
r e re ectlOn 111 the hyperpla th
to r. This is the linear map defi1 d b ( ) ne or ogonal
. I (Ie y Wr r = - rand w ( ) - r
WIt 1 r, x)=O. If x is any Vecto . '"\, h r x -x lor a]l x
r 111 v we ave
Wr(X)=X-v-.:._2 r
(r, r) .
Defiitiol1 2. I . 1. A subset <I) of 19 is caJJed a '. .
followl11g axioms are satisfied: system of roots 111 V If the
) <1'> is a finite set of non-zero vectors.
(11) <J) spans V.
(iii) If r, SE <}) then Wr(S)E<P.
(iv) If r, SE <I) then 2(r )/( ). . .
(v)+ If /.. ,S r, r IS a ratIOnal lI1teger.
+ r, r E <I), where A E!FR, then A = :i: 1.
We observe that if rE <I> tl - I' .
H',.(r) = _ f. lcn rE <) ellso. This follows from (iii) since
t ion (v) is sometimes omitted in the definiti
definItion IS the most convenient for 0 on of a root system, but the Present
ur purposes.
12
""...WiIf'.,'W.I#_:.VJ.
Let <I) be a root system. We denote by W(<1») the group generatea oy
the reflections Wr for all rE <I). W(q») is called the Weyl group of <I). It
is dearly a group of orthogonal transformations of V. Each element of
IV transforms (1) into itself, by axiom (iii). W operates faithful\y on <1>, by
axiom (ii). Since (1) is a finite set, W is therefore a finite group.
Now although <{) spans V, <{) is not linearly independent. Thus <D con-
tains a proper subset which is a basis for 1). We shall show that <D contains
a subset n satisfying the following conditions:
(i) n is linearly independent.
(ii) Every root in (1) is a linear combination of roots in IT with coefficients
which are either all non-negative or all non-positive.
A subset 11 of <1> satisfying (i), (ii) is called a fundamental system of
roots. J f --,-_.-_.__..-_.."---
------
-\
-"
"
IT = {rI, r2, . . . , rz}
and rE (D then we have
l
r= 1: /"iri,
i=I
where Ai E IR and either /..i  0 for all i or /..i  0 for all i.
1n order to prove the existence of a fundamental system of roots we
endow the space V with a total ordering. Let V+-be a subset of 1) satisfying
the conditions:
(i) Jf VE)}+ and /..>0 then /..VE+.
(ii) If Vi, V2 EV+ then Vl + V2 E))+.
(iii) For each V E1) exactly one of the conditions vE11+, -v E11+,
V = 0, holds.
Such a subset ':})+ can be obtained, for example, by choosing a basis
VI, {12, . . . , Vt of  and taking for 1)+ the set of all vectors
l
.  AtVi
VI
in which the first non-zero coefficient /..t is positive.
We noW introduce an order relation )- by defining Vi )-V2 whenever
VI _ V2 EV". This is a total ordering on V compatible with addition and
with scalar multiplication by positive elements of IR.
A subsct of <I) is called a positive system of roots if it has the form
<I) n''DT-roi' SO-Ile-'ttaror(1'eirn'gof If -1f:'--IT--i" a - fld01en tal system of
roots there is just one positive system of roots containing it. For we
can certainly choose an ordering for which 11 c )}+. But then any root
-"
1
.--",
)

,1
'
.
"
;1 I.
"-H 
l l
'jl1
:1 1


I'
J
I
1[
i
j
-I
j
1
I
I
14
SIMPLE GROUPS OF LIE TYPE
rE <1> of form
L; Io.tr"
T{ E IT
with Io.t  0 for all i wiJJ satisf r E'V+
aJJ i satisfies - rElJ+. Thus (/ n 17+ .' nd eah root for which Io.t O for
We now prove the existel IS eter1l1i11ed by n.
lee of a fundamental system of roots in (1).
PROPOSITION 2 1 2 E
. " very posilive I ,f
fimdamenlal syslem. sys em OJ rools in <1> con loins a
"-<"I 
i
i L
I
i
f
-I
j
! \.'1-'-"
PROOF Let (J\+ b . "
. 11 e a PositIve sy t .
ordering of'tJ for which (1) 1- = <1> n lJ e In <P. Then there is some total
the conditions: . et n be a subset of (1)+ satisfying
(0 Every root in (1)+ is a Jinear ""
negative coefficients. comblllatIOn of roots in n with non-
(if) No subst of n satisfies (i).
SUl a ubset II certainly exists, since (1)+ itself satisfies C)
e s ow that a subset n satisfyin:J (') ("") . , I.
To do this it is sufficient to h g I, 11 IS a fundamental system
we prove first that (r s):5: 0 ove t at 1I is linearly independent. Howeve;
if possible, that (r ) > TOhr any (tw)o distinct roots r, s of IT. SUppose
, . en Vr S - S Io.r Wh \ '
we have ' - -, ere 1\ > O. If Wr(S)E (1)+
r
i
Wr(S)= L; CXtrt,
T{ E IT
,.l i;
J.-\..i i
"-tl
r
l
t,-Lo
(< ./.A ::
where each CXt  O. Thus
[:
S=lo.r+ " \
 CXtrt, 1\ > 0 CXJ >- 0
r( E IT ' · ;;..-- .
The coefficient of S on the right-ha d .d
n SI e must be Jess than 1 oth .
, erWIse
Io.r+ L; CXirt-S ::::') ( j
T{ElI L \/V
would be in 1J+ Thus s ca b
.' n e expressed as a r b',
non-negative coefficients of th " . mear com IllatIOn with
On the other hand if _ '-'I (s) EI),,eWmaJlJ]mg roots in II, a contradiction.
, " r e 1<1 ve
,L,
r--
r
L
with CXt  0, whence
-Wr(s)= 2: cxir"
Tj E 11
,
........'
Io.r=s+ L; CXtrt,
T{ E IT
10. > 0, CXt  0.
r
I
4____".
WEYL GROUPS
15
The coefficient of r on the right-hand side must be less than >., thus r
can be expressed as a linear combination with non-negative coefficients
of the remaining roots in n, which again gives a contradiction. Thus
(r, s)(O.
It is now easy to prove that IT is linearly independent. Any linear
relation involving the elcments of IT can be written in the form
2: CXirl = 2: f3i.'ii,
CXi  0, f3i  0,
where rt, s(, are distinct elements of n. Let v= 2: CXtri= 2: f3tSl. Then
(v, v) = 2: cxlf3J(n, sJ) (0.
i. j
Thus v=O. Since ri, Si are in 1)-1' this il)1plies that CXt=O, f3l=O fOf all i.
Hcnce n is a fundamental system of roots. .
PROPOSITION 2" 1 .3. Every positive system of roots in <I) contains just
olle fimdamental system. Thus there is a one-one correspondcnce between
positive syslems and fundamental systems in <}).
PROOF. Suppose {rl, r2, . . . , rz} and {Sl, S2, . . . , Sl} are two funda-
mental systems in <D+. Then
l
rt = 2: CXt}s} ,
;=1
l
St= 2: f3lJr} ,
j=l
where exl1O, f3ijO. (exi}) and (f3tj) are inverse matrices. For each i there
exists j with CXt} =! 0. Since
l
2: Ci.t mf3 mk = 0
m=l
for all k=!i, we have f3}k=O for all k=!i. Hence f3Ji'f:0. This implies
similarly that CXtk= 0 fOf k =! j. Thus (CXt}) is a monomial matrix. By number-
ing Sl, S2, . . . , Sl suitably we may assume (CXi}) is diagonal. Since rt, St E<!)+
we have cxu>O for each i. By 2.1.1 (v), exu= 1 for each i. Thus s£=rt. .
COROLLARY 2" 1 .4. If n = {r), r2, . . . , rz} is a fundamental system in
(I), thell (rt, 1"j)(Ofor all if.:-j.
PROOF. This was true of the fundamental system constructed in
2. I .2, and every fundamental system arises in this way for some ordering.
We now choose a fundamental system n and corresponding positive

16
SIMPLE GROUPS OF LIE TYPE
system <I;>+, and keep them fixed in the folIowing discussion. (The relation
between t11e different fundamental systems wiJJ be described later.) The
_._'oo(..in (1)+ will be called positive ?ots md the remainder negative roots.-
The set of negative roots ""ill be clenoted by q)_.
LEMMA 2.1.5. Let rEn. Then Wr transforms r into -r but every other
positive root into a positive root.
PROOF. Let SE <1H with s=/= r. Then
s= L O:trt
ri E 11
with O:tO. By 2.1.1 (v) there is Some O:i>O corresponding to rt=/=r.
The coefficient of rt in I-Vr(s) is therefore also positive. Thus I-Vr(s) E <1>+. .
PROPOSITION 2. 1 .6. Each root in (I) is a linear cOl'nbinalio/l of rools
in fl with rational integer coefficients.
\ '
PROOF. It is sufficient to prove this for roots in <1>+., Let rE <1>+. If
rE n the result is trivially true, so we assume r  11. Then
r = L Atrt,
Ti E IT
where each At  0 and at least two Ai are positive. Now there is some
rt E IT for which (rt, r) > O. For if all (rt, r)  0 we would have
(r, r)= L Al(rt, r) O,
a contradiction. Choose rt E IT with (r.i, r) > O. Then
2 (rt, r)
wri(r)=r--_ ------- rt.
(ri, rt)
wr/r) differs from r in only one coefJkient when expressed as a linear
combination of mots in n. Thus at least one coeflicient of W,,(r) is positive,
and so Ulri(r) E (1) /-.
We define h(r) , the height of r, b'2 =}; ,. Then h(w,,(r» < It(r)
Thus for each positive foot which is not (u-nualTIental there is another
positive root of smaller height. Therefore (he positive roots or minimal
height are the funcl<lI11ental roots, which have height l.
We prove our result by induction on her). It is true forroots of height I.
Given r E <»+ with r I' n, choose r, as above. Then w,,(r) is an integral
WEYL GROUPS
17
. . f t. IT by induction. Thus, is also, since
combinatIOn 0 roo SIn,
2(rt, r)
r= w, (r)+--------)-- r1
t (ri, rt
.--.",
and 2(rt, r)/(rt, 't) is a rational integer, by 2.1.1 (iv).
, b . II Oif't) each element
1 7 With respect to "Ie aSlS - 
COOLLARY 2t' /. bY' a matrix with rational integer coefficIents.
oj W IS represen " .
PROOF. If riEII, w(rt)E(I;> so is an integral combination of the basIs
elements.
. h. e of some root in
2 1 8 (i) Every root in (1) IS t e Imag
PROPOSITION . . .
II under some element of Wfi' d tal reElections Wr for r E II.
(ii) JY is generated by the un amel1__.___:.._______._._._-_.,---_._--
--'-._----- f W generated by the reflections
PROOF. Let Wo be the subgroup . <I> has the form w(s) for some
Wr for rED. We show that ever oo m n certainly be expressed in the
WE JVo, sElL Let rE <1>-1-. I h(r)-. ' r ca '() If h(r) > 1 there is a root
se mductIOn on r1 r . . h
required form, so we u h ( ) E <1>+ and h(wr (r» < her), as m t e
r,E n such that (r" .r) >0: T en C:)': w'(s) for some 'E WO, SE n. Hence
proof of 2.1 .6. By mductIOn Wrt . ots can also be expressed m
r= WrtW'(s) and Wrt' E 'Yo. The negatIve ro
the required form smce
- r= WrtW'( -s) = Wrt W 'wls)
.
......."'....,
 I'
and wrjW'WsEWO. S. W l'S generated by the reflections
th t W; = W. mce .
We now show a 0 W; Now r= w(s) for some
for rE<I) it is sufficient to show that W,.E Fo.
Wr, w-I or
d w It foHows that Wr= WWs .
SE II an WE rro.
( 2(s, W-I(X» S)
WWlJw-1(x)=w w-1(x)-- (s, s)
2(r, x)
-x-------. r
- (r, r)
= Wl.(X),
"""'.
lienee WrE Wo and Wo= W.
.

10
SIMPLE GROuPS OF LIE TYPE
.
"
2,2 The Length Function
Having established that every element of tl
of fundamental reflections n 1e Weyl group rV is a product
1 th f w" r E , we denote b I() h
eng 0 any expression of w of tf' f< y w t e minimal
for rEll. We shaH show that I() 1Is, orm. Thus 1(1)=0 and I(w,) = 1
W w can a so be described' .
way.  define for each WE W' . 10 qUIte a different
. an lI1teger n(w) gIven by
n(w) = I <1->+ n w-1($-) I .
Thus new) is the number of positive ro
roots. We first derive some eleme tots trans. formed by w into negative
nary propertIes of the function new).
LEMMA 2.2.1. Let rEn and WE W 'T'z
( ') ( . .I, zen
, n w,w)=n(w)+1 if w-1(r)E $+
) n(w,w)=n(w)_1 if W-1(r)E$-'
(ll) n(ww,)=n(w)+ 1 if w(r)E <p+'
(IV) n(ww,)=n(w)-1 if w(r)E <P<
PROOF. By 2. 1 .5 w, changes the si f
Thus n(w,w)=n(w) + 1 and ( gn 0 only two roots rand -r.
- n wwr)=n(w)+ I N ( )
and only if rEw(<P+) thus . . -' ow 11 w,w =n(w)+ 1 if
if and only ifw(r)E<D pro .pro()g (I) nd (ii), and n(wW,)=n(w)+1
, vmg m and (IV). .
We now show that I(w)=n(w).
THEOREM 2.2.2. The minimallen th Oif .
of fundamental reflections is equal tg tl an expressIOn of w as a product
fi d b 0 Ie number Oif P 'f'
orme y w into neuative ro t OSl lVe roots tralls-
0' 0 s.
L
[
-.
V
I
--.
.("
;.....!
f"';'
, ,{
-J
PROOF. Let w by an element
expression of W with I(w)=k. Then w has an
f--;
;L
By 2.2.1 we have
w= W'l wr2 . . . W'k'
rtED.
r'
n(w)n(w'lw)+1 n(Wr2wrlW)+2 ... k.
Thus new)  I(w).
SUppose, if possible, that n(w) < k Then b 'J '"
j  k -1 such that . y _.2.1 (m) there is an integer
W'l W'2 . . . w,lrj+l) E CP-.
i "
""'''''''
J
1
1r--
'!I
r-
i
\VEYL GROUPS
19
It follows that there is some integer i j such that
Wrl+l . . . H'r/rj+l) E (j)+,
H'rlH'rl-fl . . . H'r/rJ+l)E<!>-.
Since WI', changes the sign of only ri, - rl, we have
WrHl' . . wT/rj+1)=rt.
Ir follows that
w" = W'i+l . . . Wr/Il''j-f] WTj . . . W't+l
and so
W'i H . . . wrj I] = H'TI . . . wr)"
We can now make use of this relalion to shorten the original expression
for w. .
W=WTl . . " wrk
= wrl " . . W'iWr. . . . WTjWTj+2 . . . wrk
= WTl . . " Wri-l WrHl . . " wrJwrj+2 . . . wrk"
Thus we have an expression for w as a produce of k- 2 fundamental
reflections. Thus l(w)<k, a contradiction. Hence n(w)=l(w). II
COROLLARY 2.2.3. If WE W satisfies w(D)= IT then w= 1.
PROOF. If w(IT)=IT then w«D+)=<1>+ and so n(w)=O. Thus l(w)=O
and so w= 1. .
We now discuss the way in which different fundamental systems JI1
£1> are related to one another.
THEOREM 2.2.4. If IT is a fundamental systern in <D, so is w (D) for
Qny WE IV. Given any two fundamental systems fIb D2 ill <I> there is exactly
vile elem£'llt WE W such that w(nj)= H2.
PROOF. Let <1>+ be the positive system containing IT. Then <1)+=<D nV+
for some total ordering on V. w(l)t) also defines a total ordering on V
anJ the positive roots with respect to this ordering are
w «1>+) = <D n W (1)1-).
w(ll) is clearly the fundamental system contained in w(I>-!-).
\Vc now show that for any two fundamental systems ITl, IT2 there

20
SIMPLE GROUPS OF LIE TYPE
is some e1ement WE1'V for which W(Hl)= fI2. Let <Di, <Dt be the positive
systems containing HI, n. We use induction on 11= I <I)t n<l); I . If
n=O we have (l)/=<l)l and so JI1=fI2. Thus we mayassumen>O. Then
HI n <j)i is not empty. For if every root in HI were in lJ)l the same would
apply to every root in <Dt. Let r E HI n (1)2' Then wrC<1>t) is the set of
roots obtained from <1>/ by replacing r by -r. Hence
I wr(ll)i'-) n<J)il =n-l.
Now wr(nl) is the fundamental system contained in w,(<Di) so, by induc-
tion, there exists w' E v with w'wr([h)= 112. Thus w(fI1)= fI2 where
W=w'wr.
Finally we prove the uniqueness of w. Suppose wl(fI1)= fI2 and
W2(111)=fl2. Then w21w1(fl\)=1I1 and so by 2.2.3 wi1Wl=l. Thus
there is a unique element of J'V transforming HI into rho II
COROLLARY 2.2.5. The /lumber of fundamental systems in <1> is equal
to the orda of W.
PROOF. Let IT be a fixed fundamental system. Then well) gives each
fundamental system exactly once as w runs through I-'V. II
PROPOSITION 2.2.6. Let <1>+ be a positive system in <D and (1)- the
corresponding negative system. Then there is a unique element Wo E W such
that wo(ll)-I-) = <1>-. Jyloreover Wo is all element of order 2.
PROOF. <1>+ is the positive system in <D corresponding to some total
ordering of V, and (1)- is the positive system corresponding to the reverse
ordering. By 2.1.3 and 2.2.4 there exists a unique element \-VOE W with
wo«I>+)=<1>-. Since w6«j)+) =(j)+ we have w= 1, so that Wo is an element
of order 2. .
2.3 A Geometrical Interpretation of the Fundamental Systems
For each root r E <I) we denote by Hr the hyperplane orthogonal to r.
Thus xEHr if and only if (r, x)=O. Now the hyperplanes are closed
subsets of 1), thus
u Hr
r E <1>
is closed and its comp]cment
1J - u Hr
rEc.I>
WEYL GROUPS
is open in 'P. It is disconnected, two points being in the same connected
.........
component if and only if they lie on the same side of each reflecting
hyperplane. The connected components of
lJ - u Hr
rE
are called chambers.
Let C be any chamber in 1) and o( C) be the boundary of C. Then the
hyperplanes llr such that Hr n o( C) is not contained in any proper
subspace of /Jr are called the bounding hyperplanes, or walls, of C.
lct II = {rl, r2,.. . , n} be a fundamental system in <D. Then the set
C of vectors XE1) with (Jt x»O for i=l, 2, ..., I is a chamber. For
if IIr is any ref1ecting hyperplane, we may assume rE<D+, and so (r, x»O
for each XEC. Thus all vectors in C lie on the same side of each re11ecting
hyperplane. We show that the bounding hyperplanes of Care
Hrl' ]fr2, . . . , Hr"
Now lIr,no(C) conists of the vectors x for (rt, x)=O and (rj, x)O
for all J=I=i. Since rl, r2, . . . , r, are linearly independent, Ffr, n o(C) is
not contained in any proper subspace of If,,. On the other hand, let r
be a positive root which is not fundamental. Then
I
r = L: A-Crt,
i=l
where each AtO and at least two Ai are positive. If xEHr no(C) then
1
 Al(rt, x) = 0,
i=l
whence (rt, x)=O whenever Ai>O. Thus H, no(C) is contained in a
proper subspace of Hr. Therefore the bounding hyperplanes of Care
IIrl' IJr2, . . . , Nrl and the roots r1, r2, . . . , rz may be characterized as
those roots in <I> which are orthogonal to the bounding hyperplanes
of C and point into C (viz., ri lies on the same side of the hyperplane
IIr, as C docs).
The following proposition describes the relation between the set of
chambers and the set of fundamental systems.
PROPOSITION 2.3. 1. The roots orthogonal to the bounding hyperplanes
nf a chamber alld pointing into the chamber form a fundamental system.
J/orevrl!l", every fundamelltal system arises in this way from some chamber.
u
21
r,  Jr ::,"'-
v r..... t). .\  .....
{,
I
I
I
i
(
I
i
I
I
i
I
I
I ''\
l. S"
j''O/
J '
I
I

22
SIMPLE GROUPS OF LIE TYPE
Li
PROOF. Let w be an element of W. Since \1(Hr)= Hw(r) , w permutes
the rel1ecting hyperplanes. It follows that if C is a chamber so is w(C!,
Thus the Weyl group operates on the set of chambers. We show that this
operation is transitive.
Let IT be a fundamental system in cJ) and C be the chamber defined by
(rt, x»O for all riEIT. Let C' be any chamber and v be a vector in C/.
Let cP+ be the positive system containing IT and let CP-'-=<!j nV+,. where
cP+ is the set of positive elements with respect to a total ordenng -<.
Consider the set of transforms w(v) of v by elements of W. Let v' be the
greatest of all these transforms with respect to the ordering -<. Then we
have
'.......
') , 2(rt, v')
Wrl(v = V - ,-- rt,
(r,t, ri)
and since Wr,(V')  v' we have (rt, v')  O. Since this holds for all rt E IT,
v' must be in the closure C of C.
Let v'=w(v). Then v' is in the chamber w(C'). Howeve the only
chamber intersecting C is C. Thus w(C'), which intersects C, must be
equal to C. It follows that vVoperates transitively on the set of chambers.
Now since C' = w-1(C), the roots orthogonal to the bounding hyper.
planes of C' and pointing into C' form the set w-l(IT), which. is a funda-
mental system. Conversely, any fundamental sytem has thIS form for
some WE W by 2.2.4. ·
ft E IT,
[
I
I
. I
COROLLARY 2.3.2. Given any two chambers Cl, C2 there is a unique
element WE W such that W(Cl)=C2.
;[
PROOF. Let TII, IT2 be the fundamental systems corresponding to
CI, C2, as in 2.3.1. Then W(ITl)=IT2, and w is unique by 2.2.4.
. f -
.!it".
COROLLARY 2.3.3. The nwnber of chambers is equal to the order
ofW.
"
"; j L("
!"!
. . .
! o-
j'
 r"
': (
PROPOSITION 2.3.4. Let C be a chamber in £>. For each v Eli} there
is a unique vector v' E C such that v can be transformed into v' by some
element of W.
,
--
PROOF. Every vector in 1;) lies in the closure of some chamber. Since
W operates transitively on the set of chambers, the given vector v cn
L
,  t---.
(j
1
,
\VEYL GROUPS
23
be transformed into C by some elemcnt of v. Suppose WI(V) = VI,
\\'j.")=U2, where VI, V2EC. Then WH'I-](VI)=v2'
We shall show that if Hi (V1)= V2, where VI. V2 E C, then Vi = V2. We
prove this by induction on l(w), the result being clear if l(w)=O. Suppose
I (I\') > O. Then there is some r E n such that W (r) E(!J-. Hence
O(Vl, r)=(v2, w(r))O.
It follows that (VI, r)=O, whence Wr(Vl) = VI. But now WWr(Vl)=V2 and
/(I\,"'r)=/(w)-1 by 2,2. L Thus Vl=V2 by induction.
COROLLARY 2.3.5. If C is a chamber ill V, its closure C cOlltains
just one element oj each orbit of V lI/ula JY.
Whcn we are considering a fixed fundamental system, as is the case
in rllOst of the discussion to follow, the corresponding chamber will be
called the funda_11eItal chamber. Corollary 2.3.5 shows that its closure
1S-a--'fll11tfa'I-iiental region' for V under the action of w.
2.4 Definitions by Generators and Relations
In the present section we describe two quite distinct ways in wl1ich a
Wl:yl group can be defined by generators and relations. Let n be a funda-
mental system in (1). Then the fundamental rel1ections Wr, r E n, generate
H' ny 2.1.. Let r, SE II and J11rs be the order of WrWs. In particular J11rr= 1
fllr each r En. Then (WrWs)mrs = I is a set of relations in W, and we show
it is in fact a system or defining relations.
TIIEOf{EM 2.4.1. vV is defined as an abstract group by generators
h'r. rL I J. slllject to rclations (II'rw....)7Urs = ].
PROOF. The following beautiful proof of this theorem is due to
R. Steinberg. Suppose the result is false and let
Wr1Wr2. . . wrk= 1,
rt E IT,
be a relation of minimal length k which is not a consequence of the given
h:l.11 it)(1s. Now each reflection has determinant - 1, thus det Wr = - 1
,illd Jet (II'r1"'r2 . . . wr/J=( - l)k. It follows that k is even, and we write

24
SIMPLE GROUPS OF LIE TYPE
k=2m. Thus
WrlWr2" . WTm'H'r1ll+l=Wr2111Wr2171-1" . wr17l+2
and so I (WTlWr2 . . . Wrm+1) < m + 1. By 2.2.1 there. is an iteger jm.
such that WTl WT2 . . . wT/r:/+1) E (.b-. Therefore there IS some mteger I J
for which
WTH1 . . . WTj(rj+1) E <D+,
Wr,wTl+l . . . WT/rj+1) E <D-.
Since rt E n this implies
,J ,'.
-- \" \
WTHI . . . WTj(rj+1) = ft.
_1----
Thus
.-----
WT, = WTl+l . . . WTj\-VrJ+IWrj . . . WTf+l
and therefore
WTHI . . . WTj_l_l = WT, . . . WTr
This is a relation of length 2(j-i+ 1). If this length .is les than m, the
relation is a consequence of the given relations. By lIS111g this relatIOn and
the relations w; = 1 we have
WT1 . . . WT" = WTl . . . WTt-1 WTHI . . . wTj WTJ+2 . . . wr".
. W W W = 1 has length less
The relatIOn WT1'" WT1-1 WTI_I_l . .. rj Tj+2'" r"
than k, so can be deduceJ from the given relations, and hence
WT1 . . . WT" = 1
can be deduced from the given relations also. This. is a contradiction,
and so we must have 2(j- j + 1)=2111, whence j=m, 1=1. It follows that
WT2' . . Wrm(rm+l)=rl. .' .
However the relation WTl'" WT" = 1 IS eqUIvalent to the relatIOn
1 TI {:" W (r +2) = r.). Hence
WT2' . . WT"WTl = .. 1erelore WT3 . .. Tm+1 m ,_,______ .
, -
WT2 = WT3 . . . Wrm+l Wrm+2wrm+l . . . WT3
and so
Wr3' . . Wrm+2=wr2 .. . Wrm+1.
Now this relation cannot be deduced from the given relations. For, .by
. .t w can be Proved equal to the shOrler expressIOn
USlllg 1, WT1'" r" W. = 1 can
'v W HI Wr ., . WT" and WrlWT3'" WTm+lwTm+3' .. 1" .
"1 r3'" Tm+l ",+3 h th latlOn
be deduced from the given relations. Thus we av.e ano e re
of length 2m which cannot be deduced from the given relatlOns, and
WEYL GROUPS
may be written in the form
Wr3WT2Wr3Wr4' . . Wrm+1Wrm-+2W'm+1 . . . Wr4 = 1.
As before we obtain WT2' . . wrm(rm+1)=r3. However we already have
)fr.... Wr17l(rm+l)=rl, therefore r1=r3.
We may now write the relation WTl . . . wrk= 1 in equivalent forms by
cyclic permutation of r1, . . . , rk, and deduce that
r1=r3= .. . =r2m-1,
r2=r4= . . . =r2m.
Putting rl=r and r2=S, the relation Wrl'" wrk=l becomes (wrWs)m=1.
Thus m must be a multiple of the order mTS of WrWs. But then (WTWS)tn= 1
is a consequence of (wTWS)mr8= 1, which is one of the given -relations.
Thus we have a contradiction, and the theorem is proved. .
Dtfinitioll 2.4.2. A group defined by generators and relations
G=(a,; (atG})m'j= I), where mu= 1, is called a Coxeter group.
Thus every Weyl group is a Coxeter group. Most but not all of the
finilc Coxeter groups are Weyl groups (cf. Bourbaki [1]).
The second definition of W by generators and relations involves the
set of all reOections WT, rE <D. The proof is also due to Steinberg.
THEOREM 2.4.3. W is defined as an abstract group by generators
Wr. rE <I), subject to relations w; = 1 Gnd wrwswr= wWr(s),
PROOF. We show first that the abstract group defined by the given
generators and relations is generated by the elements Wr for r E 11. This is
certainly true of the Weyl group W. In fact if r is any root in <D we have
r= WT1wr2 . . . Wrk(S)
for suitable roots rl,.. ., rk, SEn, by 2.1.8.
,- Thus
Wr=W'l'" WTkwswrk' . . W'l'
2-
oCt.'\.-
We show that this relation, which expresses an arbitrary reflection as a
product of fundamental reflections, is a consequence of the given relations.
-I his is clear, since the given relations imply
WrkWsWrk=WSl' where sl=wTis),
WTk_1Ws1Wr,,_l =ws2, where S2=Wrk_lWrk(s),
25
----
.=:;"";
2 1 q
_  u
'J t
0'
-:
,f L"
,

26
SIMI>LE GROUPS OF LIE TYPE
"
and hence
Wrl . . . WrkWSWrk . . . Wrl = Wr.
.J
Thus the group defined by the given generators and relations is generated
by the Wr for rED.
Now any relation involving the generators Wr, rE<1>, can be expressed,
using the given relations, in the form
WrlWr2' . . wrk= 1,
nEn.
We show by induction on k that such a relation is a onsequence of the
given relations. This is clear if k = 0, so assume k > O. Smce
1(wrlwr2'" wr,J<k,
[:
there exists an integer j  k -1 such that
l(wrl' . . WrjWrj+1) = 1 (Wrl . . . Wrj)-l.
Then by 2.2.1 we have wrl'" Wrj(rj+1)E<D-. Hence there exists an
integer i j such that
'",--
W (r ) E <1'+
wrHl . ., rj j+1 ,
[ ,
 ,"
_"_.:1
( ) E <{>-
wriwr'+1 . . . Wrj r1+1 .
Since rt E n this implies that wrHl . . . Wrj(rj+1) = rt, and therefore
i
I
r:;,
;
 -'
Wri = WrHl . . . WrjWrj+1 Wrj . . . wrHl'
Now this relation is clearly a consequence of the given relations. Thus so
is the relation
1.-",
f.)
1 '\..._.;
S;>_ 
j
Wri' . . Wrj = WrHl . . . Wrj+1'
However the relation Wrl . . . wrk = 1 can be proved equivalent, using this
relation, to the shorter relation
Wrl' . . wrl__lWrHl . . . WrjWrj+2 . . . Wrk= 1.
This can be deduced from the given relations by induction, so
- i
Wrl . . . wrk = 1
r -
,
F
!
-. I/. ./
can be deduced from the given relations also, and the theorem is proved. ·
-,.
h

i
WEYL GROUPS
27
2.5 Parabolic Subgroups of a "Veyl Group
We shall now discuss certain subgroups of a Weyl group which are them-
dves Weyl groups of certain subsystems of the root system <1>. Let n
by a fundamental system in <1) and <1>+ be the corresponding positive
ystem. Let J be a subset of II. We define V.] to be the subspace of V
panncd by J; <1>.] to be <J> n 'V.]; and v.J to be the subgroup of f'V generated
hy the fundamental reflections Wr with rE J.
PROPOSITION 2.5. 1. (I>.] is a system oj roots in V.]. J is a fimdamcJ1tal
system ill <I).]. The Weyl group of <1>.] is V.].
PROOF. It is clear that (!).] spans V.I. If r, SE (!).], then
2(r S)
wr(s)=s-' r
(r, r)
'-:p_r
, J
'-
v .
/ -/..
I
(
is in <I).] also. Thus cD.] is a system of roots in V.].
Now J is a linearly independent set and every root in (!).] is a linear
combination of elements of.l. Since J is a subset of II the coefficients in
such a linear combination must be all non-negative or £111 non-positive.
Thus J is a fundamental system in <!).]. The Weyl group of (I).] is generated
by its fundamental reOections W", rE J, so is V.]. II
The subgroups fV.! and their conjugates in Ware called parabolic
ubgroups of W.
Pi{OPOSITION 2.5.2. The subgroups W.] Jor distinct subsets J oj n
Oft' all distinct.
PROOF. Suppose J, K are distinct subsets of n and that V.] = W]{.
Assume, without loss of generality, that there is a root r in K but not in
J. Then
2(r X)
wr(x)-x= - ' r.
(r, r)
Now WrEJ.V.] and so Wr(X)-XEt>.] for each XE't>. Choosing x so that
(r. x) O, we have rEl'.!. This is a contradiction since the fundamental
rt)ols are linearly independent. .

28
SIMPLE GROUPS OF LIE TYPE
LEMMA 2.5.3. Let v be a vector in C, the closure of the fundamental
chamber, and let w be an element of rV such that w(v)=v. Then wEJf/J,
where J is the set of roots in II orthogonal to v.
PROOF. We use induction on l(w). If l(w)=O the result is clear.
If I (w) > 0 there is a root r E II such_ tlt } r) E <1-. 1el, _ r\. ,. ( l " '-::'.
O(r, v)=(w(r), v)O
since VEC. Hence (r, v)=O and Wr(V)=V. But now WWr(V)=V and
l(wwr)=/(w)-1
by 2.2.1. Thus WWrE WJ by induction, whence WE WJ.
COROLLARY 2.5.4. Let v be any vector in 17 and W be an element of
W such that W (v) = v. Then W is a product oj reflections corresponding to
roots orthogonal to v.
PROOF. Choose a fundamental system so that v is in the closure of the
fundamental chamber, and apply 2.5.3. .
THEOREM 2.5.5. Let WE J;V and 'ill be the subspace of 1) of all vectors
fixed by w. Then W is a product oj reflections corresponding to roots in the
orthogonal complement 'full. of 'QU.
PROOF. Let v!, . . . , Vk be a basis for 'lL We must show that W is a
product of reflections corresponding to roots orthogonal to VI, . . . , Vk.
We shall prove more generally that if W fixes any finite set of vectors
VI, . . . , Vk then HI is a product of reflections corresponding to roots
orthogonal to VI, . . . , Vk. We proceed by induction on k, the result being
true for k = I by 2.5.4.
Choose a fundamental system IT so that Vk is in the closure of the
fundamental chamber, and let J be the set of fundamental roots orthogonal
to Vk. Consider the decomposition
V=1)J EBVj-
and let vi=v;+v;, where V;EVJ, V"E)3j-. Now WEWJ by 2.5.3, and
therefore w(v)=v. Since w(v.t)=Vt we have also w(v;)=v;. Now fVJ
is a Weyl group operating on VJ and the element w of WJ fixes
" ,
vI> V2, . . . , Vk-I'
WEYL GROUPS
29
Thus w is a product of reflections corresponding to roots in 'PJ orthogonal
10 v;, . . . , V_I' by induction. These roots are all orthogonal to
.........
VI' v2, . . . , Vk,
so the theorem is proved.
.
THEOREM 2.5.6. Let J, K be subsets of n. Then:
(i) the subgroup of W generated by WJ, fVK is WJU K
(ii) IV./ n WK= U7JnK.
PROOF. The statement (i) is clear from the definitions of WJ, WK.
In statement (ii) it is clear that WJnK f; JVJ n WK. Let WE WJ n WI(.
We must show that WE WJnK. Now w(v)=v for each V EYj-, also for
c.tch II EV.k- Thus W fixes every vector in 1Jj- + 'Vi.
Now
'Vj- + 'Vi f; 1) j- n K"
Also
dim (t>j- + 'Vi) = dim 'V} + dim Vi - dim (17j- n 17i)
= dim Vj- + dim Vi - dim (19 J + V x)l.
=l-dim l>J+I-dim 'PK-I+dim ()3J+19K)
=I-dim ('VJ nVK)=I-dim VJnK
....->jltr.'.\
=dim'VinE'
I-fence Vj- + Vii: = V.t n K'
Now H-' fixes every vector in tJj- n K' so is a prod uct of reflections cor-
rcponding to roots in 'VJnK by 2.5.5. Thus WEWJnK.
COROLLARY 2.5.7. The subgroups WJ form a lattice of 2l subgroups
in H'.
PROOF. This foHows from 2.5.2 and 2.5.6.
.
We now show that there is a natural way of choosing a system of
rcprcscn ta ti ves of the left cosets W W J of W J in W. Let D J be the set
£C!.._.:.._W cl?_thatcld19I-qJ1}':_slpJ is a subset, although
not generally a subgroup, of W. ---.-. .-----------. .....-.-. ....--.----.-...-
..ortb>,...

t"
\m.!_ 
l
f:J
f 
f
'i,
:
t
'-.''llil''--
[
f'c;
1
,._
[
 J
1= J'
i \,
/ -'--..
t 
f \.1
\"
\
r
f --
t__
30
SIMPLE GROUPS OF LIE TYPE
THEOREM 2.5.8. Let J be a subset of IT. Then each element of J¥ has
a unique expression of the form w = dJw J, where dJ ED J and W J E JrV J.
Furthermore, we have I(w) = I (dJ) + I (WJ).
PROOF. We show first that each element of W can be expressed in the
form w=dJWJ, where dJEDJ, WJE WJ and l(w)=/(dJ)+/(wJ). If
l(w)=O, 1 = 1.1 is the required factorization, thus we assume l(w»O
and use induction on I(w). IfwEDJ then w=w.l is the required factoriza-
tion. If wr;DJ there is a root rEJ such that w(r)E<!)-. Then
I (WWr) = I (w ) - I
by 2.2.1. By induction we have wWr=dJwJ with l(dJ)+/(wJ)=/(wwr).
Thus w=dJWJwr, where dJEDJ, WJWrE WJ and I (dJ) + (l(WJ) + 1)=/(w).
Now I (WJWr)  I (WJ) + 1; however if I (wJU',)< l(wJ)+ 1 there would be
an expression for W as a product of fundamental reflections of length
less than l(dJ)+/(wJ)+ 1, a contradiction. Hence I(WJWr)=/(WJ)+ 1
and l(dJ)+/(WJWr)=/(w).
We now prove the uniqueness of the factorization w=dJwJ. Suppose
we have
dJwJ=d5w5,
where dJ, d5EDJ and WJ, WEWJ' Then d5=dJwJ(w.;)-1. Suppose
WiW)-l:f: 1. By 2.5.1 WJ is the Weyl group of the root system <})J with
fundamental systemJ. Thus there exists rEJ such that wiw)-l(r)E <p,/,
Thus dJwiw)-l(r) E <D-. However d5(r) E <D+, and so we have a contra-
diction. This shows that w=wJ and d=dJ' .
COROLLARY 2.5.9. /n each coset W WJ there is a unique element of DJ.
The length of this element is smaller than the length of any other element
in wWJ.
The elements of DJ are called distinguished coset representatives of
WJ in W.
2.6 The Coxeter Complex
The parabolic subgroups of a Weyl group W can be described geometrically
in terms of the Coxeter complex of W. This is a family of subsets of 11
defined as follows. We introduce an equivalence relation on 1} writing
x"'y if, for each hyperplane Hr, rE <}), the points x, yare either both in
WtYL GRUUPS
31
II, or both not in Hr but on the same side of Hr. The points in 't) fall
inro equivalence classes with respect to this relation, and the set of these
t:quivaknce classes is the ..Qter complex.
J f, for each root r E <})+:- we dfi- .-- --" - ..-
IIr+ = {v; (r, v) > O},
11; = {v; (r, v) < O},
HT=H={v; (r, v)=O},
t ..'
then the Coxeter complex is the collection of all subsets of V of the
form
n H;T,
Tefl)-!-
€r = +, -, or O.
Observe that every element of the Coxete.r compJex lies in the closure
of some chamber. Choosing a fundamental system n, the elements of
the Coxetcr complex contained in the closure of the fundamental chamber
an: those of the form
CJ=(v' (v, r)=O for rE J )
, (v, r»O for rEI1-J '
where J is any subset of Jl.
Let IV be an element of W. Then w transforms each reflecting hyper-
plane into another, so transforms each set JJr into a set H8. Thus if
:k is an clement of the Coxeter complex so is w(1!.z). We therefore have
an operation of the Weyl group on the Coxeter complex.
PROPOSITION 2.6.1. The stabilizer of CJ in W is WJ.
PROOF. It is clear that an element WE rVJ stabilizes CJ. In fact, since
CJ is contained in the hyperplane fIr for all rEJ, w fixes each vector
in CJ.
Suppose W is an element in Wsuch that -w(CJ)=CJ. We may write
w in the form w=dJwJ, dJEDJ, WJE WJ, as in 2.5.8. Then
CJ= w (CJ) =dJWJ(CJ) =dJ(CJ).
Suppose dr=l= 1. Then there is a root r E n such that dAr) E <1)-. r cannot
be in J, by definition of DJ. Thus rE n -J. Let v be a vector in CJ. Then
(p, r»O and so (dJ(v), -dAr)) <0. However dJ(V)ECJ, so lies in
the closure C of the fundamental chamber, and -dJ(r)E <1>+. Thus
("Av),-dAr))O and we have a contradiction. Hence dJ= 1 and WE J;VJ.
..

32
SIMPLE GROUPS OF LIE TYPE
COROLLARY 2.6.2. Every element of W which transforms CJ into I
itself transforms every vector i/1 CJ into itself. " /"_, , ,! J-;:: ).-_;( r!:,_t
PROPOSITION 2.6.3. Each element of the Coxeter complex can be
transformed into exactly one element CJ by all element of the Veyl group.
PROOF. Let 1ft be an element of the Coxeter complex. 1Lt is contained
in the closure of some chamber C' and there is an element WE W such
that w (C') = C, the fundamental chamber. Hence W (1ft) s C, and so
w(1k)=CJ for some subset J of IT.
In order to prove the uniqueness of J it is sufficient to show that
w(CJ)= C., impliesJ=J. We write w=dJWJ as in 2.5.8. Then dACJ)=Cj.
Suppose dr/ol. Then there is a root r E IT -J such that clJ(r) E <D-. Let v
be a vector in CJ. Then (v, r»O and (dJ (v) , -dJ(r)) <0. But div)ECJ
so is in the closure of the fundamental chamber. Also -dJ(r)E <1>+, there-
therefore (dJ(v) , -dJ(r))O and we have a contradiction. Thus dJ= I
and CJ=CJ. .
We may now obtain the required geometrical description of the para-
bolic subgroups of the Weyl group.
PROPOSITION 2.6.4. The parabolic subgroujJs of Ware the stabilizers
in V of the elements of the Coxeter cornplex.
PROOF. Let IfZ. be an element of the Coxeter complex. Then 1L\.=w(CJ)
for some WE}V and some subset J of n. Now w'(lk)=1l\. if and only
if w-1w'w(CJ)=CJ, which holds if and only if W'E wWJw-1 by 2.6.1.
Thus the stabilizer of Itz is w WJW-1, a parabolic subgroup of J--V. Con-
versely, any parabolic subgroup of W has form w WJw-l, so is the stabilizer
of some element of the Coxeter complex. II
,
..... \ ..././. v.
: _I.
 ,
CHAPTER 3
Simple Lie Algebras
The simple groups with which we are concerned in this volume are defined
as groups of automorphisms of Lie algebras. Before introducing them we
need certain introductory material on Lie algebras, in particular a know-
ledge of the structure of simple Lie algebras over the complex field.
Each such Lie algebra determines a root system and a Weyl group, to
\\hich the results of the preceding chapter apply. In th present chapter
\\c summarize the properties of Lie algebras which we shall need. Proofs
of all the properties which we describe can be found, for example, in
Jacobson's book [I].
3.1 Lie Algebras and Subalgebras
f Rf!'1mYj
A Lie algebra is a vector space JL over a field K on which a product opera-
lIOn [xy] is defined satisfying the following axioms:
(i) [xy] is bilinear for x, y E JL.
(ii) [xx]=O for XE 1L.
(iii) ![xy]z]+ [[yz]x]+ [[zx]y]=0 for x, y, ZEJL. .
AAioll1 (iii) is called the Jacobi identity. We note that [[xy]z] IS not
n(cssarily equal to [x[yz]], .thus Lie multiplication is not in general
aociative. As a simple consequence of axioms (i), (ii) we have
0= [x+ y, x+ y]= [xx] + [xy] + [yx] + [yy] = [xy] + [yx]
Thus [yx] = - [xy] and Lie multiplication is anticommutative. In the
prescnt work we shall be concerned only with finite-dimensional Lie
algl.:bras.
Let 1L be a Lie algebra and J1f[, jF). be subspaces of 'E. We define [j1fIjf2]
lo be the subspace of JL spanned by all elements of form [xy] for  E 1,
)"i: J1. Since [yx] = - [xy] it is clear that Ljj1;iM] = Ul11.1Ft]. Thus multiplica-
tIon of subspaccs is commutative.
A subdgebra of JL is a subspace ;iM such that /jlfIfll] s, and an
idc1 of JL is a subspace :m such that [JNJL]  J1f1. Since (finr..] = [JLJ1f[]
Ihac is no distinction in the theory of Lie algebras between left ideals and
ngh.t ideals. Every ideal is two-sided.
33
"'-"'1..,\ 
'-
;
#
;
"""""", '

34
SIMPLE GROUPS OF LIE TYPE
"
'"
For each element x of a Lie algebra JL we define a map ad x of JL into
itself by
'-
ad x .y= [xy],
Y E JL.
ad x is a linear map, and also satisfies the condition
ad x. [yz] = [x [yz]] = [[xy]z] + [y[xz]] = [ad X.y, z]+[y, ad x.z].
A linear map 8 of JL into itself satisfying
8[yz]=[8y, z]+[y, 8z], y, zEJL,
is called a derivation of JL. Thus ad x is a derivation of 1L for each x.
We note further that
f '
ad x.ad y-ad y.ad x=ad [xy].
For, given Z E Jr., we have
iil_ -:
f'->

id
(ad x.ad y-ad y .ad x) .Z= [x [yz]] - [y[xz]]= [[xy]z] =ad [xy] .z.
An important role in the theory of Lie algebras is played by a scalar
product called the Killing form. For each x, y E JL we define the scalar
product (x, y) by
-....
(x, y)=tr (ad x.ad y).
As the trace of the linear map ad x. ad y, (x, y) is an element of the field
K. The scalar product defined in this way is certainly bilinear, and is also
symmetric, since tr (Ocp)=tr (cpfJ) for any two linear maps 0, cp of JL into
itself.
After these introductory definitions we turn to a consideration of the
simple Lie algebras over the complex field C. A Lie algebra is said to be
_I2kif it has no ideals other than itself anb.,.___PftCe. The
1-dimenswIiaILie alge"bra--over-anyflefcfTs'-certainly simple and is called
a trivial algebra. We are concerned with simple non-trivial Lie algebras,
and we begin with an example which illustrates clearly the main features
of the general theory. We observe first that any associative algebra can
be made into a Lie algebra by defining the Lie multiplication by
[xy]=xy-yx.
For [xy] is clearly bilinear, [xx] =0 and
[[xy]z] + [[yz]x] + [[zx]y] =(xy- yx)z-z(xy- yx) +(yz-zy)x
- x(yz - zy) + (zx - xz) y - y(zx - xz)
''c-_I
:!I 
.1; I
'Li _''';J,(
,I;
.:1 f""'
i I
''-'''
!,
:1
r---
f
L
'",=-
t 
E
1
"..--.........
=0.
(
i
I-.

SIMPLE LIE ALGEBRAS
35
Consider the algebra of all (/ + 1) x (/ + I) matrices over C. This algebra
has dimension (/+ 1)2 and may be made into a Lie algebra as described
above. The matrices of trace 0 form a subalgebra of this Lie algebra of
dllncnsion (1+ 1)2-1 =/(/+2). For we have
tr (x+y)=tr x+tr y=O,
tr (AX)=A tr x=O, AEC,
tr [xy] = tr (xy- yx)=O.
The Lie algebra o_!' J.-t I) x (/+ I) matrices of trace 0 is in fact simple.
3.2 The Cartan DeompositioD
Tl1C classification of the simple Lie algebras over C was obtained by
W. Killing [I] and E. Cartan [I]. This c1assi[]cation is achieved by decom-
posJng such an algebra with respect to a certain type of subalgebra, now
l".dkd a Cartan subalgebra. A subalgcbra J[) of the Lie algebra JL is caJIed
t Carlan subalgebra if it satisfIes the following two conclitions:
(i) [[[:tL)J!)]:IL)]... ]=0 for some r.
+-- r-)o
Subalgcbras satisfying this condition are called nilpotent.
(ii) If [xII] E J[) for ,all hE J[) then x E ;I!).
This condition means that J[) is not contained as an ideal in any larger
uhaJgcbra of JL
It can be shown that anl: algebr_.<?_!:.,_}1as Cartan subalgebras
and any tW0 Cartan subalgebras-are'Isomorphic.l'ii-'-fac("gl'veI1"a11y'-two'
Cartan subalgebras of JL, there is an automorphism of JL which trans-
iorms one into the other. The dimension of the Cartan subalgebras ofJl
Ii lIcd the rank of JL, and will usuiTy"be-'(fenoted by J. -. ----... " .- -"--..-.
.. If thc algebra-1£..'" is-'simpre"ovei-C'-'the-"-"(:arian''iijl)91gebras actuaI1y
_ tj")J}!.2J. -although ii1T--'is--t-"tr-h1g-;;ral f;;--;';pl
aribri.ls. Thus for a simple Lie algebra, Lie multiplication inside a Cartan
.ublgcbra is trivial.
b.tl,!....bI11p,I£.,.Qver C and JI,) be a Cartan subaJgebra of JL. Then 1L"
C.U1 he decomposed into a direct sum oO!) with a number of I-dimensional
iub:lp.KCS all invariant under multiplication by 1£). Thus
JL = 31) EB JLTI EB J!..T2 EB . . . EB 1Lrk'

36
SIMPLE GROUPS OF LIE TYPE
where dim JLr. = 1 and [%;J[r,] = JLr., for each i. This is called a Cartan
decomposition of JL.
For example, if J[ is the algebra of all (l + 1) x (l + 1) matrices of trace 0,
it is easy to see that the diagonal matrices of trace 0 form a Cartan sub-
algebra 'J[}. Then we have
J[ = JI EB :2: Cel},
i*j
,".
where elj is the elementary matrix with 1 in the (i, j) position and 0 else-
where. This direct decomposition is a Cartan decomposition. For let
h = diag (Ao, AI, . . . , Al). Then
[hetj] =heij- eijh = P'l - AJ)eij.
Hence the I-dimensional subspace Ceij is invariant under Jf.
I
I
i,
3.3 The Roots of a Simple Lie Algebra
Let JL be a simple Lie algebra over C and
, JL = J!} EB J[rl EB . . . EB l!.rk
be a Cartan decomposition of 1.. In each I-dimensional subspace JLr we
choose a non-zero element er. Then, for each hE J!}, [her] is a scalar multiple
of er, and we write
[her] = r(h)er.
The map r : :If)----..C defined in this way is certainly linear, so is an element
of the dual space of J[}. The maps r1, r2, . . . , rk from 'j[) into Care cailed
the roots of JL and the su bspaces JLrl' J[r2' . . . , J[rk are called the root-
spaces of 1. (relative to the given Cartan subalgebra jt)). This terminology
originated from the fact that r(h) is a root of the characteristic equation
\_ of the map ad h. The roots rl, 1"2, . . . , rk are in fact all distinct and all
- -' non-zero. Thus the zero map is not a root. --- " ;-;- fo' -'  .<" V-( I i
Although the rot are defie? as elements of the dual space of J[) 'r .{' ,,, 
they can, by consldenng the Ktllll1g form, be regarded as elements of 1£) \.- 
itself. It can be shown that the Killing form of a Lie algebra JL is non-
singular if and only jf J[ is scmi-simple, i.e. has no proper ideal in which
the Lie multiplication is trivial. ]n particular every simple non-trivial
algebra is semi-simple, and so the Killing form of JL remains non-singular
when restricted to the Cartan subalgebra of 'j[} (although the Killing form
of J[) itself is identically zero). Thus each element of the dual space of]I)
SIMPLE LIE ALGEBRAS
IS expressible in the form h----..(x, h) for a unique element x E 1!}. The element
x associated with the map h----..r(h) may be identified with the root r. Thus
r can be regarded either as an element of 'Ji) or an clement of its dual
space; the relation between these two being given by
r(h)=(r, h),
h E JI.
Considering the roots as elements of J,J;), let <I> be the finite subset of 
obtained in this way. It can be shown that <I> spans J!), and that if we
choose any subset of <I> which is a basis for 1£) then each element of <1>
i a linear combination of the roots in this subset with rational coefficients.
Also (r, s) is rational for all r, SE cI). We denote by iij the set of all elements
of J1) which are linear combinations of elements of <I> with real coefficients.
By the preceding remarks it is evident that j[1R is a real vector space of
the same dimension as the complex dimension of 1Q. Also one can show
that, if XEjI)I!i' then (x, x)O and (x, x)=O only if x=O. Thus the Killing
form is positive definite on J.Qiij and' so iij may be regarded as a Euclidean
space. In particular we can define the length of an element XE JQIR by
I x I = Vex, x)
and the angle () between x, Y E J!)IR by
(x, y) = I x II y I cos 8.
Now it is shown in the theory of simple Lie algebras that the subset
 of the Euclidean space 1!}1R forms a system of roots in the sense defined
in 2. I . I. In particular, 2(r, s)f(r, r) is a rational integer for all r, SE <1>.
We give an interpretation of this integer. Suppose or, s are linearly inde-
rx:ndcnt. Since (D is finite there exist intergers p, q  0 such that ir + s E <1>
for -piq but -(p+ I)r+s and (q+ l)r+s are not in <D. The sequence
of roots <----_._-_.,-.. ---
-pr+s,... ,s,... ,qr+s
will be called the r-chain of roots through s. Now the reflection Wr in
, -the hyperplane orthogonal to '-'can 'be---sho' to permute the elements
of (1).)n fact it has the eITect of inverting each r-chain5)f.roQ_ts.. In particular
it lra'I;;'r111S---::'=p;+;--into--l};-+-;-a-d so -p;+j:,- q';+s are mirror images
In the hyperplane orthogonal to r. Hence
« -pr+s) + (qr +s), r)=O.
37
........',1)..".
"-
A"")
-\
)
(
./l
,:
,i
"',,",

38
SIMPLE GROUPS OF LIE TYPE
i&..
It follows that
2(r, s)
------- = P - q
(r, r) .
For each pair of roots r, SE <D we define
L
2(r, s)
Ars=-------.
(r, r)
Thus Ar8 is a rational integer which satisfies Ar8=p-q and
Wr{s)=s-Arsr.
By 2. I .2 the root system (I> contains a subsystem n which is a system
of fundamental roots. We shall denote such a subsystem by
TI={PI, p2,... ,Pl}.
Every root in <D is an integral combination of roots in n with coefficients
which are aJl non-negative or all non-positive and we denote by (j) 1-, (1)-
the sets of positive and negative roots with respect to the fundamental
system n.
We illustrate the situation by means of an example. Let 1L be the Lie
algebra of all (/ + 1) x (/ + 1) matrices of trace O. We have seen that the
diagonal matrices in JL form a Cartan subalgebra ]I), and that
JL = p;) EB 1: Cetj
Nj
is a Cartan decomposition. Let h be the diagonal matrix
diag (Ao, AI, . . . , Al).
r":

r
c
....
: .{
i".......=1"'"
'. r
"- I "-..>j
I
. I''''
, \-,...
Then
[hetl] = (At - Aj)etj
and s? the root corresponding to the subspace Ce'lj is the map h>'t- AJ
of ]I} mto C. Let PI, P2, . .. ,PI be the roots defined by
PI : IzAO->'I,
P2 : h>'I- >'2,
pi :. /i X,- :... A,.
Then PI, P2, . . . , PI form a system of fundamental roots, and the other
roots have form
1: (PHI + . . . +Pl), i <j.
The positive roots of this Lie algebra are therefore the sums of consecutive
fundamental roots.
---",.
L.,
I r
I
"It.1III>' \1 \,

".''-----
\
\\
\. \.
SIMPLE LIE ALGEURAS
.1':1
3.4 The Dynkill Diagram
Let Pt, Pi be distinct fundamental roots of a simple Lie algebra JL and
tIll be the angle between them. Since -Pi + Pl is not a root, pj is the first
member of the poi-chain of roots through it. Using the relation
2(r, s)
(r;r)" =p-q,
tkrived in section 3.3, and noting that P =0, we see that (Pt, Pl)  0. Thus
the angle between two distinct fundamental roots is obtuse.
Thac arc only a few possibilities for the value of this angle. For
2(pl. P})/(pl, Pl) and 2(p1, P'l)/(Pj, Pl) are both integers, and so
iEL'_u!?})=4 cos2 elj
(Pi, pd(pj, fJ}) .
is an integer also. Since 0  cos2 ell  I, we have 4 cos2 etl = 0, I, 2, 3 or 4.
SIIh:C Ol} is obtuse, eu is one of 7T/2, 27T/3, 37T/4, 57T/6 or 7T. The fact that
I'" Pi arc linearly independent excludes the possibility D[j= 7T. Thus
tJ1J = 1T/2, 27T/3, 37T/4 or 57T/6. We dcllne an integer 1111 by Ilij =4 cos2 ell.
Thus 1It}=O, 1, 2 or 3 if i=j:- j. ntl admits a factorization
2(pl, Pj) 2(p}, Pl)
nil = ---,---- -- . h_ -----
(p-l, Vi) (Ph Pj)
into a product of two non-positive integers. \Ve consider this factorization
an the dillerent cases which can arise.
(a) If I1t}= 1 the factorization must be 1 = -1. -I. Thus (Pi, Pt)=(Pl, Pl)
null the roots PI, Pl have the same length.
(b) If 11l}=2 the factorization must be 2= -1. -2. Thus one of pi, Pl
is \/2 times as long as the other.
(c) If 111] =3 the factorization must be 3 = -1. - 3. Thus one of pi, Pi
t \-/3 times as long as the other.
(ll) If 111]=0 we obtain no information about the relative lengths of
PI, Pl.
We observe that the pl-chain of roots through Pl has length 1, 2, 3 or 4.
For
2(pl, Pj)
(p, pI) =P -q,
\10 here p, q are the integers defined as before by the p'l-chain of roots
through Pl. We have seen that
2(/, Pl! =0 -1, -2 or -3;
(pl, Pt) ,

40
SIMPLE GROUPS OF LIE TYPE
furthermore P =0 since Pi begins the pt-chain through it. Hence q  3,
and so the Pt-chain through Pi contains at most four roots.
The same argument shows in fact that any r-chain has at most four
roots. For let s be the first root in some r-chain. Then
2(1', s)
Cr, r) =p-q
and, as before, this must take one of the values 0, -1, -2, -3. Since
P =0 we have q  3, and so the r-chain has at most four roots.
We now define the I?ynin diagram of th_Lie algebra JL This is a
graph with 'Tllodes,' one ,ss<?a.t'_.' Wlt- ead; fup-damentl root pi, such'
that the jth node is joined to the jth node by a bond of strength llij,--.M
For example, in the Lie algebra of all (1+ iYx (/+ I) matflces of tnice 0
it can be shown that the fundamental roots PI, P2, . . . , Pl all have the
same length. Consecutive roots Pi, PHI are inclined at an angle 271/3
whereas fundamental roots which are not consecutive are orthogo1"\al to
one another. Thus the Dynkin diagram of this algebra is
1 2 L-1 t
0-----0- - - - --cr-------o
Now the possible Dynkin diagrams of simple Lie algebras can be
enumerated, using the classical results of KiHing and Cartan. It can be
shQwn .tl1..(t_. the Dynkin diagram of a simple Li_.algebra must be on-
}JGt_cJ._gEE.:M Furthermore the only''cted'-'-graj)hs-whfch;;;-
Dynkin diagrams of sim pIe Lie algebras are tbe ones in the following list
A (/  1 )
,
s, (f  2) }
C, ( /  3)
q (/  4)
G2

E
6
0-------0---- - - - - ----0----0
0--------0- - -
r)
0--0------_-<

o-----a::::::::

E7
E
8
The diagrams are usuaIIy named as shown, the suffix denoting the rank
(i.e. the number of nodes in the graph). The reason that the second type
SIMPLE LIE ALGEBRAS
41
of diagram is given two different names is that the Dynkin_4igr!p....}:s
nul always detex:mipe the simple Lie algebra to within isomorphism.
{onsiJcrtT-i prble;;-;rreco'venngfhe'cotig';;tion formed by the funda-
mental roots from a knowledge of the Dynkin diagram. In the diagram
1 2 L-1 L
0---0- - - - ---a::=:::D
it is evident that the corresponding fundamental roots PI, P2, . . . , pl-I
all have the same length, but PI is either '\1'2 times shorter or '\1'2 times
longer than the remainder. If Pl is shorter the system of fundamental
roolS is said to have type Bl, and if Pl is longer the system has type Cz.
If 1=2 there is no distinction between Bz, Cl as we can obtain either by
numbering the nodes suitably. For the same reason there is only one funda-
mcntal root system of type G2 and one of type F4, since the diagrams are
i) Il1mctric. In all remaining cases the Dynkin diagram contains only
,wglc bonds-thus all the fundamental roots have the same length and
thc configuration formed by the fundamental roots is uniquely determined
by thc diagram.
Now it is possible to recover the complete system of roots (as linear
combinations of the fundamental roots) from a knowledge of the relative
lengths of the fundamental roots and the angles between them. For if
e know the configuration formed by the- fundamental roots, we know
the fundamental reflections w,., r E IT. Since the fundamental reflections
generate "V, the Weyl group is known. Finally, each root is the image
of some fundamental root under an element of the Weyl group, thus
Ihe complete root system is determined.
-.--.\
-
3.5 The Existence and Isomorphism Theorems
We have seen that every simple Lie algebra over C determines a root
) tCI11, and we now consider which root systems arise from simple Lie
\Jbcbras, and to what extent a simple Lie algebra is determined by its
fout system.
A root system <1) is said to be indecomposable if it cannot be decomposed
1n10 two non-empty complementary subsets (VI, (1)2 such that (r, s)=O
fur all rE (1)1, SE (1)2. Two root systems <1>1, (1)2 are said to be equivalent
If thac exists a bijection a : (1->1 <l>2 such that
r.
)f
.
(a(r), a(s)) = .\(r, s),
r, s E <1>1,
....hcre .\ is some positive real number independent of r, s.
f
t.



42
SIMPLE GROUPS OF LIE TYPE
t-
fft---,

L
f'
I
ti
.... '
L
i<"'_....

l
,
!""
l....
-
I .-,
i
I
L
!
L.

j
J
!
I
I
\ '"
Now the root system determined by a simple Lie algebra is indecom-
posable. This follows from the fact that the Dynkin diagram of a simple
Lie algebra is connected, using 2. 1 .8.
THEOREM 3.5.1 (Existence theorem). Let <D be an indecomposable
root system. Then there exists a simple Lie algebra over C which has a
root system equivalent to <1>.
A proof of the existence theorem using the concepts we have outlined
can be found in Tits [15].
We now consider the relation between two simple Lie algebras which
have equivalent root systems. We first describe some properties con-
cerning the muhiplication of the root spaces in a Cartan decomposition.
Let
JL =}[ Ef) z: JLr
rE \D
be a Cartan decomposition of JL. Then, for any pair of roots r, s E <1> we
have:
(i) [JLrJL8] = lLr+8'
(ii) [JLrJL8] = 0,
(iii) [1LrJL-r]=Cr.
(iv) [l£)JLr]=JLr.
In (iii) r is interpreted as an element of . Instead of considering
the root r E  it is often convenient to take a scalar multiple hr of r, defined
by
if r+s E <D.
if r+s  <1>, r+sO.
Since we have
2r
Izr = -.
. -7 (r, r) .
[n e I"J:  1) C ' ')  ')
2{r, s)
[hres] = -( ) eB = ArSe8'
r, r
( P\x,,:'
1) \'\\) -- 
.0 _, f. _
::. 1- <" t)) f
.2;: "J
\ -? I "'"{Ii ""1. ( ) \.
,DJ! j - -'J 
I J
it is evident that ad hr transforms e8 into an integral multiple of itself.
By property (iii) above we can find, for each er  0 E JLr, an element
e-rE 1!..-r such that [ere-r]=hr.
We can now state the isomorphism theorem for simple Lie algebras.
l
"I;;;!4!1I1"'"
THEOREM 3.5.2 (Isomorphism theorem). Let 1L, JL' be simple Lie
algebras over C with Cartan subalgebras 1£}, ' of the same dimension I.
I ,) /
Let PI' P2p . . . , Pl; and PI' P2' . . . ,Pl be sets of fundamen ta roots for
SIMPLE LIE ALGEBRAS
43
I, 1.' and let
A. = (!?!p;).
'J (p, p)
2(Pt, Pj)
A tj = --- --.--. -,
(pi, Pi)
l..('(
2pt
hpt = ---. ----,
(Pi, /Ji)
amI leI CPt EJLpt> e-Pt E JL-Pt be chosen so that [eple-pt] = hpt. Define
II j',. (> pi c-TJi similarly in 1L'. ..... I
Suppose A tj = A:j for all i, j. Then there IS a till/que Isomorplusm 8 . 1L -4 JL
JUdl Ihat e(hpi)=hpi, 8(ep)=elJi, 8(e-p)= e-1Ji.
//1 particular allY tlVO simple Lie algebras ouer C with equivalent root
Jyj(I.'/IIS are isomorphic.
r\ proof of the isomorphism theorem can be found in Jacobson [1],
p.I27.
3.6 Description of the Simple Lie Algebras
It follows from what has been said in sections 3.4 and 3. 5 tht the slple
L1C algebras over C are the ones shown in the 'stanard Irst' exhibited
lxlow. We have given for each algebra the dimensIOn, the rank, tle
number N of positive roots, the order of the Weyl group, and the Dynklll
dlJgram.
J. dim 1L rank 1L N [WI Dynkin diagrarn
1(/+2) l -PU + 1) (/:+1)! 0---<>- .-.-0-----<>
A,(l;;:.I)
1(21+1) 1 [2 21.l! 0-----0- - - --<r===D
l1ill 2)
1(21+.1) 1 1:3 21.l! 0------0- - - . -a::=::=D
CI(1 3) 0--0----<
/J,(l 4) 1(21- J) 1 1(/ - J) 21-1.1!
2 6 ]2 
(;3 14
24 27.32 0 0
h 52 4
1:6 78 6 36 27.34.5 
/:"; 133 7 63 21°.3'1.5.7 
r.. 248 8 ]20 21-1.35.52,7 
The matrix (At}) defined in 3,5.2 is called the Cartan. matrix o JL.
The isomorphism theorem shows that the C:art.aI matlx determl11es
lhe Lie algebra 1!... The Cartan matrices of the mdlvIdual simple algebras

44 SIMPLE GROUPS OF LIE TYPE
SIMPLE LIE ALGEBRAS 45
are shown below. (- -1)
G2: 2 ' --.""
2 -1 l
-1 2 -1 [- -1 0 -!]. -...,
-1 2 -1 0 F4: 2 -1
-1 -2 2
Al: 0 -1
-I 2 -1 0 0 0 0
0 -1 2 -I -1 2 -1 0 0 0
-1 2 -1 E6: 0 -1 2 -1 -1 0
0 0 -1 2 0 0 ,
-I 2
0 0 -I 0 2 -1
2 -I 0 0 0 0 -1 2
-1 2 -1
-1 2 -1 0 2 -1 0 0 0 0 0
-I -1 2 -1 0 0 0 0
Bl: 0 -1 2 -1 0 0 0
E?: 0 0 -] 2 -] -] 0
-1 0 0 0 -1 2 0 0
0 -1 2 -1 0 0 0 -1 0 2 -]
-1 2 -1 0 0 0 0 0 -1 2
-2 2 2 -1 0 0 0 0 0 0 Rrl:1\t,
r
2 -1 -1 2 -1 0 0 0 0 0
'tl,
-1 2 -1 0 -1 2 -1 0 0 0 0 1.
,
-1 2 -1 0 0 0 -1 2 -1 0 0 0
-1 £8: 0 0 0 -1 2 -1 -1 0
Cl: 0 0 0 0 -] 2 0 0
-1 0 0 0 0 -] 0 2 -1
0 -1 2 -1 0 0 0 0 -0 0 -1 2
-1 2 -2
-1 2 We also give a description of the indecomposable root systems. We
h<:gHl with the systems of rank 1 and 2, viz., systems of type AI, A2, B2, G2.
2 -1 I'lgure 1 shows the roots expressed as integral combinations of funda-
-1 2 -1 mental roots.
-1 2 -1 0 In order to describe the root systems of higher rank it is convenient
-1 to use an orthonormal basis of the vector space containing the roots.
Dl:
-1 (i) Type Al. Let eo, el, . . . , el be an orthonormal basis of a Euclidean
0 -1 2 -1 -1 )pacc of dimension 1+ 1, and let't) be the subspace of vectors
-1 2 0 l l
-1 0 2 L: A.lei with 2.: Ai =0. - .......
£=0 i=O

46
SIMPLE GROUPS OF LIE TYPE
I,,
-0
o
o
o
A1
b
o+b
b
a
t
b
-0
A2
r:
D
c
b
o+b
[
82
-0
o
t_
["--
.'  .
. .'"
-a-b
-b
3a+2b
(
G2
.r
. 1
-3a-2b
Figure 1
..
Then the fol1owing vectors in lJ form a fundamental system of type AI.
e -e e -e e.
o 1 1 2 /-2 -"l-1 "l-1 -e,
0---0---- - --0---..0
The full system of roots with the above fundamental system is given by
<D={et-e' ;-.t.. . '-0 1 I}
j, -r-J,I,J-, ,..., .
.......A
.1;:1J.l:J-
L
srMPLE LIE ALGEURAS
47
(ii) Type Br. Let el, e2, . . . , el be an orthonormal basis of a Euclidean
p.lCC 1'. The following vectors form a fundamental system of type Bl.
e1 -e2 e2 -e3 e3 -e4 1--1 -fj e,
0-----0------0- -- - --a::=::=D
The full system of roots with the above fundamental system JS given
by
(:tet:tej; ii=j, i,j=I, 2,... '1
w= .
:t et; i = 1, 2, . . . , I
(iii) Type C/. Let el, C2, . . . , el be an orthonormal basis for 11. Then
the following vectors form a fundamental system of type C/.
e1 -e2 e2 -e3 e3 -e4 tt-1-t.( 2el
0------0------0 - - ----a:==:=D
The full system of roots with this fundamental system is given by
( :tei:t ej; i i= j, i, j = 1 , 2, . . . ,/]
$= .
:t 2et; i = 1, 2, . . . , I
(iv) Type Dl. Let el, e2, . . . , el be an orthonormal basis for 'P. The
following vectors form a fundamental system or type Dl.
e1 ___ '1-2:0--0 e'_1-e,
tj_1+et
The full system of roots with this fundamental system is given by
(I)={:l::ei:!::ej; ii=j, i,j=I, 2,... ,/}.
(v) Type G2. This has already been described.
(vi) Type F4. Let el, e2, e3, e4 be an orthonormal basis for V. The
following vectors form a fundamental system of type FI1'
e1-e2 e2-e3 e3 (-e1-e2-e3+e4)
o-----a:::::=:=--o
. The full system of roots is
(:tei:tj; ii=j, i,j=1,2,3,4j
$ = :t ei; 1 = 1, 2, 3, 4 .
-H :tel:t e2 :t e3 :t e4)

48
SIMPLE GROUPS OF LIE TYPE
(vii) It is convenient to describe next the root system of type E8. The
systems £7, E6 are then easily obtainable as subsystems.
Let el, e2, e3, e4, e5, e6, e7, es be an orthonormal basis for 't1. The foHow-
ing vectors form a fundamental system of type Ea.
8
e1-e2 e2-e3 e3-e4 e4-eS e&;.-e6 e6+e7 _i 2: e
oJ 2 i=1 i
e6 -e7
The full root system is
(:t: ed: ej; i f= j, i,j= 18' 2, 3, 4, 5, 6, 7, 8).
<D= 8
t l: £tet; q = I 1, IT £t = I
i=1 i=1
(viii) Let et (i= I, 2, . . . 8) be as in (vii). Then we have a fundamental
system of type E7 given by
e2-e3 e3-e4 e4 -es e5 -e6 e6 +e7
1 8
-- 2: e.
2 i =1 I
e6 -e7
These vectors lie in the subspace of elements
8
l: >"tet satisfying >"1 = >"8.
i=1
The full root system is
IetIej; if=j, i,j=2, 3, 4, 5, 6, 7
:t: (el + e8)
<D=
8 8
t l: £tet; q = II, £1 = £8 = I, IT £t = I
i=1 i=1
8 8
- t L (et; £t = I I, €I = £8 = 1, n £t = 1
i=1 i=J
SIMPLE LIE ALGEBRAS
(ix) Let et(i = I, 2, . . . 8) be as in (vii). Then we have a fundamental
s)'!.tcm of type E6 given by
1 8
e3-e4 e4-eS eS-e6 e6+e7 -2/1e,.

e6-e7
These vectors lie in the 6-dimensional subspace of elements
8
 >"tet satisfying >"1 = >"2 = >"8.
i=1
The full root system is
:!:etIej; i-:/=j, i,j=3, 4,5,6,7
8 8
t  €tet; £t = II, €I = £2 = £8 = I, 11 q = 1
i=1 i=1
8 8
- t l: £iet; £t = ::!: 1, €I = £2 = £8 = 1, n £i = 1
i=1 i=1
<D=
For further information about these indecomposable root systems, the
rC41Jcr is referred to Jacobson's book [I].
Let (1) be any indecomposable root system. For each root r E <D define
2r
Irr=----.
(r, r)
h, is called the co-root corresponding to r (cf. 3.5.1). Let <D* be the set
of co-roots hr for all rE <D.
PROPOSITION 3.6. 1. (1)* is also a root system. Moreover if <I> contains
roo Is of lwo different lengths, r is a short root of <I> if and only if lir is a
IOllg rool of (1J*.
PROOF. We show that (1)* satisfies the axioms for a root system
(2. I . I). Sioce IIr is a scalar multiple of r we have Wr = WIl,. Thus
2s 2wr(S)
whr(hs) = Wr -( ) ( () ( » = hWr (8)'
S, S Wr S , Wr S
Als.o we have
2(hr, hs) 2(s, r)
--=-,
(Ilr, hr) (S, S)
\\hich is an integer. Thus <1>* is a root system.
') ti t.
f
(Y'J r
.{/l
1 r ;1
Cvt-. . ,;....L' .J C!
( I__J ( ("."
'J v! 'I . ,.-
49
-"'\

:
.-....
f
o,)
pIP"",r.
 i,""
----i
"'-\' - Q. ?

f:
t
b
y
'(
c
I '
[:
t
.........11-
['"
'",v
i ,
[
"
f '''''
r'''-
f
\.,w
ir: I
["
i I
e  1 i
j [;._--
, ,
"-',(
": '-...
L
',_.,. , i
50
SIMPLE GROUPS OF LIE TYPES
Now (hr, hr)=4/(r, r) and so (r, r)«s, s) if and only if (hr, I1r) > (hs, I1s).
Thus r is a short root in <I> if and only if I1r is a long root in <1>*. "
(D* is caJIed the dual root system of <D. It is clear that (1)** =<I).lL
readily verified that the dual of a rootg!!!. .2ftype Bl is a system of
type Cl and that G2 ad F4 are selr--dflJ.,__yi,n)i"Sm-ce--.B;-C;-'-tiis'T;
-a"seifdual sys'tem" a1so.'-Tlie--(rualiiy'js''''tivial fr systems whose roots all
have the same length.
We remark that if II is a fundamental system in <I> then IT* is a funda-
mental system in <1>*.
We conclude the present chapter by giving two lemmas which will be
useful in the development to foJIow.
LEMMA 3.6.2. Any positive root rE <1>+ can be expressed as a sum of
fundamental roots
r=pi1 +Pi2 + . . . + Pit
in such a way that Pi1 + ptz + . . . + Pia is a root for all a  k.
PROOF. Let
1
r= I: J2.lP1.
i=l
be the expression of r as an integral combination of fundamental roots.
Then
(r, r)= (r, il n1Pt) = il n1.(r, Pi).
Now (r, r»O and ntO for each i. Thus there is some i for which
(r, Pt) > 0. Suppose r is not a fundamental root. Then r, pot are linearly
independent, and r cannot be the first member of the Pt-chain of roots
through it, thus r- P1. is a root. By repeating this process we obtain the
required expression for r as a sum of fundamental roots. .
LEMMA 3.6.3. Let r, s be roots such that r + s is a root. Then the integral
combinations of r, s which are roots (i.e. the elements of <I> of form jr+ js
with i, j E ) form a root system of type A 2, B2 or 02.
PROOF. The elements of <D of form ir+ js, with i, jE, satisfy the
axioms 2.1.1 for a root system in the 2-dimensional space they generate.
This 2-dimensional system is indecomposable, since it contains two
independent non-orthogonal roots, so must have type A2, B2 or 02. .
CHAPTER 4
The Chevallcy Groups
We now begin the development of the theoy of the Chevalley grops,
ilia king use of the properties of tle sinplc Lie algebras over C .descflbe.d
111 th last chapter. The information given there about these Inple _.Ie
_ I . 'bl"'1s may be regarded as classical; however we shall req Ull e certdll1
il Jili:)nal facts about them which necessitate a closer Iok at the Cart an
omposition. We shall show that if JL is a simple Lie algebra oe.r C
t is possible to choose a basis for 1£., adapted ,to a Cartan d.ecomosltlOn,
UL'h that the constants of multiplicatiun with respect to this basIs are all
rational integers.
4.1 Properties of the Structure Constants
Let
JL=) ED  1£.r
r (, <J.)
be a Cartan decomposition of JL. Let
21'
I1r='-- m_
(r, r)
be the co-root corresponding to the root r E <I>. For each root. I' let e: be
I t f']T If e- is alrc'ldy chosen for r E<l)+ there IS a ul1lque
a non-zero e emen 0 Jl-.-r. - r <- <- .
I t E 1£. SllCl1 tl\'lt [e e-r] = hr and we shall suppose e-r chosen III
c mcn e-r -r - <- r ,
\illS way. The set
{hr, r E IT; er, r E <1>}
'. b . Cor '":If It consist"-ri\;'-f;d'l'tl"co-roots hr together with
IS a aSlS I' Jl-.-. . . I . I t r tl er
the set of all root vectors er. The elements of this basIs mu tIp y oge 1
. as follows:
[l1rl1s] =0,
[hres] = A rses,
[ere-1'] = hr,
[eres] =0,
r, SEl1,
rEl1,SE<1>
rE (p,
r, SE <I>, r+s <D.
51

52
SIMPLE GROUPS OF LIE TYPE
If r, s, r + S E <D then [eres] is a scaJar multiple of er+6 since [J[r'I.s] = Irt,.
We define Nr, a by
[eres] = Nr, ser+8.
The elements .Nt. s for r, SE lD are called the SJDJt1-lJ--e constants of I.
They clearly depd--upoli.---the-- C]l-Glee of th-;oot vear1iN
task is to consider the relations betwcen them.
(i) Since [eser]= -[eres], it is clear that Na, r= -Nr, a for all r,sE<1>.
(ii) Suppose rI, r2, r3 are three roots such that ri + r2 +'3 = O. By the
Jacobi identity we have
[[ er1er2]e"3] + [[ er2er3]erl] + [[er3erl]er2] = O.
Thus
"
\
'-
Nr1. r2[e-r3era]+Nr2. rJe-r1er1] + Nr3. r1[e-r2er2] =0.
Hence
Nrlo r2hra + Nr2. r3hr1 + Nr3. r/1r2 = O.
It follows that
---..
...;:
2Nr1. r2r3 + 2Nr2. rar1 + 2 Nr3, r}r2 = 0
(f3, (3) (rI, rI) (r2, r2) .
Using the fact that r1 + r2 + r3 = 0 we have
,\'
(Nr_Nr12)rI + (If-r_1J._ Nr1)r2=0'
(rI, rI) (r3, r3) (f2, r2) (r3, r3)
Now r1, r2 are linearly independent. For otherwise r2 = :!: r1 and r3 = - 2rl
or 0, a contradiction. Therefore the coetTicients of rI, r2 in the above
equation must be zero, and we have
Nr1, r2 _ Nr2. r3 _ Nr3. r}
(r3, r3) - (1"1, rI) - (r2, r2)'
(iii) Suppose r, SE <I> are ]inear1y independent. By the Jacobi identity
we have
[[ere-r]es]+ [[e-res]er]+ [[e8er]e-r] =0.
I--Ie n ce
[hres] + N-r. 8[e-r+ser] + N8, r[er+8e-r] =0.
THE CHEV ALLEY GROUPS
53
(It is convenient here and throughout the exposition to assume that
Nr. ,=0 if r, SE <D but r+s is not a root.) Thus
Ar8e8+N-r, sN-r+s, res+Ns. rNr+s. -re8=0.
Using the relations obtained in (ii) above, we have
(-r+s, -r+s) N N (s, s) -0
ArI+Nr-a, -rN-r+s. r (S, S) + 8. r -r. -8 (r+s, r+s) .
We may rewrite this as
(s, s) (-r+s, -r+s) - A
Nr aN-r -s -( --)-Nr. -r+sN-r. r-8 (s s) r8'
. · r+S, r+s ,
-)
-''1,
\\'e definc Mr. 8 for r, s E <D by
(s, s)
Mr 8=Nr aN-r -8 -).
· , · (r+s, r+s
Then the above equation becomes
Mr. s-Mr. -r+s=Ars.
We now consider the r-chain of roots through s and apply this equation
repeatedly. Let this r-chain be
- pr+s, . . . ,s, . . . , qr+s.
-)]
y
Then we have
Mr, s-Mr, -r+s=Ar8,
Mr. -r+8- Mr, -2r+8=Ar, -r+8=Ar8-2,
Mr, -(p-l)r+s-Mr, -pr+s=Ar, -(p-I)r+8=Ars-2(p-I),
Mr, -pr+8=Ar, -pr+s=Ars-2p.
(The term Afr, -(pH)r+s does not appear in. the last equation since
_ (p + l)r+s is not a root.) Adding these equatlOns we obtam
Mr, s=(P+ I)Ars-p(p+ 1).
However, Ar8= p-q, as in section 3.3, thus
Mr, 8= -(p+ l)q.

I,
,-
f
I

r

f
II cncc
(r+s, r+s)
Nr. 8N-r, -s= -(p+ l)q (s, s) .
-"\
: 
c

54
SIMPLE GROUPS OF LIE TYPE
This expression can be simplified by the use of the following lemma.
LEMMA 4.1.1. Suppose r, s, r+sE «1>. Then
(r+s, r+s) _p+ 1
(s, s) -fJ.
[
PROOF. We consider separately the different possibilities for the
r-chain through s, bearing in mind that the length of any r-chain is at
most 4. We use the information given in section 3.4 about the relative
lengths of roots inclined at a given angle. The various possibilities for
the r-chain are shown below.
[""
.::;;;j
s r+s
0--0 p=o q=1 (s, s) = (r+s,r+s)
s r+s 2r+s
0--0---0 p=o q=2 (s, s) 2(r+s,r+s)
-r+s s r+s f
0--0---0 p=1 q=1 (s, s) = z(r+s,r+s)
s r+s
ס--ס---o---o p=o q=3 (s,s) = 3(r+s,r+s)
s r+s (5, s)
ס----ס----o---o p=1 q=2 = (r+5, r+5)
s r+s p=2 q=1 (S,5) = (r:t"s, r+s)
ס--ס---o---o
'",-
The result of the lemma is clearly valid in all cases.
.
L
Applying this result we obtain
Nr, 8N-r, -8= -(p+ 1)2.
LJ
(iv) Finally we consider four roots rl, r2, r3, r4 such that
T1 +r2+r3+r4=0
and such that no pair are equal and opposite. By the Jacobi identity we
have
f
Ld
( --
L_.
[[ erl er:a]ers] + [[ er:aerS]erl] + [[ erSerl]er2] = o.
Thus
Nrl. r:aNrl+r2, rs+Nr2, rsNr:aTrs, rl + Nra, rlNra+rl> r2=0.
Now by the formulae obtained in (ii) above we have
\
\..,;
Nrl+r2, r3_ Nra, r4
. ("4, r.J- - (rl + r2, rl-+ r2)
TilE CI-II:YALLEY GROUPS
55
and there are corresponding formulae obtained by permuting the roots.
thing these formulae we obtain
Nr!> T2NT+ Nr2. TaNT!> r4 +_!!r3. TINr=O.
(r1 + "2, r1 + r2) (r2 + r3, r2 + r3) (r3 + r1, r3 + r1)
As usual we note that a term may be 0 if the corresponding vector is not
a root. However, suppose r1 +r2E <1>. Then r3+r4E <I> since
r3+r4= -(rl+r2).
Thus the first term in the above formula is non-zero. Hence at least
on of the other terms is non-zero. Thus either r2 -I- r3 and rl + r4 are roots,
or '3+r1 and r2+r4 are roots, or both.
We summarize the results we have obtained in. the following theorem.
THEOREM 4.1.2. The structure constants of a jimplc Lie algebra J[.
ove, C satisfy the following relations:
(i) Ns. r= -NT. s,
r, s E ([>.
(") Nrlo r2 Nr2. r3 Nr3. TI
II ----- =------ ---=----------
(r3, r3) (rl, rl) ('2, r2)
If'l. '2, r3 E (1) satisfy '1 -1-'2 -I- r3 = O.
(iii) Nr, 8N-r, -8= -(p+ 1)2,
r,SE(}).
(iv) Nrl> r2NTa. r4 + NT2. T3NT1' T4 _ + NT3. rlNr2. r4 - 0
(r1 + r2, r1 + r2) (r2 + r3, r2 + r3) (r3 + r1, r3 + rl)
ifrl, r2, T3, r4E<D satisfy r1+r2+r3+r4=0 and if no pair are opposite.
II
1
r
!'
4.2 The Chevalley Basis
Formula (iii) of Theorem 4.1.2 suggests that, by a sufficiently careful
choice of the root vectors er, it might be possible to arrange that
Nr, 8= ::f: (p+ I)
for all relevant pairs r, s of roots. This is in fact so, and to prove it we
use the isomorphism theorem for simple Lie algebras, stated in 3.5.2.
In the notation of 3 . 5 .2 we dell ne 1L' = JL and p = - Pi for i = 1, 2, . . . , I.
Then it is clear that Aij=Aj' We also define c1Ji= -e-Pi and e-pi= -cPi'
f,
i
!;
:1
i\

56
SIMPLE GROuPS OF LIE TYPE
Then
[e1Jie-pi]= [e-p,ept1= -hp,=h-p,=hpi,
as required. Thus there is an isomorphism 0 of J[ to itself, i.e. an auto-
morphism of JL, such that
O(epJ= -e-Pt'
O(e_PI) = -ept'
O(l1pl)= -hp,.
Now 02 transforms epi' e-pi, hPI into themselves, so must be the identity
by the uniqueness part of 3.5.2. Thus 0 has order 2.
Now by 3.6.2 each root rE <1)+ can be expressed as a sum offundamental
roots
r=rl +r2+ . . . +rk
in such a way that rl + r2 + . . . + r a is a root for all a  k. It follows
that [[Cr1CrZ]... CrA;] ELr and is a non-zero scalar multiple of er. The
image of this element under 0 is [[ - e-rl' - e-r2] . . . - e_rk], which is
a non-zero scalar multiple of e-r. Thus er is transformed by 0 into a
scalar multiple of C-r.
Let O(er)= Ae-r. Then, since 0 has order 2, O(e-r)=.\-ler. Hence
O(J1-er)= J1-AC-r= J1-2A(J1--1e_r).
Now it is possibJe to choose J1-EC such that J1-2= _.\-1. With such a
choice of J1- we have 8(J1-er)= -J1--1e--r and [J1-er, J1--1c-r]=hr. We now
a1ter our choice of the root vectors. By choosing J1-er as the root vector
in JLr and J1--1e-r as the root vector in JL-r, and then renaming these
Cr, e-r respectively, we see that it is possible to choose erEJLr, e-rEJl-r
such that [ere-r] =hr and 8(er) = -e-r.
Now
[eres] = Nr, ser+s
whenever r, s, r+sE<D, and so applying 0 we have
[- e-r, - e-s] = - Nr, s e-r-s.
It foHows that N-r. -8= -Nr, s. However Nr. sN-r. -s= -(p+ 1)2, and
so N r, s = ::!:: (p + I).
We can now state Chevalley's basis theorem for simple Lie algebras.
THEOREM 4.2.1. Let JL be a simple Lie algebra over C and
1..=J1) EB L JLr
rEClJ
THE CHEVALLEY GROUPS
57
!j
J
,{

l --
-\1 '

rJ'
f
i
f
b-l a Carlan decomposition of JL. Let hr E) be the co-root corresponding
Iv the root r. Then, for each root rE <1>, an element er can be chosen in 1Lr
such Ihat
[ere-r] = hr,
[eres]= ::!:: (p+ l)er+s,
..-hae p is the greatest integer for which s- prE <1).
.!.!.!!..slemeI11 Jb r  r..JI;_€-(,t,-JiQll?Llt kqJi.[E._r..}!::_,"" called If-C:!!!E!!!!l!.!
J!!!:l_:.., The basis elements multiply together as follows:
[hrhs] =0,
[hres] = Arses,
[ere-r] =hr,
Jeres] =0
[eres] = Nr, ser+s
ifr+sf/3 <1>,
if r+SE <D,
i\.hae Nr. B= ::!:: (p+ 1).
111C multiplication constants of the algebra with respect to the Chevalley
hasis are all integers.
""!I!\;
PROOF. Since Ars and Nr, B are integers, it only remains to prove
thJt each co-root hr is a linear combination of the fundamental co-roots
\\ ilh integer coefficients. This follows from the fact that the co-roots
h, for rEn form a fundamental system of the dual root system (cf.
3.6. I). .
!<
L
1,
Now a simple Lie algebra JL has many different Chevalley bases, and
we consider the amount of freedom available in the choice of such a
b.l50is. In the first place, every Chevalley basis is defined relative to some
C.trtan subalgebra. ]f a Cartan subalgebra :II) of JL is prescribed, then
the root spaces JLr are determined. The elements of the basis which lie
If} 1lJ arc defined rdative to some fundamental system IT in <1). If n is
prescribed then the fundamental co-roots hr, r E IT, are determined. The
f unJamcntaJ root vectors, er, r E n, may be chosen as arbitrary non-zero
dements of the root spaces 1..r, rEU. The remaining positive root vectors
Cf, rEtI)I-, arc now determined to within a si'gn by [eres] = :!:(p+ l)er+s,
uing 3.6.2. The relation [ere-r]=hr then determines the basis vectors
t., for rE (1)-.
'-

58
SIMPLE GROUPS OF LIE TYPE
I
...
Systems of stn/cture constants
Now every Chevalley basis determines a system of structure constants
Nr,lf given by [ere8]=Nr,lfer+If' Since Nr,8= :t(p+l), wherep is defIned
as before, the absolute value of Nr, 8 is determined. However, different
Chevalley bases will give different values for the signs of the Nr, 8. W(.
consider the extent to which these signs can be chosen arbitrarily in
determining a system of structure constants. We show that for certain
ordered pairs (r, s) of roots the sign of Nr, 8 may be chosen arbitrarily
and that the remaining structure constants are then determined.
Suppose we are given a total ordering on the space containing the
roots, as in section 2.1. An ordered pair (r, s) of roots wiH be calkd a
special pair if r+SE (D and 0-< r-<s. An ordered pair (r, s) is called extra
special if (r, s) is a special pair and if for all special pairs (1'1, SI) with
r+s=rI +SI we have r  ri. Then every root in cp+ which is the sum of
two roots in (v+ can be expressed uniquely as the sum of an extraspccial
pair. Since by 3.6.2 every root in <})+ which is not in n has this property,
the extraspecial pairs are in 1 - 1 correspondence with the roots in
CP+-I1.
Now given a Chevalley basis {hr, rEll; er, rEeD}, we may change the
sign of any subset of {er, rE(})+- ll}, and changing the signs of the c,
for r E<I)- in a corresponding way to preserve the relation [ere-r] =h"
we obtain another Chevalley basis. Thus the signs of the structure
constants Nr, 8 for extraspecial pairs (r, s) can be chosen arbitrarily.
On the other hand, we show that if the Nr, 8 are given for the extraspecial
pairs they are determined for all pairs.
J
1
,
L1
L


i '_'
!
i J
f -.
j
I
l-__
.t
/1:
ili
J1'
t--
:!t
o:! I
il ;
j L
PROPOSITION 4.2.2. The signs of the structure constants Nr, 8 may be
chosen arbitrarily for extraspecial pairs (r, s), and then the structure cons tail IS
for all pairs are uniquely determined.
PROOF. Let {hr, rEll; er, rE<D} and {hr, rEll; e;, rE<D} be two Che-
valley bases giving rise to systems of structure constants (Nr, .), (N;, ,).
Thus [ere.]=Nr, .er+. and [e;e;]=N;, .e;+..
Let e;= rer' where \#0 EC. Then r,N" ,= r+.N;. ,. Now suppose
that Nr. .=N;.. for all extraspecial pairs (r, s). Then r8=r+8 for such
pairs. By 3.6.2 it follows that if r E (1)+ and r = nIPl + . . . + nIpl, then
t : 
!J:
!
,= ;; . . . ;;.
The same equation holds for negative roots since _,= ;1. As a conse-
quence of this we have rs = r+8 for all pairs (r, s) of roots, and so
")
i.  i
I .
j
t
THE CIIEVALU'Y {.!,()UI'S
,"-t'r. ,=N;, If for all pairs. Thus the structure constants are uniquely
dtarnined by their values on the extraspecial pairs. .
It is of interest to point out that the values of the structure constants
N, , can be derived from the structure constants on the extraspecial
pairs by the relations (i), (ii), (iii), (iv) of 4. 1 .2. To prove this consider
thl: !let of ordered pairs (1', s) of roots such that r+ SE <}). If (r, s) is such
a pair, so arc thc following twelve pairs or roots:
(r, s), (s, r),
(s, -r-s), (-r-s, 5),
(-r-s, r), (r, -r-s),
(-r,-s), (-s,-r),
(-s, r+s), (r+s, -s),
(r+s, -1'), (-1',1'+5).
Since r+s+(-r-s)=O, either two of r, s, -r-s arc positive or one is
pu:.itive. Jt follows that of the above twelve ordered pairs of roots, exactly
on is a special pair. Moreover relations (i), (ii), (iii) of 4. I .2 determine
all the structure constants in terms of the Nr, s for special pairs (1', s).
It rcmains to show that the Nr,8 for special pairs can be expressed
in terms of the Nr, 8 for extraspecial pairs. Suppose (1', s) is a pair of
roots which is special but not extra special. Then there is a unique extra-
rx:cial pair (n, SI) such that r1+sI=r+s. Since r+s+(-r])+(-sI)=O,
rdation (iv) of 4.1.2 gives
!!!0..N-rlt -.1'1+ Ns. -rlNr, .!+ N-r1, rNs. -.1"1 -0.
(r+s,r+s) (s-rI,S-rI) (-1'1+1', -rI+r)
Now the roots r, s, r1, -1 are ordered by
0-< rl -< r -< s -< SI.
Thus the special pairs associated with the pairs
(-rI, -SI), (s, -(1), (r, -51), (-rI, r), (s, -SI)
respectively are
(r1, SI), (1'1, s - rI), (SI - r, 1'), (r - rI, n), (SI - S, s).
However for each of these special pairs (r, s) we have f+s -< r+s. Thus,
by using relations (i), (ii), (iii), (iv) of 4. 1 .2, Nr. s can be expressed in
tcrms of Nrlt 81 and various terms Nr. I' where (f, s) is a special pair with

60
SiMPLE GROUPS OF LIE TYPE
f+s-<r+s. By using induction on the sum r+s it can be seen that Nr,.
is determined by the values of the structure constants on the extraspecial
paIrs.
4.3 The Exponential Map
We shall now show how to construct certain automorphisms of a Lie
algebra, using the exponential map. We recall that an automorphism of
a Lie algebra Jf. is a non-singular linear map 8 of JL into itself such that
[8x, 8y]= 8[x, y].
The set of all automorphisms of J[ clearly forms a group.
LEMMA 4.3.1. Let J[ be a Lie algebra over a field of characteristic
o alld 8 be a derivation ofJL }v/ric/z is nilpotent, i.e. .satisfies 8-n9 for some
11. Th e 11 ---._...,...,-
82 8n-1
exp 8=1+8+---+ ... +--
2! (n-l)!
is an automorphism of JL.
PROOF. exp 8 is a non-singular linear map, its inverse being exp (- S).
Now we have
8 [xy] = [8x,y]+[x, 8y],
x, YEJL.
Thus
8r [xy] = '£ ()[81X, 8r-1y],
i=O 1
8r r [81 8r-1 ]
-1 [xy] = iO iT x, (r- i51 y
= L [, x,  Y] .
i, j 1. ] .
'i+j=r
Therefore
exp 8 [xy] = h L
r> 0 1.1
i+j=r
[8 v  )J]
" 01\, "'
l. ] 0
[8f 8J]
= L: h --x-y
iO j;;:O i! ' j!
= [exp 8 oX, exp 8.y].
II
THE CHEV ALLEY GROUPS
61
Now let JL be a simple Lie algebra over C with Cartan decomposition
....,.....l"';"
1L= EB  J[r
,el!>
"
{
and Cheval1ey basis {hr, rEn; er, rE<I>}. Then the map ad er is a derivation
of 1. (cf. section 3.1) and this derivation is in fact nilpotent. For we
have
ad er 0 j!} = 1.r,
(ad e,):;:-:J {J". 9 oltjz ,,_, l' C "
LJl.Q.- Q .r v rr.--Y)--J '. ! I - '\
{Ll (0/1;( Q "--
O I, (, -" ........-}ad er.1.r=O,
'1/\ >ANt  \'.... u,,
[Qft Qt)::'-'J ader.1.-rJ[), (ader)3.1.-r=O.
, t - .. \ r -. '" j- I
r JQ. 1. ., '-; !;. Q I  1\ ""t. Ie"!p '- =,"; ".\ :- r
(ad e,)<lH.1.s=O If r, 's are hnearly independent, since (q+ l)r+s is
not a root, Th us
(ad er)n.J[=O
for all sufficiently large values of n.
Let' EtC. Then ad aer) =  ad er is also a nilpotent derivation of JL
Thus exp a ad er) is an automorphism of J[. We write
xra)=exp a ad er).
t
We now consider the effect of the automorphism xr( C on the elements
of the Chevalley basis. We have
xra) . er = er,
Xr( ) 0 e-r = e-r + hr - 2e"
Xr()' hr=hr- 2er.
Also, if r, s are linearly independent,
xr( () .Izs = I1s - Asrer,
x,.(().es=e8+Nr.ser+s+ Nr, sNr. r+82e2r+s
+ . . . + :,- Nr, sNr, r+s . . . Nr. (q-l)r+8qeqr+8'
q.
.,
;:L!
We write
1
lvlr, s, (=:-;- Nr, sNr, r+s . . . N" «-1)r+8'
1.
Then
q
xr(C.es= 2: Mr, 8, t(etr+8'
i=O
--..

!ryll"', GH.c ,UPS CF ;..,.IE. T"'IF1".E
" \,
;,!
(We define .N1r  .-,= 1 ) TTsi.,,," +1-,,,, + +'nat J\T . ( . ")
.0. '--' ./ '---' 'Ui::> ..u. .i.a....l. l LYy. 8=::!: p-t-l v;e see that
--'",
Mr.., j= + (p+ l)(p+  . . __ (p+i) :!: (P7').
In particular Mr. B, i is an integer. Thus the automorphism xr({) transforms)
each element of the . Chevally basis into a linear combination of basis ./
elel!1ents: the coeffi,czents bemg non-negative integral powers of , Wilh \
ratlOnalmteger coefficients. \
It is this property which enables us to define automorphisms of this
type over an arbitrary field.
":J

4.4 Algebras and Groups Over an Arbitrary Field
Let 1[. be a simple Lie algebra over C with ChevaHey basis
{hr, r En; er, r E <D}.
We denote .by 1[.z the subset of 1[. of all linear combinations of the basis
elen:nts WIt? coefficients in the ring 7L of rational integers. 1[z is an
ddl.tlve abehall gr9,up. By 4.2.1 the Lie product of two basis vectors
hs III 1[2, thus 3Lz is dosed under Lie multiplication. J[z is therefore a
Lie algebra over 71...
Now let K be any field. We form the tensor product of the additive
group of K with the additive group of J[z, and define
.I r"' _I 1.11.--- Jl. /1 r:-
1[K=K@JLz. ! (/}-V' '/l O'[/to 'i l '-. -/i-1" -'.'
i . r f . .. I r .... _...".. '.",....
Then IK is an additive abelian group. Let IK be thlrln'ti el{;t' f''K. ',-
Then every element of JLK can be written in the form
 A,.(lK@hr)+  IL,{IK@er),
Tell re(D
where AI', ILl' EK. We write
hr= I]{@hr er= IK@er.
Then 1LK is a vector space over K with basis
{hI" rE n; er, rE (D}.
We now define a Lie multiplication on 1[.K.
· i
,
-..c.,
_l6;-..F_--......
; t
! -. .
,-
I
i ,i
j
!.I
f
. i
f
'-n_.
PROPOSITION 4.4.1. Let x, y be any two elements of the Chevalley
, ,
i
Tiii: Ciii;VALLEi: _
63
tillis 0./1[. Then the 171.wtiplicatioll 0/1 1£.K defined by
[1)(@x, l)(@y]=l)(@[xy]
ami extended by linearity makes JL]( into a Lie algebra over K. The multi-
r!!...ioll cO/._t!1J1ts of JLK wilh respect _!!LJJJ[:'- hJJ.$is {hr_ r En; er,-;--<i;Y'ar
cht: 1J/-UTilpficatioi-i-cons;iQlits'''o/1L'-,itlz respect to the basis -- ---.- ------.--.-.-.
{hr, r E 11; er, r E <D}
il/lerpreted as elenzents of tire prime subfield of K.
PROOF. This is clear, since by 4.2. 1 the multi plication constants of JL
arc in 71.. ..
Having introduced the Lie algebra JL[(, we shaH define automorphisms
oflKanalogous to the autornorphisms x;(O ofJL. Let Ara) be the matrix
representing Xra) with respect to the ChevaIJey basis of JL. We have
x:cn that the coefficients of Ara) have the form a'i, where aElL and
i;; O. Let t be an element of K and Ar(t) be the matrix obtained from
AI( 0 by replacing each coeOlcient a,t by liti E K, where ii is the element
of the prime field of K corresponding to aE71... We now define Xr(t) to
be the linear map of JLK into itself represented by the matrix A,{t) with
rcs pect to the basis {hI" r En; e r, r E (I)}.
PROPOSITION 4.4.2. Xr(t) is an automorphism of JLK for each rEeD,
tEK.
PROOF. Observe first that xr(t) is non-singular. Since Xra)Xr( - {) = 1,
it follows immediately from the definitions that X,{t)Xr( - t) = 1. Thus
.\;,-( -1) is the inverse of Xr(t).
To show that -Yr(t) is an automorphism of JLK we consider the effect
of .\'r(t) on the basis of JL)(. Let VI. V2,... be the Chevalley basis of
I and VI, V:!" .. . . be the corresponding basis of JLK. Suppose
[ViVj] = L: YijkVk,
k
[VlVj] = L: Y£jkVk,
k
where YtJkE71.. and Yijk are the corresponding elemnts of the prime field
of K. Now we have
Xr({). Vi = L: Ara)ijVj,
j
Xr(t). Vt= L: Ar(t)ijVj.
j

oq
SIMPLE GROUPS OF LIE TYPE
Thus Xr(/) is an automorphism of J[K if and only if
,, Ar(t)tt'Ar(t)jj'YI'j'k= L YiJk'Ar(t)k'/c
1,J k'
for all i, j, k.
However, Xr( t) is an automorphism of JL for all t E Co Thus
.,, Ar{t)l-£'Ar(lJjj'Yt'j'k- 2: YlJk'ArCth'k
1.J k'
is a polynomial in Z[ t] which vanishes for all t EC. It is therefore identicall
zero. J t follows that the polynomial Y
., Ar(t)u,Ar<I)jj'jit'j'/c- L: jitj/c'.Ar(/h'/c
1 . J k'
;:%(1] is identica1Jy zero. Thus Xr(/) is an automorphism of JLK for all
.
. N0.w that we hve established the results of 4.4.1 and 4.4.2 we shall
simplIfy the notatIOn. We shaH write IIr for h e for e- X (I) f' '"c; ( )
d A ( )  - . .. r, r r, r lor A, t ,
an. r t . or Ar(t). ThIs omISSIon of the bars will not lead to confusion
or InCOnsIstency sInce the objects originally called hex (1) A (t)
special cases of hr, e" X,-(/) , Ar(t) when K = C. r, r, , , rare
We shall now den_He the C;hcvalley groups. The Chevalley grou of
type JL ove the field K, enot_"l?J.'_()_._ _efined t- b-the giou-of
automorphlsms of the LIe algebra ':If t d b -', 't:"-...--.-...----- ---- .----
-. 1 "'M._ Jl..-K genera e y tIle Xr(/) for all
rE o.,_t EK. . - .. -
Tbe genera.tors of JI.(K) operate on the elements of the Chevalley bsi
of JI.K accordmg to the formulae:
X,-(/). er = er,
xr(t). e-r= e-r + thr- t2e"
Xr(t) . Its = h8 -Asrler,
rE<1:>,
rE <1:>,
rE (]), sEll,
fJ
Xr(t). es = L: Mr. s. ,ttetr+s
i=O
if r, SE<1:> arc linearly independent.
PROPSITlON .4. 3. The group J[(K) is determined Lip 10 isomorphism
by Ihe sImple LIe algebra JL over C and the field K.
PROOF. We must show that JI.(K) is inde.Dcn.dent of the choice of the
THE CHEV ALLEY GROUPS
Chcvalley basis of JL Let {It" r En; er, r E <I)} be a Chevalley basis ofJL.
We must show that any ChevalIey basis of JL can be transformed by an
aUlomorphism of JL into one of the form {hr, rEI1; fer, rE <D}. Firstly,
incc any two Cartan subalgebras of JL can be transformed into one
another by some automorphism of JL (3.5.2) we may restrict attention to
Chcvalley bases corresponding to a fixed Cartan subalgebra J[). The
isomorphism theorem 3.5.2 also shows that there is an automorphism
of JL which transforms any set of fundamental roots into any other;
and also that for a given system of fundamental roots there is an auto-
morphism of JL which transforms any set of fundamental root vectors
l'r, rE n, into any other. Thus we may restrict ourselves to Chevalley
bases in which JL), 11 and the er for r E n are fixed. Since the structure
constants N" s are all determined to within a sign we see, using 3.6.2,
that each root vector Cr is determined to within a sign. Thus any Chevalley
basis of JL can be transformed by an automorphism of JL into one of the
form {hr, rEll; fer, rE<I>}. It is now evident that the group generated
by the elements Xr( t) is independent of the Chevalley basis. For if - er
is chosen as a root vector instead of er, the generator xr(t) is simply
replaced by x,( - t), its inverse. Hence the isomorphism type of the group
l.(K) depends only upon JL and K. .
4.5 The Groups A1(K)
The simplest examples of Chevalley groups are the groups AI(K). We
shall show that these groups are isomorphic to the linear groups PSL2(K).
We note first that the simple Lie algebra Al over C can be represented
as the algebra of 2 x 2 matrices of trace 0 under Lie multiplication
[xy]=xy- yx. For if we define
hr=( _),
er=( ),
e-r= ( ),
we have
[Jrrer] =2er,
[hre-r] = - 2e-r,
[ere-r] =hr.
These are the relations satisfied by a Chevalley basis of the simple algebra
A 1. Note that the root vectors er, e-, are represented by nilpotent matrices.
We now require the fol1owing lemma.
LEMMA 4.5. 1. Let JL be a simple Lie algebra over C and suppose we
have a representation of JL by matrices Linder Lie multiplication. Suppose
65
-".........
()
I
/
I
!
I
!
""\
'f
!
';
'1
.,
:.;
i!
;
---..) -..,
';
.
Ii-
j;'
I
I
i
I
i


I

l
-
[
r>
L
f'

I
'\ -;-
r--
\'l"...."..
L-.---
....J
L
66
SIMPLE GROUPS OF LIE TYPE
{1f.1L iSdrepresented by a nilpotent matrix. Then ad y is a nilpotent derivation
{J/ an
cxp (ad y).x=cxp y.x.(cxp y)-l
rr all XE1L. Thus the image of x under the automorphism exp (ad y) is
gIVen by transJorrning by exp y.
PROOF. We have
ad y.X= [yxl= yx-xy,
(ad y)2
2f -.x=-Hy2x-2yxY+Xy2).
We show

(ad y)k yt ( - y)1
-k' .X= L -:- x --:----, .
. i j Z! ).
i+J=k
This is true or k=.I, 2 and we prove it by induction. Assuming the above
formula, by mductIon we have
a_y)k+l x- (yHl (- y)J y' (_ Y)1+1)
(k+ I)! . -k+ 1 . i!x+i' x----;-,-
i+!.k '. } .
ym (_ y)n (m+n)
.2; --x----
1n,11 m! n! k+(
m+11=k+l
.2;
ym ( _ y)n
m! x nf--'
In, n
m+n=k+l
Now y is nilpotnt ad so ((ad y)k/k!)x=O for sufficiently large values
of k. Thus ad y IS a mlpotent derivation of JL. Also we have
exp (ad y).x=  ? J.'_ x
k=O k!
co
 . ( - y)1
.2; x ----
i.j i! j!
i+j=k
=.2;
k=O
=   ?x (y)1
i=Oj=OZ! )!
=exp y.x.(exp y)-l.
.
TIlE CHEYALLEY GHOUl'S
67
PROPOSITION 4.5.2. A1(K) is isomorphic to PSL2(K).
j
I
I
I
r
J
j
I
!
!
PS<OOJl. We first apply 4.5.1 to the Lie algebra Al over C. If tEC
and er, e-r are the matrices defined above we ha ve
xr(0.x=exp aCr).x.exp {Scr)-l,
x-ra).x=exp (Sc-r).x.exp (SC-r)-l.
We now pass to the Lie algdJra JLK for an arbitrary field K. JLK is
isomorphic to the Lie algebra of 2 i< 2 matrices over K of trace O. The
Chevalley group JLK js gC/laaled by the elemL:nls x,(I), x-ref) as t runs
through K. Now Xr(t) is given by transformation by
exp (ter) = ( )
and X-r(t) is given by transformation by
exp (te-r) = C ).
But the matrices
( ;), G )
generate SL2(K) as t runs through K. (A proof of this well-known fact
will be given in section 6.1.) Thus there is a sUljective homomorphism
SL2(K)Al(K)
under which the image of m ESL2(K) is the automorphism xmx m-1
ofJLK. Under this homomorphism we have
( ;) xr(t),
G ) x-r(t).
The kernel of the homomorphism consists of aU m ESL2(K) such that
m commutes with all XEJLK, and this is easily seen to be {:t h}. Thus
AI(K) is isomorphic to PSL2(K). .

CHAPTER 5
Unipotent Subgroups
5.1 The Subgroups lJ, V
Let G = J[(K) be the Chevalley group of type JL over K. G is generated by
elements Xr(t) for all rE <I>, tEK. Now we have
Xr(tI).Xr(t2)=exp (tl ad er).exp (t2 ad er)
=exp (tl + t2) ad er
=xr(h + t2).
Let Xr be the group generated by the elements Xr(t) for all tEK.
Xr is a subgroup of G isomorphic to the additive group of K. For the
map tXr(t) is an epimorphism from K to Xr; but if X,.(/) is the identity,
t must be 0, as can be seen [rom the operation of xr(t) on the basis elemcnts
of JLK. The subgroups Xr are Gll1ed the root subgroups of G.
Let U be the subgroup of G generated by the elements Xr(t) for rE (1)+,
tEK; and V be generated by the xr(t) for rE <1>-, tEK. Then U and V
clearly generate G, and U is generated by the root subgroups Xr cor-
responding to the positive roots, while V is generated by the negative
root subgroups. We shall elucidate some of the properties of U, and
analogous properties will clearly hold for V also.
We show first that the elements of U and V operate on JLK as unipotent
_!ieCl--_transformations. A ]inar tansformation U of a vecto -sp__L9--
itself is saId to be imipotent if u-l is nilpotent. et ---- -----.-...-,-. -----, --.-.--.
JLK = j!) (f) b JLr
rE(l>
be the decomposition of JLK correponding to the Cartan decomposition
of JL. We define JLo = j£} and
JLt= b JLr
h(r) =i
for each i =f- O. Then
JLK= EBJLi.
i
68
I  t::
UNIPOTENT SUBGROUPS
Now if rE <D+ and XEJ(t it is evident that
Xr(t).X-XE  1[-1.
;>i
Since each element of U is a product of elements Xr(t) for positive roots r,
we have for each UEU, XE1Lt:
U.X-XE L JLj.
j>i
Thus l/-1 is a nilpotent linear transformation of JLK, whence u s uni-
potent. Similarly each element of V is a unipotent transformatIOn of
l.x.
The foIlowing lemma will be very useful in the subsequent development:
LEMMA 5.1.1. Let JL be a simple Lie algebra over C. Let y be an
clem/!nt ofJL sllch that ad y is nilpotent and let e be an automorphism ofJL.
Theil
B.exp (ad y)B-I=exp (ad By).
PROOF. Let XEJL. Then we have
O.exp (ad y) B-l(X)OCo h [y,. .. [y, 0-l(Xm)
0:> 1
= L -:-, [Oy, . . . [By, x]]
i=O 1.
= exp (ad B(y)). x.
It follows that
B.exp (ad y) B-l=exp (ad By).
We now require a more technical lemma about certain nilpotent linear
transformations of a vector space, which is a special case of the CampbeII-
Hausdorff formula.
LEMMA 5.1.2. Let V be a vector space over a field of characteristic 0
a//(I t, 7] be linear transformations of V into itself such that t,7] and
[, 7]] = t7] -7]t
are nilpotent and [t, 7]] commutes with  and 7]. Then t+7] is also nilpotent,
and
exp (t+7])=exp t.exp 7].exp (--H, 7]]).
0'-
7
t
"
(
.
;
>--"""1'"\
, '
ti j
;/ !i
 
;1 i
 \
 ,
r 
i. :
r
);
--.
r
\

\'iW..::iiIo' 'I"
L
("-
:1
-..
'
r-
L,
r<
l"",
L
i .
I
L__
[
.li1a9""
[
I
I
L-
70
SIMPLE GROUPS OF LIE TYPE
PROOF. We shall show that
(t + 7})n
1i!--- =
tt 7}1 [t, 7}]k
. ''2k kl.(-l)k.
1,J. k I . ) . "
i+i+2k=n
This is true for n = 1 and 2, since
!(t + T))2 = t2 + t7}2 + t7) - H, 7}].
To prove it in general we require the formula
7}t'=gt7}-itt-1[t,7}J.
This is true for i = 1 and its proof by induction is evident.
We assume the required result for (g+7})n-lj(n-I)!. Then
(g+7})n _ g+
n! - n
g' 7}1 [g, 7}]k
.  7f .  . -2-k k 1 . ( - 1) k
'. J. k I . } . ..
i+J+2k=n-l
2:
n i.i,k
i+J+2k=n-l
gHl 7}1 [g,7}]k
--;"'2k kl'( -1)k
l. }. ..
1 g' 7}1+l [g,7}]k
+- h '--;-;-'-2k-k,.(_I)k
n i,i, k I.} . ..
i+i+2k=n-l
tt-l 7}1 [g, 7}] k+ 1
+- . ('-1)1 .. 2k k' .( - 1)k+1
n 1, J. Ie I , ) . ..
i+i+2k=n-l
 r "2!. !{- (-l)k (+-L+)
i,i, Ie i!"j !"2k.k! . . n n n
i+i+2k=n
2:
i.i, k
i+i+21e=n
t' 7}1 [g,7}]k
''2-k kl .(-I)k.
I . ) . "
Since g, 7} and [g, 7}J are all nilpotent,
(g+7})n =0
n!
for sufficiently large values of n. Also
exp (t+7})=exp g.exp 7}.exp (--![g, 7}]),
as required.
.
UNIPOTENT SUBGROUPS
71
5.2 Chevalley's Commutator Formula
In the present section we shall derive a formula, due to ChevaIley, which
exprcsses the commutator of two generators of G as a product of genera-
tors. We work first over the complex field iC and transfer at a later stage
to an arbitrary field K.
Let JL be a simple Lie algebra over C; r, s be linearly independent
roots of JL, and t, u be elements of C. Consider the expression
Xr(t)Xs(U)Xr(t)-l.
By 5.1.1 we have
Xr(t)Xs(U)Xr(t)-l=X,{t) exp (ad lles)Xr(t)-l
=exp ad (xr(t).uelJ)
=exp ad CO llir, s, ttiuelr+s)
= exp (.f Mr. 8. ttiu ad elr+s).
1=(J
Now jf rl, r2 are linearly independent roots sllch that rl + r2 IS not a
root, ad erl and ad er2 commute. For
ad erl.ad er2-ad er2.ad erl=ad[erler2] =0.
Thus if r+s is not a root ad (Ier) commutes with ad (ues) and hence
xr(t) commutes with xs(u). On the other hand, if r +S is a root, the integral
combinations of r, s which are in (1) form a root system of type A2, B2
or G2, by 3.6.3. We restrict attention to the roots of the form ir+ js
V.ilh i,j>O, and distinguish between a number of special cases.
(i) Suppose r+2s, 3r+2s are not roots, Then if ir+jsE(D and i>O,
j> 0, we must have j= 1. This means that all the transformations ad e-lr+s
commute. Thus
q
Xr(t).:rs(U)Xr(t)-l= 11 exp (!vfr. 8, tttu ad etr+s)
i=O
q
= n xir+s(Mr, s, ttiu).
i=O
(ii) Suppose r+2s is a root and the integral combinations of r, s in

72
SIMPLE GROUPS OF LIE TYPE
UNIPOTENT SUBGROUPS
73
)...,
<1> form a root system of type 8z. Then
x,(t)xs(u)x,{t)-l=exp (u ad es+N,.. slu ad e,.+8).
Let t=u ad es and 7}=N,., slu ad e,.+s. Then
[t, 7}J=N,., sNs, ,.+stu2 ad e,.+28'
Thus [t, 7}] is nilpotent and commutes with t and 7} (because r+ 3s, 2r+3.s
are not in (1»). Applying 5.1.2 we obtain
X,.(t) xlu) Xr(t)-l
= exp (ll ad es). exp (Nr, stu ad er+s). exp ( - -tNr. sNs, r+stu2 ad er+23)
=xs(u) xr+s(N,., stu) x"+2s(Ms. ,., 2IU2).
r+s,
2r+s,
3r+s,
3r + 2s.
[, 'I] is again nilpotent and commutes with t, 7]. Thus
xr(t) xsCu) X,.(t)-l
=exp (u ad e8+M,., 8, 1M,., 8, 2t3u2 ad e3,.+28)
x exp (--tN,.+s, 2r+sM,., s, IMr. 8, 2t3u2 ad e3,.+28)
x exp (M,., 8, 3t3u ad e3r+s)
x exp ( - -tNs, 3r+sM,.. s, 3t3u2 ad e3,.+2S)
=xs(u) xr+s(M,., s, Itll) x2-r+s(Mr, s, 2t2U) X3,.+s(Mr, 8, 3t3u)
X X3r+2s(( - -tNr+s, 2,.+sJr, 8, 1M,., 8, 2 - -tN8, 3,.+sMr, 8, 3)t3U2).
Observe that the coefficient of X3r+28 is
(- -!N,.+8, 2r+sMr, 8, 1Mr, 8, 2 - -tNs, 3r+sM,., 8, 3) t3U2
=( -tNr+s,2,.+sN,., sN,., 8Nr, ,.+s-tN8, 3,.+sM,., 8,3) t3u2
=(-tMr+s, ,., 2 - tNs, 3,.+sM,., 8, 3) t3u2,
since Nr s=:!: 1.
This oefficient can be simplified by the following lemma:
'-
(iii) Suppose the integral combinations of r, s in <D form a root system
of type G2 in which 3r+2s is a root. Then the integral combinations
of form ;r+ jSE cP with i> 0, j> 0 are
Then
Xr(/) xs(u) xr(t)-1=exp CO A1, s, 1t1U ad e1r+s).
LEMMA 5.2.1. Ns, 3r+sAlr, 8, 3=-}M,.+s. r. 2.
Let
2
t= 1: Mr, s, trtu ad e1,.+8,
i=O
PROOF. We use the relation
s+(3r+s)+( -(r+s))+( -(2r+s))=0.
By formula 4 .1. 2 (iv) relating the structure constants we have
N8. 3,.+sN-(,.+s). -(2"+8) + N-(,.+s) sN3,.+s, -(2r+s) =0.
3 '
Using formulae 4.1.2 (ii), (iii) we obtain
-N8, 3,.+sN,.+8. 2r+s Ns, ,.N-(2,.+s), -,. -0
-------+ '3 '
7}=Mr. 8, 3t3u ad e3r+s.
Then
[t, 7}J = Ns, 3r+sll-lr. 8, 3t3u2 ad e3r+28.
Thus [t, 7}J is nilpotent and commutes with t, 7}. By 5.1.2 we have
Xr(t) x8(u) X,.(t)-1 = exp CO Mr., s, 1tiU ad e1r+h) . exp (M,., 8, 3t3u ad e3r+8)
x exp (- tNs, 3r+slvfr, 8, 3t3u2 ad e3r+2s),
We now make a second application of 5.1.2. Put
whence
N8, 3,.+8N,.+8. 2,.+8 = N8, rNrt 2r+8.
1
t= 2.: Mr, 8, 1tiU ad e1r+s,
i=O
7} =A1r, s, 2t2U ad e2r+s.
· It follows that
Ns, 3r+811-1,.. s, 3= iNs. 3r+sN,., sN,., r+sNr, 2r+s
= -iNr, ,.+sN,.+s, 2r+S (since N;, 3r+,=I)
= iNr+s, ,.N,.+s, 2r+s
=t1\.1,.+s,,.,2.
--.
Then
[g, 7}]=N,.--\s, 2,.+sMr, s, l11-1r, s, 2t3u2 ad e3r+2s.
.

\....
'1
l:
f'"
L,
..,
.'
t
L...
ft----
J
l,,,-,
""'-
i ,-."
 -
1
j
.( -
--\. ..'
74
UNIPOTENT SUBGROUPS
75
SIMPLE GROUPS OF LIE TYPE
Applying 5.2. 1, the coefficient of X3r+2s in the expression for
xr(t) xs(u) Xr(t)-l
2r+s
IS tMr+B, r, 2t3u2. Thus
Xr(t) xs(u) Xr(t)-l= xa(U) xr+s(Mr. s, l/u) x2r+s(Mr. s, 2t2U)
X x3r+a(Mr, s, 3t3U) X3r+2.s(-!-Mr+s, r. 2t3u2).
(iv) Suppose the integral combinations of r, S in cD form a root system
of type G2 in which 2r + 3s is a root. Then the integral combinations of
the form ir+ jSE<P, with i>O,j>O are
-2r-s
r+s,
r+2s,
2r+3s.
Figure 2
r+3s,
The formula of 5. 1 .2 can no longer be used to calculato
Xr(t) x.s(u) xr(t)-I.
We use instead an indirect method based on (iii). We have shown In
(iii) (interchanging the roles of r, s) that
Xa(U) xr(t) xs(u)-l=Xr(t) xs+r(Ms, r, lUt) x2s+r(Ms, r. 2U2t)
X X3s+r(.Nfs, r, 3u3t) .X3s+2rGMs+r, s, 2u3t2)
=xs+r(Ms, r, lUt) x2s+r(Ma, r, 2U2t) xr(t)
x X3s+r( 1\fa, r, 3u3t). xr(t)-lX3s+2r(-tMs+r, s, 2u3t2) xr(t)
=xa+r(Ms, r, lut) x2a+r(Ms, r. 2u2t) X3s+r(Ms, r. 3U3t)
X X3s+2r«-!-Ms+r, s, 2 + Nr, 3a+rMs, r, 3)U3t2) .xr(t).
form ir+jsEc1> with i>O,j>O are r+s, 2r+s, 2s+r. The root system in
this case is shown in Figure 2. Then we have
-"r(l) Xa(u) Xr(t)-l
=Xr(t) exp (11 ad es) Xr(t)-l
=exp ad (_ /lIr, s, itillCir+s)
&-0
=exp ad (lies + Nfr. 8, 1(lICr+8+1vlr, a, /211C2r+s)
=exp ad (ues+ Mr, s, Itller+s) exp ad (Mr, B. 2tZllC2r-I-8)
Now by 5.2.1,
=exp ad (ues) exp ad (Mr. s, ltllCr+s)
X exp ad ( -1Mr, a, INs, r+sfu2er+2S) exp ad (Nfr, a, 2t211Czr+s)
=Xs(u) Xr+a(M,., 13, I/U) X2r+s(Mr, s, 2t2U) Xr+2.s(i\I.s, r, 2/112).
Nr, 3s+rMs, r, 3 = tMs+r, s, 2.
Thus
Xs(u) Xr(t) Xa(U)-I=Xs+r(Ms. r, lUt) X2s+r(Ms, r, 2u2t) X3s+r(Ms. r, 3li3t)
X X3s+2rG-Ms+r. s, 2u3t2) .xr(t).
Replacing u by -u in this formula gives
Xr(t) xs(u) Xr(t)-I=xs(u) Xs+r( -Ms, r. Iut) x2s+r(Ms, r, 2u2t)
X X38+r( - Ms. r, 3u3t). X3s+2r( - tMs+r, 8, 2u3t2).
Now the cases (i), (ii), (iii), (iv), (v) considered above exhaust all possi-
bilities. All the cases except Gz are (kall with in (i) or (ii), nd. the _three
dilTerent ways in which r, s can be placed inside G2 not satlsfymg (I) are
dealt with in (iii), (iv), (v).
We may summarize the results of (i)-(v) in a single formula as follows:
Xr(t) xs(u) Xr(t)-I = xs(u). n Xir+Js( Curstiu!),
i,;>O
(v) Suppose the integral combinations of r, s in  form a root system
of type G2 in which r+2sE<D, 3r+2sf<D, 3s+2rf. Then the roots of

76
SIMPLE GROUPS OF LIE TYPE
where
CUrs = Mrs1,
Cllrs=( -l)lM15rl,
C32rS= lMr+s, r, 2,
C23rS= -j-lYfs+r, 8, 2.
The product is taken over all pairs i> 0, j> 0 such that ir+ js E <1), the
order of the terms in the product being such that i + j is increasing. t
The numbers Ctlrs defined here are all rational integers. This is clear
for Cilrs and C11rs, while for C32rs and C23rs the absolute value of
Mr+s, r, 2, lvf15+r, s, 2 can be obtained by examining the root system of G2.
These both have value :t 3. Thus C32rs o-=: :t 1 and C23r15 = + 2. Therefore
in all cases we have CtJrs= ::J: 1, ::J: 2 or ::!: 3. -
Now it follows from results established in chapter 4 that the validity
of the equation
xr(t) xs(u) Xr(t)-I=Xs(u). II Xir+Js(Ctlrstiuf)
i,;>O
is equivalent to the vanishing of certain polynomials fk(t, u)=O, where
fk{X, y) E[X, y]. Let K be an arbitrary field and t, II be elements of K.
Then the validity of the equation
Xr(t) xlu) Xr{t)-I = xs(u) II Xir+ls{ CilrstiUJ)
i,;>O
in G = JL{K) is equivalent to the vanishing of the polynomials .fk(t, u) =0,
:vhere.fk(' Y)EK[x, y] is the polynomial obtained fromfk(x, y) by replac-
mg each mtegral coefficient by its image in the prime subfield of K under
the natural homomorphism. However, we have shown that fIc(t, u)=O
for all t, UEC. Thus each fk is identically zero. It follows that each.fk is
identically zero. Thus!k(t, u)=O for all t, UEK. We have therefore estab-
lished the following theorem, known as Chevalley's commutator formula.
THEOREM 5.2.2. Let G = JL(K) be a Cheva/ley group over an arbitrary
field, 1', s be linearly independent roots of JL and t, u be elements of K.
Define the commutator
[xs{u), Xr(t)]=Xs{U)-IXr(t)-IX8(u) Xr(t).
Then we have
[xs{u), Xr{t)]= IT X1r+ls(Cilrs( -t)iUl),
i,j>O
t This condition does nt dteminc the order uniquely in the situation of case (v).
However, the two terms wIth 1+ J = 3 commute, so both orders give the same result.
UNIPOTENT SUBGROUPS
77
,,,here the product is taken over all pairs of positive integers i, j for which
ir + js is a root, in order of increasing i + j. The constants Cilrs are given
by
"
i
Cnrs=Mrst,
CIJr8={ -l)lMsrJ,
C32rs = !-Mr+8, r, 2,
C23rs= - iMs+r, 8,2
Each Cilrs is one of ::!: 1, ::J: 2, ::!: 3.
COROLLARY 5.2.3. If r, s E <D+ are linearly independent then
[xs(u), Xr(t) I = IT Xir+Js( Cljrs( - t )iuf),
i,j>O
where the product is taken over all positive roots of the form ir+ js, i> O,j>O,
ill increasing order.
5.3 The Structure of U and V
We shall now use Chevalley's commutator formula to investigate the
structure of U. We first require two lemmas.
..-."
1
"
LEMMA 5.3. 1. The ordering of the roots can be chosen compatible with
the height function, i.e. so that r-<.s implies h(r)h(s).
PROOF. Let 11 be the real vector space spanned by <D and let h : ---+ IR
be the linear map which coincides with the height function on <1>. Let
1)0 be the null-space of h. Then dim Vo = I-I. Let VI be an element
of V with h(Vl) = 1 and V2, V3, . . . , Vz be a basis of o. Then
hCl AtVi) =>\1.
If we define 1)+ to be the set of non-zero vectors whose first non-zero
ocfficient Ai is positive we obtain an ordering compatible with the height
function. ·
LEMMA 5.3.2. Let rE<D. Then the element hrEJLK is non-zero.
PROOF. This is true by definition if r E TI. The co-roots hr EJL form

'I.
J
I
78
79
SIMPLE GROUPS OF LIE TYPE
UN[POTENT SUBGROUPS
the dual root system <1>* by 3.6.1, and TI* is a fundamental system in
<D*. Thus
Now elements of the form xiu) generate Um and elements of form xr(t)
generate Un. Thus each element of [/7/1 commutes with each eIcment
or Un in VfUm+n. l--Ience [a, b]EUm+n for all aEUrn, bE Un. Thus
[Um, Un] S; Um+n.
In particular [U, Un] S; Un+1 for each 11. Thus each element of Un/Un+1
commutes with each element of U/Un-H, and so Un/Un+1 is in the centre
of U/Vn+I. Therefore we have a central series for U.
Now each element of U is a product of elements Xr(t) with r E <1)+.
If in such a product we have a pair of consecutive terms xs(u) x,.(t) with
r -<s we use the formula
hr=nIhpl + . . . +ndlpl'
where IT = {PI, P2, . . . , pe} and nt E 7L.
Now it is only possible for hr=O in '1LK if K has characteristic p and p
divides each nt, i= 1, . . . , I. We suppose that this is so. Now r can be
expressed in the form Wept) for some WE Wand some i. Then w(hpt) =hr.
The element W is represented with respect to the basis hpl, . . . ,hPI by a
matrix with coefficients in 7L, and one of the columns of this matrix is
(n1, n2, . . . , nl). Thus det w is divisible by p. Since det w = ::!:: 1 we have a
contradiction. .
Xs(1I) x,.(t)=Xr(t) xs(lI) n Xlrl}.5(Ctjrs( - t)lUJ).
i,j >0
THEOREM 5.3.3. Let G = J[(K) be a Chevalley group, U be the subgroup
\J of G generated by the root subgroups Xr with rE ct>+, alld Um be the sub-
l  group generated by the Xr with rE <!J+and her)  m. Then:
J' (i) U is nilpotent and
U = UI ::> U2::> ... ::> Uh::> 1
t

to,,,,
f --
t-,
r '..
!
L...
r
f
,
"""'J'
The terms Xr(t) xsCu) now occur in their natural order and all new terms
introduced satisfy l1(ir+ js)I1(r)+I1(s). Thus the process of rearranging
lhe terms in their natural order by means of the commutator formula
lcrminates after a finite number of steps, and each element of U is expres-
sible in the form
is a central series for U, where h is the greatest height of a root of JL.
- (ii) Each element of U is uniquely expressible in the form
IT Xr,(f.t),
n e IJI+
Xrltl) Xrit2). . . XrN(tN),
where 0 -< r1 -< r2 -< . .. -< r N are the posi tive roots of 1L.
We now prove uniqueness. We shall prove by descending induction
on m that
where the product is taken over all positive roots in increasing order.
n xit:)
heR)  m
PROOF. Observe first that Um is normal in U for each m 1. For if
SE<I>+ and h(s)m we have
Xr(t) xs(u) Xr(t)-I E Um
for an rE<D+, by the commutator formula. Thus xr(t) UmXr(t)-l s; Um.
Replacing t by - t we see that Xr(t) U111Xr(t)-1 = Um. Since the Xr(t)
for r E <D+, t E K generate U, it folIows that Um is normal in U.
Let [Um, Un] be the subgroup generated by all commutators [a, b]
for aEUm, bE Un. Let r, SE(j)+ satisfy h(r)l1, h(s)rn. Then
h(ir+ js)m+n
fr xitg) =
Il(g)  m
implies t8= t:, where the products are taken over all positive roots of
height at least 11"1 in increasing order. The statement is trivially true for
m=h+ 1, since Uh 11 = 1. We assume it true for 111 + 1. Let
u= 11 XsCtg)= IT xit:)
h(s) ;-. m I/(s)  VI
for all i, j> O. Thus
and let r be a positive root of height m. Consider the element u. e-r of
1..}{. If h(s»m and -r+sE<I), then -r+sE(I)+, since the ordering of
the roots is compatible with the height. If l1(s)=m then -r+s cannot be
 root, since its height would be O. Thus
u .e-r=e-r+ trhr+x,
[Xs(u), Xr(t)] E Um+n,
using the commutator formula. This means that the images of Xr(t), xs(u)
under the natural homomorphism commute in the factor group UjUm-l-1l'
where
XE L JLr;
reiD

80
SIMPLE GROUPS OF LIE TYPE
and similarly
u .e_,=e_,+t;h,+ y,
where
YE L: J[r.
,E t;I>+
But JLK has the direct decomposition
J[ /{ =:If) EB 2:: l£r EB L: l£r
rE (J)+ rE <.1>-
and hr =I- 0 by 5.3.2. Thus t,=/; and x=y. Hence
u= l n Xit8)' n xaCtlJ= 11 ;}(lt8).
1(8)=711 11(8)11I+1 It(8)=m
It follows that
IT xit:).
It(8)m+l
(II xaCt8»)-1.u= IT x(t)= IT x (t')
It(8)=m It(8);;'m+l 8 8 11(8)m+l J 8 8'
whence ts = t: by induction, and the proof is complete.
It is cleaf that similar results hold for V in terms of negative roots.
.
i
f
!
1
I
j
f
t
I-
f
J
J
1
t
i
f
i
!
f
f

!
i
1
f
i
I
i
f
-"\
CHAPTER 6
The Subgroups < Xr, X -r>
In the preceding chapter we considered relations involving generators
Xr(t), xlu) of a Chevalley group corresponding to linearly independent
roots r, s, and showed in particular that two such elements generate a
nilpotent subgroup. We shall now investigate relations involving generators
or the form xr(t), X-r(Ll). We do this by determining the structure of the
subgroup (.X r, X -r > generated by two root subgroups corresponding to
opposite roots. This subgroup turns out to be closely related to the group
SL2(K); in fact there is a homomorphism from SL2(K) onto <Xr, X-r).
We prove the existence of this homomorphism first for the complex
field. Regarding Jl as a module for the group (Xr, X-r) , we show that
the operation of (Xr, X-r) on certain submodules of JL is the' same as
the operation of SL2(C) on the space of homogeneous polynomials in
two variables of a suitable degree.t
--.)
6.1 The Group SL2(K)
LEMMA 6. 1 .1. Let K be all arbitrary field. Then the group SL2(K) is
generated by the elements
( :).
G )
as t runs through K.
PROOF. Let
(: 
be an element of SL2(K).
1.1' y-f- 0 we have
(: )=( {a-!!jy-l)C )( (S-:}rl).
: This method of establishing the existence of the homomorphism was described to
(he author by C. W. Curtis.
--....
81

82 SIMPLE GROUPS OF LIE TYPE
, If fJ =I- 0 we have
(: :)  (8-:)P-l )( K'-:)P-l }
If fJ=y=O we have
( ,,,) = C-L 1 )( :)Cl )( - "-}-
il
J
[
p..
[
l
l
[
r:

t- -
l
L
f
,
I
L"
It is dear from these relations that SL2(K) is generated in the manner
descri bed.
Let R=C[x, y] be the polynomial ring in two variables over C and R
q
be the subspace of polynomials which are homogeneous of degree q.
Rq has a basis
XfJ, xfJ-1y, xfJ-2y2, . . . , yfJ
and dimension q+ 1. An element
(: 
of SL2(K) operates on R by the transformation
xca+yy,
yf3x+ oy,
nd Rq is a representation space for SL2(C) for each q under this opera-
tIOn.
LEMMA 6.1.2. Let Vi =xiyg-l, ;=0, I, . .. , q. Then the images of v,
under transformation by
( :) and C )
are given by:
(
C
t) g-,( .)
. Vi =  q -: ' t1vH/,
I J=O ]
0) i ('J
. Vl=  '. tlvl_I'
I J=O )
II
THE SUBGROUPS (Xr, X -r)
83
PROOF. Since
i
1.
I
I
f
J
1
f
t
f
!
t
t
j
i
I
j
I
I
1
I
f
f
( :)
transforms x into itself and y into tx+ y, and
C )
transforms y into itself and x into x+ty, the result is clear.
II
6.2 The IIomomor!)hism from SL2(C)
We now consider the operation of the subgroup (Xr, X-r) of G='3L(C)
on the Lie algebra JL. Each r-chain of roots gives rise to a submouule
of J[ under (Xr, X-r) as follows. Let s be a root ]inearly independent of
r which begins an r-chain. Let
s, r+s, 2r+s, . . . , qr +s
be the r-chain of roots through s. Let :fl!ls be the subspace of 1L defined
by
:fl!1s = JLs EB JLr+s EB . . . EB 'JLqr+s.
Then JJ[s is invariant under (Xr, X-r) , by the results of chapter 4. Let
j/..)r be the I-dimensional subspace of J[> spanned by the co-root hr and
1Q.t be the subspace of ) of elements h with (11, hr)=O. Then the sub-
spces 1., EB J!)r EB JL_, and 'Ji-)f are invariant under the action of (Xr, X -r)'
and Jl is the direct sum of these subspaces together with the spaces j1I1s
for the various r-chains of roots.
LEMMA 6.2.1. Let r, s be linearly independent roots sllch that the
r-chain through s is
s, r+s, . . . ,qr+s.
Let ftl, = 1L, EE> JLr+B EB  JL qr-l--s and write ji = e1r+s. Then Jo, fl, " . ,Jq
is  basis Jor :iMs and we have
Xr(t).ji= gi EtEHI. . . €H1-1 (i+J) tiftH,
;=0 }
i (q-i+j) j
x-r(t).ji= 2: Et-lEt-2... €1-1 . t'fi-j,
;=0 }

84
SIMPLE GROUPS OF LIE TYPE
where the numbers fO, E1, . . . are al/ i 1 and are defined by
Nr, 1,.+8 = £i(i + I).
(Note that Nr, t,.+8= :!:: (i+ 1) by 4.2.1.)
PROOF. We know from chapter 4 that
q-i
Xr(t). etr+8 = L: M,., 1,.+s, jlje (1+j)"+8
j=O
q-i ("+.)
=.L: £t £1+1 . . . £1+j-1 Z . ] tj e (1+j)"+8"
J=O j
Similarly we have
i
X-,.(t).et,.+8= "L M-r. tr+8, jtj e(t-j)r+8'
j=O
However, using relations 4.1.2 (i), (ii), (iii) between the structure constants
and also 4.1.1 we have
I
1.1-,., tr+8, j=- N-r. tr+sN-r, (t-l)r+8' . .
J.
1 q-i+1 q-i+2
= ---;-, N,., (1-1)r+s. - . Nr, (t-2)r+8. -----;--1 ...
J. 1 l-
= £t-l(q- i + 1). £t-2(q- i+2) . . .
] .
(q-i+j)
= £1-1£1-2" . . £t-j j .
Thus
i (q-i+j)
X-,.(t).etr+8= "L £1-l€t-2... £'£-j . tj e(t-j)r+s.
j=O ]
..
We now show it is possible to choose a basis for Rq so that
( :).
G )
operate on this basis in essentially the same way that Xr(t), X-r(t) operate
on the basis fo, Ji, . . . ,fq of fils.
THE SUBGROUPS <Jrr, Jr_r)
85
LEMMA 6.2.2. Let Vt =xtyq-t ERq and £0, £1,..., €q-1 be integers
equal to :t 1. Define
-----."
Ut= £0£1" . . £t-l(i) Vt,
i=O, 1, . . . , q.
Theil
(
G
t) q-i (i+j)
"Ul = "L £t €t+1 . . . £Hj-l . tj U1+f,
1 j=O ]
0) i (q-i+j)
.u£ = "L £t-l £t-2 . . . £t-j . tj Ut-j.
1 j=O ]
PROOF. By 6. 1 .2 we have

(1 t) q-i (qj i) (J) £0£1. . " £t-l
.Ut = "L tj UHj
° 1 j=O (. q .) £0£1.. . £1+j-l
Z+j
q-i (i+j)
"L £i£1+1... £Hj-l . tj U1+j.
j=O ]
(:
0) i () (i) £O€! . . . £t-l
.Ut= "L t1 Ut-j
1 j=O (.q .) £0£1... £t-1-1
Z-j
!-, (q-i+j) j
=  . £t-l £t-2 . . . £t-jt Ut-1'
j=O ]
.
We next compare the action of (Xr, X-r) on JLr EB j[Jr EB JL-r with
the action of SL2(C) on R2. We recall the relations:
Xr(t) . er = er,
Xr(t) .hr =hr-2ter,
Xr(t). e-r = e-r + thr - t2er;
X-r(t). er = er - thr - t2e_r,
X-r(t). hr = hr + 2te-r,
x-r(t).e-r=e-r.
"""'''\
D

J _...;"

'-
l...
'..n,
[
[,
r-'
'JY
L_
[
86
SIMPLE GROUPS OF LIE TYPE
87
TI-IE SU13GROUPS <Xr, X -r>
LEMMA 6.2.3. Consider the basis _X2, 2xy, y2 of R2.
PROPOSITION 6.2,5. There is a homomorphism from SL2(C) onto the
subgroup (Xr, X -r > of G = JI..(C) undcr Il'hiclz
( :) and C )
( ;) ->x,(t),
C ) ->x(t)
operate on this basis as follows:
J
(01 It) ..
-xz..... _X2,
2xy 2xy - 2t. ( - X2),
y2y2+t.(2xy)_t2.( _X2);
PROOF. Write jtlIr= '1Lr EB;If)r EB'1L-r. Then we have a decomposition
of J£.
(t1 01) ..
-X2 _x2- t . (2xy) - t2.y2,
2xy 2xy + 2ty2,
JL = ;If) - EB $1;1 r EB  jt.18
8 E I'
into a direct sum of submodules with respect to (Xr, X-r), where r
is the set of roots which are independent of r and begin an r-chain. Now
(Xr, X-r) operates as the identity on 'J[)f. Lemmas 6.2.1,6.2.2, and
6.2.3 show that there exists a module .ffi for SLlC) , which is a direct
slim of spaces of homogeneous polynomials with a trivial module, such
that
y2--+y2.
PROOF. This is clear. Thus
( :).
C )
( ;),
(: )
operate on the basis -x2, 2xy, y2 of R2 in the same way that xr(t), X-r(t)
operate on the basis er, hr, e-r of JLr EB r EB JL-r. .
operate on Jlf[ in the same way as Xr(t), X-ref) operate on JL. Since the
elements
We shall also require the fol1owing lemma.
( :),
G )
LEMMA 6.2.4. Let
Then
(: ;) E SL2(C).
for tE C generate SL2(C), there is a homomorphism from SL2(C) onto
(Xr, X-r) under which
( ;)->x,(t) and C )->x_,(t).
(:  . 2xy=2xy+2( -{3.( -x2)+{3y.(2xy)+y.(y2)).
..
PROOF. This is evident since 0'.8 - f3y = 1.
.
6.3 The Homomorphism from SL2(K)
We can now prove the existence of the required homomorphism from
SL2(C).
We now transfer from the complex field to an arbitrary field K and establish
.the following result.

88
SIMPLE GROUPS OF LIE TYPE
THE SUBGROUPS (Xr. X_r>
89
THEOREM 6.3. 1. Let K be any fie/d. Then there is a homomorphism
Ironz SL2(K) onto the subgroup (Xr, X-r) ofG=JL(K) under which
( :)-X,(t),
C ) H(t).
with which we are concerned. Thus we may assume that hr is a funda-
mental co-root.
Now the remaining elements of the Chevalley basis which lie in 1!)
are the other fundamental co-roots. Let hs, SE n, be one of these. Then
we have
hs= >...hr+ y,
where Y E ]I,;- and
PROOF. We already have the existence of such a homomorphism
when K=C. Consider the module jl11 for SL2,(C) constructed in scction
6.2. Yt!l is a direct slim of modules Rq with a trivial module, and we obtain
a basis for j'U by combining the natural bases of the componcll ts RfJ
with a basis of the trivial module. With respect to this basis of ;fliI an
element
(: )
>..='-) =).
(Izr, hr) (s, s)
In particular, 2>" is an integer. We note that if OE<Xr, X-r) then we have
O(l1s) = >..O(hr) + y=hs+ >"(O(hr) -hr).
Now consider the action of SL2(C) on JlIl. Let mr, Ins be basis elements
of J-1;1 corresponding to hr, hs E JL. Let
T  (:  E SL.(C).
";;\:,...
of SL2(C) is represented by a matrix whose entries are polynomials in
a, {3, y, 0 with coefficients in 7L. This is clear because each component
Rq gives such a polynomial representation. This basis for .1fb1 corresponds
to a basis for JL as in section 6.2. This basis for JL is adapted to the decolll-
position
Then we have
(: )
T(ms) =ms + >... (T(mr) -Inr).
By 6.2.4, T(mr)-mr is a linear combination of basis .elements. of :ffi
with coefficients which are polynomials in a, {3, y, 0 wIth even mteger
codlicicnts. Since 2>" is an integer the entries in the matrix F(a, (3, y, 0)
arc polynomials in a, {3, y, 0 with coefficients in 7L. .
\Ve also observe from 6.2.4 that the polynomials in a. {3, y, ooccurnng
as coefficients in T(mr)-mr are alI homogeneous of degree 2. Since
0.8 - (Jy = 1 we may write
T(ms) =(ao - (3y) ms + >... (T(mr)-mr).
Thus T(flls) is a linear combination of basis elements of.:ffl with coefficients
which may be taken as homogeneous polynomials in 0:, (J, y, 0 of degree 2.
We now transfer to the field K and define a map
1. = J[);. EB Jtf[ r EB 2: Jtf[ /I
BEl'
of 6.2.5, but is not necessarily a Chevalley basis of JL However, it difTers
from a Chevalley basis only on jl). It wil1 be necessary to show that when
we change to a basis of.fllI corresponding to a ChevalIey basis of JL, the
element
of SL2(C) will still be represented by a matrix F(o:, (3, y, 0) whose entries
are polynomials in 0:, (J, y, 0 with coefficients in lL.
With this in mind we first transform the basis elements of J[ which
lie in jI) by an element of the Weyl group which transforms I1r into a
fundamental co-root. Since the elements of the Weyl group operate on
j[) by integral unimodular transformations, this will not affect the property
(:  --> F( ", p, y, 8),
where a, (J, y, 0 E K. This maps SL2(K) into a group of matrices over K
.obtained by replacing the variables in the polynomial entries of F(a, (J, y, 8)
-'"

...
[
I
l-d
t '
i
'J
t._.->
i ..

1
 __J
;'--
\
i .,,-
90
TilE Sl dlGROUPS <-..\'r, X -r>
91
SIMPLE GROUPS OF LIE TYPE
by elements of K. The condition for this map to be a homomorphism is
the vanishing of the matrix
Since .:!1 is a module for SL2(C), we have
Gi(a, f3, y, 8, a', f3', y', 8') =0
whenever ao-{3y=1, a'o'-{3'y'=1. Since Gi is homogeneous in a, {3, y, 8
and also in a', fJ', y', 0', we have Gi =0 whenever aO - {3y= [, a' 8' - {3'y' = t
for all f, t #0. Let pea, {3, y, 0, a', {3', y', 8') be a matrix coeHlcient of Ct.
Then the polynomial
pea, {J, y, 0, a', {3', y', 0'). (ao-{Jy)(a'o'-{3'y')
vanishes for all a, {J, y, 0, a', {3', y', 8' EC, so is identically zero. Since
0.8 - {3y and a' 0' - {3'y' are not identically zero, P is identically zero. Thus
the matrix G is identically zero and we have
F(Ci, {3, y, 8) F( a', f3', y', 8') == F( aa' + {3y', Ci{3' -+- fJo', ya' + oy', y{3' + 08')
in the sense that equating matrix coefficients gives a polynomial identity.
This remains true when we transfer to the field K. Thus the map
F(aa' + {3y', a{3' + {38', ya' + 8y', y{3' + 00') - F(a, {3, y, o)F(a', {3', y', 0')
for all a, {3, y, 0, a', {3', y', 8'EKsatisfying a8-{3y=1, a'8'-{3'y'=1. This
matrix certainly vanishes for all a, {3, y, 0, a', {3', y', 0' E C satisfying
a8-{3y=l, a'0'-{3'y'=1, and we shall prove that the same is true
for K.
\Ve take a Chevalley basis of Jl and decompose it into disjoint subsets
consisting of:
(i) the root vectors corresponding to the roots in a given r-chain.
(ii) the remaining set er, e-r, hs for SE n. The basis obtained fvr j1:1
corresponding to this Chevalley basis of JL decomposes in an analogolls
way, and the submoduies spanned by the basis vectors in each individual
subset are invariant under SL2(C). The matrix representing
(: 
(: :) -> F( ", fJ, y, 8)
on a su bmodule corresponding to an r-chain of roots of length q + 1 has
as its coefficients homogeneous polynomials in a, {3, y, 8 of degree q,
and the matrix representing
with a, {3, y, 8 EK is a homomorphism.
Let
(: 
Ara)=F(1, ',0,1),
A-r(')=F(1,O, ',1), 'E C.
Then Ar( 0, A-ra) are the matrices representing xr( '), x-ra) with resect
to the Chevalley basis of Jl. For each tEK Jet Ar(l), A -r(l) be the matnces
obtained by replacing the complex variable , in Ar(), A-ra) by the
clement t of K. Then A1{t), A-r(t) are the matrices representing the
ckments Xr(t), X-r(t) of Jl(K), by definition of the latter. We have
Ar(t)=F(1, t, 0,1),
A-r(t)=F(1, 0, t, 1),
on the submodule corresponding to {er, e-r, hs; SE TI} has coefficients
which are homogeneous polynomials in a, {3, y, 8 of degree 2, as has been
pointed out above. Thus the matrix F(a, {3, y, 8) representing
e :)
on JAIl is a diagonal sum of matrices Fi(a, {3, y, 0) in which the coefficients
are homogeneous polynomials in a, {3, y, 0 of constant degree. Also the
matrix
for IE K.
Now SL2(K) is generated by the matrices
G(a, {3, y, 8, a', {3', y', 8') =F(aa' + {3y', a{3' + {38', ya' + 8y', y{3' + 8S')
- F(a, {3, y, S) F(a', f3', y', S')
is a diagonal sum of matrices Gi(a, {3, y, 8, a', {3', y', 8') in which the coeffi-
cients are homogeneous polynomials of constant degree in each of
a, {3, y, 8 and a', {3', y', 0' separa tely.
( ;). C :)
for tEK and so the group of matrices F(Ci, (3, y, 0) with a, {3, y, 8EK and
o.o-f3y=1 is generated by A1-(t), A-r(t) for tEK. This group of matrices

92
SIMPLE GROUPS OF LIE TYPE
is therefore isomorphic to the subgroup (Xr, X-r) of JL(K). Thus we
have a homomorphism from SL2(K) onto <Xr, X-r) under which
( :)-+x,(t) and C )-+x-.(t).
This completes the proof.
.
6.4 The Elements "r(A) and nr
We shall denote by CPr the homomorphism from SL2(K) into <Xr, X-r)
whose existence has just been proved. We consider next the images under
CPr of the diagonal matrices
( -,)
in SL2(K) and the monomial matrix
c  ).
The images of these matrices play an important role in the sequeL
PROPOSITION 6.4. 1. Let
h,(A)=, ( _,).
Then I1r(>...) operates 011 the Chevalley basis of JLK in the following manner:
hr()...).hs=hs,
SE IT,
SE.
hr()"'). es = ,.\Ar.es,
PROOF. Suppose s is linearly independent of r. Let s be the root which
begins the r-chain through s, and let s = ir + s. Then hr()"') operates on
es in the same way that
( -,)
operates on
U,= '0" . . . ,,-,() VI,
THESUBGROUPS(Xr.X-r)
93
''-'"''''
where Vi=XfyfJ-i. Now
(
o ) .X"yq-"=,.\2i-qx(yq-,.
)...-1
Thus we have
(
o ). Ui = )...2i-qU,
)...-1
and so
hr()...). es = )...2i-qes.
However, since the r-chain through s is
-ir+s, . . . , s, . . . , (q-i) r+s,
we have Ars=i-(q-i)=2i-q by section 3.3. Hence
hr()"'). es = )...Ar.es.
We know also that hr()"') operates on er. hr, e-r in the same way that
( -,)
,........,..-J}..
operates on -x2, 2xy, y2. Now
( :_,): -X2-+A2( -x2)
2xy-+ 2xy
y2-+)...-2y2.
Thus h,.()...).er=)...2e,., hr()...).e-,.=)...-2e-,.. Finally, h,.()...) operates trivially
on "r and 'J!);, so operates trivially on the whole of jf}. ·
PROPOSITION 6.4.2. Let
nr=cpr( 0 1).
-1 0
Then nr operates on the Chevalley basis of JLK as follows:
n,.. hs=hwr(s),
.....rI!,\
nr.e8=7)r, sewr(s),
where 7)r, 8= :t 1.

94
SIMPLE GROUPS OF LIE TYPE
TilE SLJl1GROUl'S <Xr, X -T >
95
PROOF. Suppose s is linearly independent of r and Jet s= -ir+s be
the root beginning the r-chain through s. Then I1r operates on e8 in the
same way that
PROPOSITION 6.4.3. The llumfJers Yjr, s defined in 6.4.2 sati.5fy the
conditions:
(-  )
7')r. ,. = - 1,
7')r, -r = - 1 ,
"
operates on
7')r, s7')r, wr(s) =( -1)Ar8,
7')r. S 7')r, -8 = 1 .
-
Ut = E'OEl . . . E'i-l (j) Vi,
where Vt=X1yq-l. Now we have
PROOF. It has been shown in 6.4.2 that 7')r.r=-l and 7')r.-,.=-1.
We therefore suppose that r, s are linearly independent. Let
Thus
( ) xtytz-t=(-l)iXtJ-tyl.
- :
( ) Vi=( -l)ivq_i
- >
( ). Ul=( -1)1 E'0E'1... £1-1 Uq-(.
-  . <0<1. . . <,-<1
-pr+s, . . . , s, . . . 1 qr+s
la
be the r-chain of roots through s. Then
r ,
t.."
_( 1) E'O€!... E'p-l
7')r. s- - p -----,
E'OEl . . . E'q-l
and so
where
Nr, (i-p)r+s= E'i(i+ 1).
The above chain is also the r-chain through Wr(S)=(q-p)r+so Thus
""".if"-""-
It follows that
LJ
where
/1r.es=7')r, se(q-t)r+I=7')r, sewr(s),
( ) E'O€!. 0 . E'q-l
7')r. Wr(S) = -1 q ----------.
E'O€! 0 0 0 E'p-l
Hence
(
I
l _
_( 1)1 £0£10. 0 £(-1 - 1
7')r s - - - + .
, E'0E'1. 0 0 Eq-i-l -
7')r, s7')r, wy(s) =( -l)P+f1=( -l)P-q=( -1)Ar.o
Now consider the r-chain through -so This is
Also /1r operates on er, hr, e-r in the same way that
-qr-s,..., -s,. o. ,pr-so
i
C
(-  )
Therefore we have
I
I
i__,...
operates on _X2, 2xy, y2.
Now
EOEI . 0 0 Eq-l
7')r -8 = ( - 1)1] ----- --,
, EOil 0 . 0 ip-l
.......
i
i
i
,..-....
-X2 _y2,
01) .0
2xy -2xy,
y2x2.
Thus l1r.er= -e-r and I1r.e-r= -er. On J!) we have nr.hr= -/1r and
I1r.h=h if (I1r, h)=O. Therefore I1r.hs=/tWr(6) for aU SE <P. .
where
C
Nr, (i-q)r-s = €i(i + 1).
However, the relations between the structure constants proved in 4.1.2
show that
Nr, jr+8 _ N-(j-H)r-s, r _ _ Nr<!.--I:!l
(0+ l)r+8:-(}+ l)r+s) - (jr+s,jr+s) - (jr+s,jr+sf
L

96
SIMPLE GROUPS OF LIE TYPE
Since each €, i. is just the sign of the corresponding structure constant it
follows that
CHAPTER 7
Ej+p = - iq-j-l,
j=O, I, .. ., (q-I).
Thus
The Diagonal and Monomial Subgroups
'TJr, -8=( -l)q O:_q-l=( -I)p Ep+q-l€p+q-2 ._.:._.
EO E1 . . . Ep-I £p+Q-1 Ep+Q-2 . . . EQ
7. 1 Properties of the Subgroup H
It follows that
'TJr, s'TJr,-s=(-l)p £0£1... Ep-I.(_I)P £p+q-I... Ep_l.
£OEI . . . Eq-I £p+q-I . .. Eq
\Ve have introduced the elements /rr(A) of the Chevalley group G=JL(K)
as the images of the diagonal matrices
.
We shall also need some properties of the elements
( :-1)
nr(t) =1>,-( 0 Ot),
-t-I
which are stated in the following lemma.
under the homomorphism from SL2(K) into (Xr, X-r). We define H
to be the subgroup of G generated by the elements hr(A) for all r E cD,
A =I- 0 E K.
Now hr(A) operates on the ChevalIey basis of 1LJ( by
LEMMA 6.4.4.
I1rCA) .118=l1s,
hr(A) .es= AAr8es.
(i) nr(1)=nr,
Hence each element of H is an automorphism of 1LK which operates
trivially on 1L)K and transforms each root vector es into a multiple of
itself. The coeHicients arising in this way define naturally a character
on the additive group generated by the roots, as we shall now describe.
Let P=Z<l), the set of all linear combinations of elements of <]) with
raLional integer coefficients. P is the additive group generated by the
roots of 1L It is a free abelian group of rank 1 and has a basis consisting
of the set of fundamental roots n = {PI, pz, . . . , pz}. A homomorphism
from the additive group P into the multiplicative group K* of non-zero
elements of K is called a K-character of P. Thus a K-character of P is a
map from Pinto K* satisfying the conditions
nr( -1)=n;\
(ii) n,-(t)=xr(t)x-r( -t-I)xr(t),
(iii) IIr(t) = nr(t)nr( - 1).
PROOF. These results follow from the corresponding relations in
SLz(K) using the homomorphism CPr. .
We ote that Proposition 6.4.2 shows that the element nr of G operates
on :If) 111 the same way as the element Wr of Wand that nr also permutes
the root spaces JLs,sE <1>, in the same way as Wr. This relationship betwecn
fir and Wr has many implications which will be made use of in the de-
velopment to follow.
x(a+b)=x(a) X(b),
X( -a) = X(a)-I,
a, bE P,
aEP.
NowaK-character of P is uniquely determined by its values on the
fundamental roots. Thus if a map from <}> into K* can be extended to a
K-character of P, this extension is uniquely determined. We consider
such maps of the form
S---+AAr8,
rE (1), AEK*.
97
". -,
\ ro-,
0)
I 'J
'\...
!
""'-;

---
r
.J,!
["-
"'.
r'
t
.................
tc;:i
r
t;.,;,
f
L3
--

L..
( "
I
I
L.,
f-
I
"''v.;;:u;>"""
1
L"
r
98
SII\!PLE GROUPS OF LIE TYPE
TilE DIAGONAL AND T..1ONOI\IIAL SUBGROUPS
99
This map from <I) into K* can be extended to a K-character of P. For
let Xr, A be the map from Pinto K* given by
Xr. A(a)=A2(r, a)/(r. r).
Then Xr. A is a K-character of P which takes the value AAT3 at the root s.
Now the K-charaters of P form a multiplicative group. For if Xl, X2
are K-characters so IS XIX2, where
let Q be the set of linear combinations of {jJ, . . . ,{jl with rational
intcger coeflicients. Q is the additi\' grollp generated by the fundamental
weights and is free abclian of rank t. Since each Pi is in Q, P is a subgroup
of Q. The index I Q : P i of P in Q is 11nite. In fact, since
rJ
Z
Pi = 2: Ajiqj,
j=l
(
XIX2(a) = XI(a) X2(a), a E P.
Moreover, each K-character X of P gives rise to an automorphism h( )
of JLK defined by X
h(X)' hs = h8, h(x). e8 = X(s) ea.
Observe that if X= Xr, A then h(x)=hr(A). The automorphisms of JLK of
t"_!_!!m a subgroup H of the flina'utolr'PhTs-;;;'-g-roup of JLK:
For ._u._._.__..-______.. ..--....--- .... -.-..-.
h(xI). h(x2) =h(XIX2).
Also the map x---+h(x) is an isomorphism from the group of K.:.cl1aracter.s.....
of P onto the group Ii. --
ow, as H Tsthesubgroup of G generated by the automorphisms
hr(A) for all r E<l>, A E K*, H is a subgroup of fl. Each element of JJ there-
for has the form hex) for some K-character X. We shall now consider
whIch K-characters X give rise to automorphisms hex) which are in H.
Let qI,q2, . . . , qz be the basis of j[} dual to the basis h, h 'I
f f d PI' P2'...' f, Pr
o un amental co-roots. Thus qI, . . . , qz are defined by
(h )_{l if i= j,
Pjtqj - 0 if i=l=-j.
Th non-singularity of the Killing form shows that qI, . . . , qz form a
basIs for If}. They are calId th f!!!ldamental weights of 1£... In particular,
each fundamental root Pt IS a lmear combination of qI, . . . ,qz. Let
p.I. = }LilqI + . . . + }Luqz.
Then (hp1, Pi) = fLij. Thus we have
fLij = 2 !!.!il = A P = Ajl
(Pi> Pj) :J PI
in the notation of 3.5.2. Thus
we have I Q : PI =det (Ajt). We shall write I Q : P I =. The values
of /J. for the difTerent types of algebra may be calculated from the Cartan
matrices shown in Section 3.6, and are as follows.
JL: Az Bz
t..: /+1 2
Cz Dl G2 F4 E6 E7 E8,'
2 4 1 1. 3 2 1.
(/J. is equal to the order of the fundamental group of the Lie group of the
given type. See Adams[l].)
Now every K-character of Q gives rise to a K-character of P by restric-
tion. However, not every K-character of P need be the restriction of some
K-character of Q.
Wf:, can now describe which K-characters of P give rise to elements
of ll.
THEOREM 7.1 . 1. II is the subgroup of G consisting oj all alltomorphisms
hex) oj J[.]( for which X is a K-character oj P which can be extended to a
K-character oj Q.
PROOF. H is generated by the elemenls h(Xr, A) for all r E cD, A E K*.
Now Xr. A can be extended to the K-character of Q given by
Xr. A{a)=A2(r. a)/(r, r),
aEQ.
(Note that 2(r, a)/(r, r)ElZ. since 2r/(r, r) is a co-root and aEQ.) Thus
each element of H has the form h(x), where X is a K-character of Q.
Conversely, let X be any K-character of Q. Let
X(qI)=AI, X(q2)=A2,. . ., X(ql)=AZ.
Now XPIo At takes value At at g-l and value 1 at qj for all j#- i. For
XPi. Alqj) = A;(P,. qj)/(Pjl Pi) = AJI.
l
Pi =  Ajiqj
j=l
and Pi is actuaHy an integral combination of qI, . . . , qz.
Thus we have
X= XPt. "lXP2. A2 . . . X1JI, ",

100
SIMPLE GROUPS OF LIE TYPE
THE DIAGONAL AND MONOMIAL SUBGROUPS
101
and so
Now let e.es=es+Y. Suppose sEcD+. Then
Y EJ'f} EB  JLr
'-<8
hex) = h(XPI. AI) h(xpz. AZ) . . . h(xPI. AI)'
Therefore h(X) EH and the theorem is proved.
.
since () E V. Also
We now consider the relation between H and the subgroups V and
V of G. Note first that Ii normalizes each root subgroup Xr. For
Y E b JLr
rp'
hex) Xr(t) h(X)-l=h(x) exp ad (ler) h(X)-l
=exp ad (h(x).ter)
=exp ad (x(r) ter)
since e E VII. But we have
1=1f) EB 2: Ir EB  11"'
,-<B rpB
h(x) xr(t) h(x)-l=x,.(x(r)t),
hencey=O. Thus e.es=es.
A similar argument applies if SE cD- and so ()= l.
Note. A similar argument shows that VH n V = 1. In particular we
have Jf n V = 1 and II n V = 1. These results clearly apply also when
H is replaced by fl.
.
by 5. 1. 1. Thus
whence
hex) Xrh(X)-l = Xr.
It follows that H normalizes V and V, which are generated by root
subgroups. In particular, UI1 and VH are both subgroups of G.
COROLLARY 7.1.3. VH n VH=H.
PROOF. UH n VH=(Ull n V) H=H.
""'I.
Note. The same argument shows that the group fl also normalizes
V and V. Although fI is a group of automorphisms of JLJ(, we have not
shown that f1 is contained in G, and in fact this is not in general so.
However it is evident now that if normalizes G in the group of all auto-
morphisms of JL](, since G is generated by U and V.
7 .2 Properties of the Subgroup N
Let N be the subgroup of G generated by H and the elements nr fr .a11
rE <I>. In investigating the properties of Nwe begin with a lemma descnbll1g
the way in which the elements nr transform the root subgroups.
LEMMA 7.1.2. VlI n V= 1.
LEMMA 7.2.1. Let r, SE <1>. Then
(i) I1r,xit).n;1=xwr(8)(YJr. st),
(ii) nrXBn;l= XWr(B).
PROOF. Let 0 E VH n V and consider the effect of 0 on the Chevalley
basis of JLK. Since 0 E V we have
O.hs=/zs+x, , where XE 2: JLr.
r E <1>-
XEJI) EB  J[r.
rE Il>-
PROOF. I1rXs(t) 11;1=11, exp ad (te,) n;-
=exp ad (nr.tes)
= exp ad (YJr. stewr(S»)
=XWr(8)(YJr. st),
using 5.1.1 and 6.4.2.
The most important property of the subgroup N is the one we sha11
now prove.
fGC't
Since ()E VII we have
However
.
JL = JI) EB b JLr EB 2: JLr
r E <1>+ r E <1>-
and so x=O. Thus 0.hs=h8'
""",,,,,-
\

. .
'"
'-
I:
-."1;
[
r>
L
[
''!i
L
[
f"
I
i
i
L.
L"
i
I
J
t__
'-
L_-
102
103
SIMPLE GROUPS OF LIE TYPE
HIE DIAGONAL AND MONOMIAL SUnGRQUPS
THEOREM 7.2.2. There is a homomorphism from N onto W with kernel
H Ufler. which nrWr for all r E <1). TIlLIS II is a normal subgroup of N alld
Nj H IS Isomorphic to JfT. If n EN, heX) E 11, we have
nh(x) 11-1 =h(X'),
where X'(r)=X(w-1(r)), w being the image of n under the Clbove homo-
morphism.
Since hwr(s)(7]r s) E H we have
I1rH. nsH. (I1r!-I)-1 = I1wr(8) H.
Since n = h( - 1) E H we also have
(nr}1) 2 = II.
Now W is defined as an abstract group by generators iVr, rE <1\ subject
to relations w;=l, l\irWsw;l=wwr(S) (2.4.3).
Hence there is a homomorphism from W onto NjH under which H'r
is mapped to I1rH. Since this is the inverse map of the one obtained above,
it is an isomorphism. Thus Nj H is isomorphic to J-JI. ·
PROOF. By 6.4.2 lir operates on the ChevalJey basis of 1£K by
nr.hs=hWr(B),
nr.eS=7]r,8eWr(S)'
Thus I1r transforms the root space 1£s into 1£wr(s), and so l1r permutes
the root spaces in the same way that the element Wr of W permutes the
oots. FUfthermor.e, each element of H transforms each root space into
Itself. Thu there IS a homomorphism of N onto v which maps I1r ;nto
Wr and which contains H in its kernel.
We show next that H is a normal subgroup of N. Let n EN and hex) EH.
Then we have
COROLLARY 7.2.3. N nfl=H.
PROOF. Let nEN nfl. Since nEH we have I1.JLs=JLs for each sEct).
Thus 11 is in the kernel of the homomorphism mapping N to J;V, and so
Il E J-l. III!
nh(x) n-l.hs=nh(x). (n-l!zs)=n. (n-1hs) =hs,
nh(X) n-l. es = nh(x). (7]ew-l (S»),
where w is the image of n under the above homomorphism and 'J1 = + 1
Thus -, -'
I1h(X) n-l. es = 7]X(w-l(s)) n. ew-1 (s)
=7]x(w-1(s)) 7]-les
= X'(s) es,
where X'(s)=X(w-1(s)). Hence we have
nh(x) n-l=h(x')
COROLLARY 7.2.4. (i) V nN= 1, (ii) VH nN=H.
PROOF. Let 0 E V n N. Then we have
e. es = "ew(s),
where" E K* and WE W is the image of 0 under the homomorphism from
N to v. However
O.es=es+x,
where
X E 3I) EB L: lLr,
r>-s
since OEV. 1t follows that x=O, "=1 and w(s)=s. This is valid for all
JE(I), and so w=1. This means that OEH. But V nJ/=l and therefore
0= 1. Hence U nN= 1.
l-'-inally we have
and so H is normal in N.
. We thus have a homomorphism from NjH onto W under which I1rH
IS mapped to Wr. We show this is an isomorphism. Let r, SEe!). Then
nrn/1;l =nrxs(1) x_8( -1) x8(1) n;1, by 6.4.4,
=XWr(s)(7]r, 8) XWr(-8)( -7]r, -8) Xwr(s)(7]r, s), by 7.2.1,
=XWr(s)(7]r, s) X-Wr (8)( -7];:!) XWr(s)(7]r, s), by 6.4.3,
=nWr(S)(7]r,8)
=hwr<s)(7]r, 8) nwr(s), by 6.4.4.
VH nN=(U nN) H=H.
.
Note. A similar argument shows that vA n N = H.

THE DRUHAT DECOMPOSITION
105
CHAPTER 8
PROOF. Ur is permutable with Xr and X-r since both normalize Ur.
Since Ur and Xr are permutable subgroups of U, UrXr is a subgroup
of U. However, it contains all the root subgroups of U, so is the whole
of U. ·
-,
The Brullat Decomposition
8.1 Bruhat's Lemma
COROLLARY 8.1.3. nr normalizes Ur.
Let B he J!.group__, UB oft._ . Ghevall,ey., group _ G.,,, In. the_. ca$__),Yb_e.IJ,,,_
the field ,j!g_rcafIy.- dosed the subgroup B and i_ conjugates in G
-are'usu'alTy ca lIed Bor-erslibg-ro ll"ps'o(G':"'ri:i-'TI;Tscase'-G"'ny"bJll'terpreted
as an algebraic grou£"ancL.th-1?ord subgrps a-eci;ely-the' maximal
_ solu b!i=<2!2!tc[_gR!:g.!l?__.?.£ ..9-: It was-proved by -BoierfflaFany'two
such subgroups are conjugate (cf. Borel [3]). We shall investigate the de-
composition of G into double cosets BgB with respect to B, following an
investigation by Bruhat of the double coset decomposition of a smi-
sinlple Lie group with respect to a maximal soluble connected subgroup
(cf. Bruha t [J D.
\Ve shall first require some additional information about the subgroup
U of G. Let r E n be a fundamental root and define
Ur= 11 Xs,
BEw+-{r}
where the product is taken over all the positive roots other than r in
increasing order. ChevaJley's commutator formula shows that Ur is a
subgroup of U and that the root subgroups in the product can in fact be
taken in any order.
PROOF. The normalizer of Ur contains Xr and X-r so contains nr,
since I1r E(Xr, X-r). ·
We define similarly, for each rEn,
V-r= II Xs,
BEcI>--{-r}
where the product is taken over the negative roots other than - r in
decreasing order. Then Xr and X-r normalize V-r and we have
V= V-rX-r=X-rV-r and V-rXr=XrV-r.
t
Furthermore l1r normalizes V -r.
We observe that the above results depend essentially upon the fact that
r is a fundamental root. They are not valid if r is an arbitrary positive
root. We shall use them to prove the following propositions.
..,.....
PROPOSITION 8. 1 .4. Let r En. Then the subset B U BnrB is a subgroup
ofG.
LEMMA 8. 1 . 1. Let r En. Then Xr and X-r normalize Ur.
PROOF. We must show that this subset is closed under multiplication
and inversion. Inversion is clear, since
PROOF. The commutator formula 5.2.2 shows that [Xr, Ur] S; Ur,
hence Xr normalizes Ur. Now cO,nsider the subgroup [X.-r, Ur]. Suppose
s is a positive root distinct from r and suppose - ir --I- jSE (1), where
i>O, j>O. Then -ir--l-jsE([>+. For when this root is expressed as an
integral combination of fundamcntal roots, at least one coefficient is
positive. Thus each coeflicicllt l1lust be l1on-negativc and we havc a
positive root. The root r itself cannot occur in this form. Applying the
commutator formula once more we have [X-r. Ur]  Ur. Thus X-r
normalizes Ur also. .
B-1 = B,
(BnrB) -1 = B-1n; 1 B-1 = Bn; 1 B = BnrB.
In order to prove that the subset is closed under multiplication it IS
sufficient to show that
COROLLARY 8.1.2. (i) U= UrXr= XrUr, (ii) UrX-r= X-rUr.
104
nrBnr  B U Bl1rB.
Now B= UI/=XrUrll, as in 8.1.2. Then we have
nrBl1r = I1rBn; 1
= nrXr UrHn; 1
= X-rUrH £; X-rB,
--'.

-
IJ
 .
t
tc.J
r'
t..J
1 '
,
:..=-!
I' -
J
l..,'
"'-'
 .-.

Ii.-",
'f -
, .1'
-----
106
TilE DRUHAT DECOMPOSITION
107
SIMPLE GROUPS OF LIE TYPE
ince nr normalizes Ur. Thus it is sufficient to show that X-r £ B u BnrB,
l.e. that x-r(t)EBuBllrB for all tEK. If t=O, X-r(t)EB. If t:fO we
have
as above. Thus
X-r(t)=Xr(t-l) nr( -t-1) xr(t-I), by 6.4.4 (ii),
=Xr(t-I) h,( - t-I) nrXr(t-I), by 6.4.4 (iii).
Thus X-r(t)EBllrB and the proof is complete.
.
BnB. BnrB = Bll'llrB. Bl1rB
= Bn' B. BnrB. BnrB
£ BIl'B.(B UBnrB), by 8.1.4,
=Bn'B uBn'B.BnrB
=Bn'B u Bn'l1rB
= Bnl1rB U BnB.
Thus
The fol1owing proposition is of crucial importance in the later de-
velopment.
BnB. Bl1rB £ Bmlt.8 U BI1B.
If w(r) E <1>-, the set BIlB. BlhB contains elements from both BI1Il,B
and BIlB. For nil, is an element in this set. from Blll1rB, and nx,{I)l1r is
an element from BIlS, since
IlXr(l) I1r=llllr.n;lxr(l) Ilr=l1l1r.x_,C-l)
PROPOSITION 8.1.5. Let r E IT, n EN and w be the image of n under the
natural 110m 0111 OIph ism from N to W. Then
BnB. BnrB £ BllnrB U BnB.
III particular, if w(r) E <1:>+ we have
=f1l1r. xr( - 1) nrxr( - I) EBn' B. Bl1rB= BnB.
.
COROLLARY 8.1.6. Let r E n, 11 EN and w be the image of n ill w.
Theil
BnB. BnrB = BnnrB,
whereas if w(r) E <1>- the set BnB. BnrB contains elements from both double
cosets BnnrB, BIlB.
Bn,B. BnB s; BllrnB U Bn B.
PROOF. LetgEBn,B.BIlB. Then g-lEBn-lB.BllrB. Hence, by 8.1.5,
g-lEBn-1nrB or g-lEBn-1B. 1t follows that gEBllrnB or gEBnB. II
PROOF. Suppose first that w(r) E <1)+. Then we have
BnB. BnrB= BIlXrUrHnrB
8.2 Groups with a (ll, N)-Pair
=BnXrn-l.nnr.n;lUrHnrB.
It is now useful to introduce the general concept of a (B, N)-pair in a
group. This concept, originally due to Tits [11], is useful not merely for
deriving further properties of the ChevalJey groups, but also in con-
nection with the 'twisted groups' discussed in chapters 13 and 14, and
with the associated geometrical structures described in chapter) 5.
A pair of subgroups B, N of a group G is called a (B, N)-pair if the
. foJIowing axioms are satisfIed:
BN 1. G is generated by Band N.
EN 2. B (IN is a normal subgroup of N.
BN 3. The group W=N/B (IN is generated by a set of elements Wi,
ieI, such that \v;= 1.
Now
nXrn-1= Xw(r) £ B,
since w(r) E <1:>+. Also
n;lUrHnr= UrH£ B,
by 8. 1 .3. Hence BnB. Bl1rB £ BnnrB. The reverse inclusion is clear.
Now suppose that w(r)EcD-. Let w'=wwr. Then w'(r)E<1:>+. Let 11'
be an element of N mapping to w' under the natural homomorphism.
Then we have
Bn' B. BnrB = Bn' nrB,

108
SIMPLE GROUPS OF LIE TYPE
BN 4. If l1iEN maps to Wt under the natural homomorphism of N into
W, and jf!l js any element of N, then
BntB. BnB s BflinB U BnB.
BN 5. If ni is as above, then niBni =/:-B.
PROPOSITION 8.2.1. The Chevalley group G = JL(K) has a (B, N)-pair.
ROOF. We show that the subgroups of G previously denoted by B, N
satisfy the above axioms. By 7.2.4 we have B nN=/I, and by 7.2.2
we know that If is normal in N and that NjH is isomorphic to the- WeyI
group W. The Weyl group is generated by the fundamental reflections
Wr, rEn, and we take these as the generating involutions in BN 3. By
8. 1.6 we have, for each rErr,
BnrB. BnB S BnrnB U BnB.
If rETI we have XrsB, but l1rXrl1r=X-r is not in B. Thus l1rBnr=/:-B.
Finally, G is generated by Band N. For G is generated by the root sub-
groups Xr, and each root r has the form w(ri) for some IV E Wand rt E n.
Let II be an elemenl of N mapping to tV under the natural homomorphism.
Then
Xr= Xw(rt) =nXrtn-1
by 7.2.1. Thus Xr is in the subgroup generated by Band N, whence
(B, N) = G. Therefore G has a (B, N)-pair. .
We now derive some consequences of the axioms for a (B, N)-pair.
PROPOSITION 8.2.2. Let G be a group with a (B, N)-pair. Then
(i) G=BNB.
(ii) For each subset J of I, let WJ be the subgroup of W generated by
the clements wi/or iEi, and NJ be the subgroup oj N mapping to WJ linder
the natural homomorphism. Then P J = BN J B is a subgroup of G.
PROOF. We prove (ii) first. BNJB is certainly closed under inversion,
and so we must prove it closed under muJtiplication. Let 11 E NJ. Then
11 may be writlen in the form n = /11f1'l . . . nk, where each Ili maps under
the natural homomorphism to a generator Wi of W with; E J. Thus
flBNJB=nln2. . . flkBNJB
S 111n2 . . . J1k-1BNJB s . . . s BNJB
THE BRUHAT DECOMPOSITION
109
by repeated application of BN 4. Hence
NJBNJB s BNJB
and it follows that
BNJB .BNJB s BNJB.
Thus BNJB is a subgroup of G.
We now take J=I. Then BNB is a subgroup of G. This subgroup
certainly contains Band N, and must be G since Band N generate G. ·
It is evident from 8.2.2 that every double coset BgB contains an
element of N. We now consider the question of when two elements of
N lie in the same double coset.
'I PROPOSITION 8.2.3. Let G be a group with a (B, N)-pair. Let n, n'
be elements of N. Then BnB=Bn' B if and only if n, n' map to the same
element of W under the natural homomorphism from N ;nto w. Tlws there
is a natural 1-1 correspondence between double cosets of B in G and
elements of W.
(We note that in the case where G is a Chevalley group and B a BorI
subgroup this result implies that the number of double co sets of B 111
G is finite, and equal to the order of the Weyl group. For a general group
with a (B, N)-pair, the group W need not be finite.)
-
PROOF. Each element of W is a product of generators Wi, iEI. We
denote by I(w) the shortest length of any expression for w  a product
of such -geije'i.t2ii:. 'Sllppose-'-JjiiB=BIl'B, where /1, n' correspond to
iv, w' -'re'spectively, and suppose that I(w)  I(w'). We show that w = w'
by induction on I(H} If 1(11')=0 we have w= 1 and BnB= B. Thus Bn' B=B
and so l1'EBnN. Hence w'=l.
Now suppose l(w»O. Then W=WiW", where ;EI and l(w")=/(w)-l.
Let nt, 11" be dements of N corresponding to Wi, W" E W. Then
ntll"B S Bn'B.
.
Thus by BN 4 we have
n"B S niBn'B S Bnin'B uBn'B.
Hence BIl"B=Bntn'B or Bn"B=Bn'B. By induction we have W"=WiW'
or w" = w'. The latter is impossible since I(w") < I(w'). Thus IV" = H'tW' and
so w' = WiW" = W.
,-.
.

--
[-
-,
€
t
f---
!
L
,--.
t
i:<.
to'
t
).
L"
r
I
L,
L
L.r
f
t.,
r
I
j
-- ):
1-
t
110
THE DRUHAT DECOMPOSITION
III
srMPLE GROUPS OF LIE TYPE
Since every double coset of B in G has the form BnB for some n EN
there is a 1-1 correspondence '
PROOF. By BN 4 we have
n-tBnt s; B U Bl1tB.
BnB(B nN)n
between double co sets of B in G and elements of W.
Since I1tB/1i =I B we have
..
I1tBni n BniB =I cpo
It follows that
The next two propositions establish analogues of the more detailed
results concerning double coset multiplication proved for Chevalley
groups in 8. 1 .5.
X PROPOSITION 8.2.4. Let G be a group with a (B, N)-pair. Let Wi. iEI,
be one of the distinguished generators of W alld w be an element of JtV
such that I(Wtw);?: l(w). Let nt, Il be elements of N mapped by the natural
homomorphism into Wt, WE W. Then Bll-tB. BII B S; BlltnB.
ntB nB/1iB/1t=lcp
nlBII n BniB/1i/1 =I (p.
Now l(wi.wiw);?:I(Wtw), thus by 8.2.4
nlB/1iIl  BIIB.
Hence /1lBn intersects BIIB and the result follows.
PROOF. We again use induction on l(w). If l(w)=O then w= I and
nEB, so the result is clear. Suppose l(w) > o. Then w = W'Wj for some
jEI, where l(w')=/(w)-l.
Suppose by way of contradiction that the result is false. Then BntB. BIIB
intersects BIlB, by BN 4. Let n', nj be elements of N mapped by the natural
homomorphism to w', Wj E W. Then
ntBn n BnB =I cp
Note. It follows from 8.2.4 and 8.2.5 that I(H'tw)=lI(w).
COROLLARY 8.2.6. Suppose the 110tation is as ill 8.2.4. (f I(Wtw) <l(w)
thelll1t EBnBIl-1 B.
PROOF. By 8.2.5 we have
I1tBI1 n Bn B =I (p.
Therefore I1tEBI1BIl-IB, as required.
iii
and therefore
ntBn' n BnBnj =I cpo
Now I(Wtw')  I(w') and so, by induction, we have
8.3 l:}arabolic Subgroups
ntBn'  Bl1tn' B.
Thus
Let G be a group with a (B, N)-pair. A parabolic subgroup of G is one
which contains some conjugate gBg-I of B. For example, the subgroup
P.T of 8.2.2 is a parabolic subgroup. We shall show that the subgroups
PJ for the various subsets J of J are the only subgroups of G containing
B, and therefore that every parabolic subgroup is conjugate to some
P J. Furthermore, the subgroups P J are all distinct and non-conjugate
in G, and _eb.. .9f thcmis equal to its_ normali?-er.
w;.--tist pro-- ,1. propositIon -'describing the subgroup generated by
B and a single element of N.
Bntn' B n BnBl1j i:- cpr
Now it follows easily from BN 4 that
BnB . BnjB S; BnnjB U BnB.
Hence Bntn'B intersects either BnnjB or BnB. Thus, by 8.2.3, WtW'=WWj
or WtW' = W. The former equation gives WiJV' = w', whence Wt = 1, which
contradicts BN 5. The latter gives WiW= w', which implies I(Wtw) < l(w), a
contradiction. .
PROPOSITION 8.2.5. Suppose the notation is as in 8.2.4. Then, if
/(WiW)  /(w), BntB. BnB has non-empty intersection with BnB.
PROPOSITION 8.3.1. Let G be a group with a (B, N)-pair and let /1
be an element of N. Let IV E W be the image of n under the natural

. ;:. J....., :
- 1:", ..... ..''''''
112
SIMPLE GROUPS OF LIE TYPE
W = Wil JIli2 . . -. HliA:'
ir, iz, . . . E I,
THE BRUHAT DECOMPOSITION 113 
Let nEN rUF).(pJ). Then we have 1flti:7;.-.
P J "2 (B, nBIl-I) = (B, n) t,:<>:::;,;":
by 8.3. I. Therefore n EPJ. It follows that P J  jI)( P J). .:>
Suppose now that P J, PI{ are conjugate subgroups of G. Let,
gPJg-I=PK where g=bnb' with b, b'EB and nEN. Then nPJn-I=PK.
Thus
homomorphism and let
where l(w)=k. Let J be the subset {iI, iz, . . . , h} of I. Then the following
three subgroups of G are equal:
(i) (B, n),
(ii) (B, nBn-I),
(iii) P J=BNJB.
PROOF. We have /1=ni/1i2 . . . nik, where l1ia EN corresponds to Wia E W.
Each Ilia is in NJ, so IlENJ. Thus we have the inclusions
PK"2 (B, nBn-I) = (B, n)
(B, nBn-I)  (B, n)  P J.
Now P J is the subgroup generated by B and the elements nil' /li2, . . . ,l1ik.
Since I(Wi1\\') < 1(11') we have, by 8.2.6, nil E(B, nBn-I). Since
I(Wi2Wi1W) < I(Wi1W)
by 8.3.1. Therefore nEPI{ and so p]{=PJ. Thus distinct subgroups
P J, PI{ are non-conjugate. .
THEOREM 8.3.4. Let G be a group with a (B, N)-pair. Theil the sub-
groups P J for distinct subsets J of I are all distinct. Furthermore we have
PJ nPj{=PJnI{. Thus the subgroups PJ form a lattice isomorphic to the
lattice of subsets of I.
we have
ni2E(B, ni1nBn-Int;.1)  (B, nBn-I)
and arguing similarly we see that each of nil' l1i2, . . . , l1ik lies in (B, nBn-I).
Hence P J is contained in (B, nBn-I) and the result follows. .
PROOF. It is necessary to show first that the elements Wi of W for
i EI form a minimal set of generators of W. Suppose we remove one
elementj from I, and assume by way of contradiction that W is generated
by the elements Wi for i E 1- {j}. Then G has a (B, N)-pair with I replaced
by J - {j}, since the axioms BN 1-5 are still satisfied. Let
THEOREM 8.3.2. Let G be a group with a (B, N)-pair. Then the sub-
groups P J are the only subgroups of G containing B.
Wj = Wi1 J1'i2 . . . Wik'
iaEI-{j},
J= U Ja.
III
be an expression of minimal length of Wj in terms of the remaining
generators. Let /1j be an element of N corresponding to Wj E W, and let
J={h, iz,..., ik}. Then, by 8.3.1, we have (B,/1j)=BNJB. However,
<JJ, II}) = B U Bl1jB by 8.3.1 applied to the fult set of generators of IV.
Thus BNJB=B uBn}B. By 8.2.3 this implies that 1-'VJ={I, II']}, which
gives a contradiction. Thus the elements l\"i, iEJ, form a minimal set of
genera tors.
Now let J, K be any two subsets of 1. P J nPJ{ is a subgroup of G con-
taining B, thus PJ nP/{=PL for some subset L of I by 8.3.2. Now
P J n J{ S; PL. We suppose by way of contradiction that P J n K i=-PL. Tllen
. certainly L is not contained in J n K and we may assume without loss
of generality that L is not contained in J. However, PL  P J and so
NL  NJ by 8.2.3. Hence WL  WJ. Let iEL- J. Then WiE WL and so
Wt E VJ. But this means that Wi is expressible in terms of the remaining
generators of 1-'V, which is impossible. Thus PJ nP/{=PJn]{.
.-
PROOF. Let Ai be a subgroup of G containing B. M is a union of
double cosets of B in G, and each such double coset contains an element
of N. Thus Al is generated by B and a cerlain set of elements of N. Let
na be any element of N. Then, by 8.3. I, the subgroup generated by B
ancilia is P.T", for a suitable subset Ja of I. Thus !vI is genera led by sub-
groups P Ja for a family of subsets Ja of I, and so M =P J, where
THEOREM 8.3.3. Let G be a group with a (B, N)-pair. [hen each sub-
group .p JPlf! is equal 10 its normalizer. Furthermore disti/1i's--;;;;g,.--;ps
P J, PK ca/1/1ot be conjugate in G.
PROOF. The normalizer .112(P J) is generated by B and elements of N.

114
SIMPLE GROUPS OF LIE TYPE
THE URUHAT DECOMI>()SITION
115
"-"
.J
We show finally that the subgroups P J are all distinct. Suppose P J=PK.
If J =f-K we may assume that J-K is non-empty. Since
P J=P J nPj(=P J 0 K,
we have WJ= WJO K, as before. This again means that some generator
Wt is expressible in terms of the remainder, which gives a contradiction.
Thus P J=PK implies that J=K. II
We define
Vm= 11 Xr
h(r»m
for III = 1, 2, . . . and also defll1e V = U1/L n VI, V = Vm n V2. We show
UIII= U.,;U1 for all m. This is clear if m is sufficiently large, and we prove
it by descendi ng induction. Assume inductively that UII-1-1 = UL/-l U, -j-I'
Now we have
[
We emphasize here that all the results of sections 8.2 and 8.3 :have
been obtained using only the axioms for a (B, N)-pair.
Vm= n Xr. Vm+1
h(r) = 1n
and Vm/Vm+1 is abelian. Thus
t
8.4 A Canonical Form
Um = U'I UI UIII-'+ I'
r-
L.
We now return to the situation in which G is a Chevalley group J1..(K). T:he
results just proved on the double coset decomposition are valid for G since
G has a (B, N)-pair. Thus every element of -G can be written in the form
bInb2, where hI, b2 EB and n EN. However, it may be possible to express an
element of G in such a form in a number of different ways. We seek a
canonical form for elements of G, i.e. a way of decomposing an element of
G so that each element has a unique expression in the given form.
A subset qr of <}) is called a closed set of roots if, whenever r, so E qr and
ir + js E (I), where i, j are positive integers, the root ir + js is in \}f also. We
consider the situation in which (J)+ is expressed as the disjoint union of
two closed subsets. Suppose <1>+ = 'Y 1 U '¥ 2, where 'Y 1 n 'Y 2 is empty and
qrI, 'Y2 are closed. Let
Since Vm+1 is normal in Vrn we have
Vm= V1;Ul;Um+l = V,,V1It+l UI
= V,;, V'I + 1 V;-I_l VI = u.,; Vl'
r:
Finally, putting 111= 1, we have U= VIU2 as required.
.
r
g
L.
COROLLARY 8.4.2. Each element LIE V is uniquely expressible in the
form U=UlliZ with lllEUI, UzEU2.
f-
t
VI= 11 Xr and V2= 11 X,.,
re':Yt re'Y.
A decof!1position of (})+ of the type we have been considering is deter-
mined by each element of the Weyl group W. Let lV E V and define
\YI={rE cI)+; w(r)E (f)+},
r
f
.. 0'_
where the products are taken over the roots in increasing order. The
commutator formula shows that VI, V2 are subgroups of Vand that the
same subgroups are obtained if the factors are taken in a different order.
'Yz={rE(!>+; w(r)E(!)-}.
Then 'If1, 'Ir2 are disjoint closed subsets whose union is (D+. We define
Vt- = n Xr,
rE '/"1
r-
L
LEMMA 8.4.1. V= VIV2, VI n V2= 1.
Vt;= n Xr.
r E 'I".
PROOF. Let liE VI n V2. Then
u= II Xrltt).
r e 1,1)+
Then V = V;% VI; and v1t n V;; = 1.
We can now describe the required canonical form for elements of G.
'-
Since UE VI we have tt=O whenever rtE'Y2. Since UEU2 we have tt=O
whenever ri E'YI. Thus tt=O for all i and u= 1.
THEOREM 8.4.3. For each WE W choose a coset representative nw EN
lrhich maps to lV lInder the natural homomorphism. Then each clement

116
SiM PLE GROUPS OF LIE TYPE
of G is expressible ill juS! one way in the form
g=bnwll,
Hlhere b E Band U E U;; .
PROOF. We shall show that llwUttn;1 S U. By 7.2.1 we have
11, X/I; 1 = X w,(S),
for all r, SEe!>. Let W=H-'rlWr2'" \I'rk be an expression of lV as a product
of reflections. Then I1w and 11'1n'2' . .l1rk both have image w under the
natural homomorphism, hence
I1w=hJ1rl . . .11rk
for some hE H. Thus
X -1 1 X-I Ih-l
nw /lw =llI1rl'" 11rk snrl '" J1rk
= hX w(s)h--1 = X w(s)
by 7.2.1. It follows that
11wU;t"11;;l=l1w. 11 Xs.n;:;;l S U.
SE'¥,
(One can prove similarly that nwU; 11;;1 S V.)
Now consider the double coset BllwB. We have
BnwB = Bl1wHV = Bn wHu"/; U;;
=BHl1wV;); V;; S BVnWV1; =Bl1wV;;.
But clearly Bl1wV;; S Bl1wB, so we have equality. Since each element
g EGis in some double coset Bn wB, g may be expressed in the form
g= bflwU, where bE B, U E V;;;.
To show uniqueness, sllppose
bl11W1l1 = b2nw'u2,
where b1, b2EB, lV, W'EW, U1EV,;, 1i2EU;;;,. By 8.2.3 we have w=w',
thus I1w=l1w'. It follows that
b2"lb1 = nwuzli11n;l.
Since 11 wU;;; 11;;/ S V, this element is in B n V, so is 1 by 7.1.2. Thus
bi = b2 and lli = liZ. II
THE BRUHA T DECOMPOSITION
117
COROLLARY 8.4.4. Each elem(,llt of G has a unique expression in the
form g = Ulll11wu, where UI E U, hE 11, w E V, U E U;; .
...........
One consequence of the canonical form we have established is the
following useful result.
PROPOSITION 8.4.5. T'l-vo eleme11ts of H whiclz are conjugate in G are
conjugate i/1 N.
PROOF. Let hl, hzEH satisfy hz=gh1g-1. Let g=uIh11wU as in 8.4.4.
Then
u1hl1wuhl = IzZUIhl1wu.
It follows that
UI . hn,)1111;; 1. llWhl1uhl = h2111h;;1 . hzh. llw' U.
By the uniqueness of expression of elements of G in canonical form we
have
UI = hzu/z;;!,
hnJl111;;; 1 = Izzlz,
\
h11Uh1 =U.
In particular, 112 =n,)11/1;;;1, so hI' h2 are conjugate in N.
.
We conclude this section with a result concerning the group fI defined
in chapter 7.
PROPOSITION 8.4.6. G nfI=H.
PROOF. Let gEG n fl. Then g=bInb2, where bI, bzEB and nEN.
Thus
n=bllgbilEBHB.
Now we have
BHB= UHHHV= vBu= viI.
Thus IlE vfl nN. But vB nN=H by 7.2.4. Hence nEI! and gEB.
Rut now gEB nB and
B nH= VH nli=(V nH) H=H,
si nce U n {I = 1 by 7 . 1. 2.
--...
.
E

-
[.
',,,",
[-.
. :::'01
(
L..
f--
I
to4
f
t
L-r
(
'"
118
THE BRUHA T DECOMPOSITION
SIMPLE GROUPS OF LIE TYPE
119
COROLLARY 8.4.7. Let (; be the group of automorphisms ofJLK generated
by G and H. Then G is a normal subgroup of G and GIG is isornorphic
to HI H.
If J is. th empty set, then P J=B. Now B admits a decomposition into
the semI-dIrect product of U and J-I. This semi-direct decomposition
generalizes in fact to any parabolic subgroup, as we shall now show.
For any subset J of n we define UJ to be the subgroup of G generated
by the root subgroups Xr for which rE (1)+ n(f>J. <1J-1 n(l)J is evidently
a closed set of roots, therefore
UJ= IT Xr,
rE + () <1>J
PROOF. Ii normalizes G by the remark following 7.1. 1. Thus G is a
normal subgroup of {; and G= GH. Also
G/G=GIi/G-;;H/G nli=H/H
by 8.4. 6.
by ChevaHey's commutator formula. (The factors may be taken in any
order.) We also define LJ to be the subgroup of G generated by }f and
the root subgroups Xr for a1/ r E <j)J.
8.5 The Levi Decomposition
We shaH now derive further information about the parabolic subgroups
of the Chevalley group G. We have seen in sections 8.2 and 8.3 that
every parabolic subgroup of G is conjugate t one of the. subgroups
P J=BNJB for some subset J of the set IT of fundamental roots. Further-
more distinct subsets of IT give rise to non-conjugate subgroups P J. Thus
G has 2l conjugacy classes of parabolic subgroups.
We first give an alternative way of describing the subgroup P J. Let
<1>J be the set of roots which are integral combinations of roots in J and
(1)J be the set of roots which do not belong to <I>J.
THEOREM 8.5.2. (i) UJ is a normal subgroup of PJ.
(ii) PJ=UJLJ and UJ nLJ=1.
(iii) P J is the normalizer of U J in G.
. The decomposition of P J into the semi-direct product of UJ and LJ
IS called the Levi decomposition, and LJ and its conjugates in UJ are
caIJed the Levi subgroups of P J.
PROPOSITION 8.5. 1. P J is the subgroup of G generated by H and the
root subgroups Xr for r E <1>+ u <1> J.
PROOF. We show that the subgroups generating P J all normalize UJ.
It is clear that II normalizes UJ. Let r be a positive root. If SE <v+ n(f)J
aJl roots of form ir + js, where i,j are positive integers, are also in (l)+ n <i>J.
Thus the commutator formula shows that Xr normalizes UJ. Now suppose
rE<I)- n<1>J. Then -r is not in <1)+ n<I>J, and if s is any root in (I)t- n<5J,
all roots of form ir + js, where i, j are positive integers, are in (1)+ n(f)J.
For ir + js involves some fundamental root not in J with a positive coeffi-
,::icnt. Hence Xr normalizes UJ in this case also, and so UJ is normal in
P J by 8. 5. 1.
Now 8.5.1 shows that PJ is generated by UJ and LJ. Since UJ is
normal in P J we have P J = UJLJ.
We now consider UJ nLJ. Let OEUJ nLJ and consider the effect
of 0 on the Chevalley basis of LIe. We have
PROOF. P J=BNJB is the subgroup of G generated by B and the
elements nr for rEJ. Since nrE(Xr, X-r) , it is clear that
PJ £ (H, Xr; rE<1>+ u<1>J).
Conversely, it is clear that H £ P J and Xr £ P J if rE <1>+. Thus we consider
the subgroups Xr with rE<l>- n<l>J. By 2.5.1 each root in <I>J is the image
of a root in J under an element of WJ. Thus
O.h8=IrS+X,
r= Wrlwr2 . . . Wr.Js),
where
where r1, . . . , rk, s E J. Hence
XE  1f.r.
rE (,l>+ () i!iJ
X -1 -1 -l_x
nrlnr2 . . . nrk snrk ... n 2 nl"1 - r
since BE UJ. But
by 7.2.1. Therefore Xr£PJ as required.
O.h8EH(f)  Jr.r,
rECl>J
.

120
. SIMPLE GROUPS OF LIE TYPE
since e ELJ. However
3L = :lL) EB L lL,- EB L lL,-
rE(jJJ rECliJ
and so O.hs=hs.
Now consider O. es. Suppose first that S E <D J. Then
O.es=e/J+x,
where
X E L: JL-r,
rE Cii J
since BE UJ. Also
O. es E ]I} EB b 1lr
rEo <!)J
since OELJ. It roHows that B.es=e/J'
Now suppose SE(l)J. Then
B.es=es+x,
where
X E JI} EB b 3L-r,
r-BEIDJ
since OE VJ. But
e.esE1lsEB 1: Jlr,
r-BEc1>J
. e EL ThllS e e - e in this case also. lIence e operates trivially on
SInce - J. . S - s
the Cheva])ey basis, so 0 = 1. .
Finally consider the subgroup ftc(VJ). This subgroup contaInS P J,
so must be one of the subgroups P Jl for som subset II ;2 J. If h -::J J,
choose a root rElI with ref-J. Then rE<l)+(j<l>J and. so X,-sVJ. Also
n,-E$).c(UJ). However l1rXrl1;l = X -r is not contained III UJ and we have
a contradiction. Therefore h=J and $lc(UJ)=PJ. ·
8.6 The Finite Chevalley Groups
We now consider the special case in which the base field K is the finie
fie1el GF(q) with q elements, where q is an arbitary primc power. G IS
then a group of non-singular linear transformatiOns 01 a space over a
THE BRUHAT DECOMPOSmON
121
finite field, so is a finite group. The ChevaJ]ey group of type 1.. over GE( q)
will be denoted by 1L(q). We shaH obtain a formula for the orders of the
groups 1L(q).
We begin with the canonical form for elements of G established in
8.4.4. Each element of G has a unique expression in the form u1hnwu,
where !lIE V, hEH, WEW, !lEV;;;. Thus we have
I G I = I: I Bl1wB I = 1: I VHnwV;; I = I V J .1 H I. 1: I V;; I .
WEW WEW WEW
N ow each element of V is uniquely expressible in the form
n x,-,(tt)
r E c1>+
with ttEGF(q). Thus I VI =qN, where N= I<D+I. Also each element
of U;; is uniquely expressible in the form
n Xr,(ti),
n E c1>+
wert) E c;I>-
and so
\
\
,
I
I
t
I
\
I V;;; I =ql(w).
We now consider the order of fl. By 7.1.1 His the set of aut om or ph isms
heX) of JL/{ where :x is a K-character of P which can be extended to a
K-character of Q. H is a subgroup of ii, the group of automorphisms
hex) as X runs through aU K-characters of P. Since P is isomorphic to Q
(both are free abelian groups of rank I), B is isomorphic to the group of
automorphisms hex) for aU K-characters :x of Q. Thus there is an epi-
morphism H- H obtained by restricting X from Q to P. The kernel consists
of the automorphisms hex), where :x is a K-character of Q which is the
identity on P. The kernel is therefore isomorphic to the group of K-
characters of the factor group Q(P.
Now I if I =(q_I)I, since the image of each generator under a K-
character can be chosen in q -1 ways. Thus
-...,
1
I H I = d (q-l)l,
j
where d is the order of the group of K-characters of Q/P. Now it has
been shown in section 7.1 that the bases PI. .. . , PI; q1, . . . , ql of funda-
! mental roots and fundamental weights for P, Q respectively are related
\
\ by
...
I
pt= 1: Ajiqj.
j=l

I.
Id
i'
i
i,;
r
L.
-f.. -
I
A
r
\d
"'-<-,
122
SIMPLE GROUPS OF LIE TYPE
Thus the factor group Q/P is generated by elements ijI, . . . , ijz subject to
relations
CHAPTER 9
l
I: Ajiqj=O.
j=1
A consideration of the Cartan integers Aij (listed in section 3,.6 for the
different types of simple algebra) shows that Q/P has the followlllg struc-
ture (If denotes a cyclic group of order i):
J(.: Al Bl Cl DZk+1 DZk G2 F4 E6 E7 E8
Q/P: l+l 2 2 4 2 X 2 1 1 3 2 1
Now the number of K-characters of i is the number of ith roots of  in
GF(q), which is (i, q-l). Thus the number d of K-characters of Q/P
has the following values:
J(.: Al BI CI D2k+1 DZk
d: (l+1,q-1) (2,q-l) (2,q-l) (4,q-l) (2,q-l)2
J(.: G2 F4 E6 E7 E8
d: 1 1 (3,q-l) (2,q-l) 1
The two cases for type D may be summarized by writing d=(4, ql-l). .
We now collect together the information obtained above and obtalll
the following formula.
Polynomial Invariants of the Weyl Group
9.1 The Algebra of Polynomial Invariants
We have seen how the Bruhat decomposition enables us to obtain a
formula for the orders of the finite Chevalley groups which involves the
polynomial
2.: flew).
weW
We shall show in the present chapter that this polynomial factorizes into
a product of terms of form
I de - 1
1 +1+/2+ .. . +ldt-1=_,
1-1
where dl, . . . , eft are the degrees of certain basic polynomial invariants
of W. The numbers (fJ,.. ., tll have other interpretations also, as we
shaH see in the following chapter, but we describe them first in terms
of polynomial invariants as this appears to be the approach which
generalizes most readily to the twisted groups to be considered in chapters
13, 14.
Let W be a Weyl group operating on the Euclidean space 11, and let
el, C2, ..., el be an orthonormal basis of 'P. Then each x E'P can be
written in the form
PROPOSITION 8.6. 1. LeI G = J[( q). Then
I G I =! qN (q-1)l L ql(w).
d WE W
Although this is a simple theoretical formul for the order of G, it is
not the best formula in practice as the expressIOn
L qllw)
we TV
is very cumbersome. We shall show in the following chapter how this
expression can be simplified.
X=Xlel + . , . +Xlel, Xt EIR.
Given any polynomial P(Xl, . . . , Xl) in Xl, . . . , Xl, P may be regarded
as a map from 11 to tR. Let  be the algebra of all such polynomial func-
tions on V. f!:p is independent of the basis chosen for 'P, For the homo-
geneous polynomials of degree one in fiv form the dual space 1> of V,
and  is then the symmetric algebra of i>; viz., the algebra of symmet-
ric element in the tensor space
IR 1 EB iJ EB CD 0 iJ) EB (V 0 iJ 0 iJ) EB . . . .
Now the action of W on 1) may be transferred in a natural way to an
action on V by defining w(f), Hi E W, f E:: l> by ,
II {f) (u'(x»= I(x) ,
123
XEl>.

124
SIMPLE GROUPS OF LIE TYPE
The action of W on V may then be extended to an action on  by de-
fining
w(P)(x) =P(w-l(x)),
PE fiv, XElJ.
A polynomial function in  is called an invariant of W if w(P)=P for
all WE W. The invariants form a subring 3f of . Now #:v is the poly-
nomial ring [R,[Xl, X2, . . ., Xl], and we shall show that its subring of
invariants is also a polynomial ring, i.e.
3J = lR[h, /Z, . . . , lz],
where II, 12, . . . , I, are certain elements of .1f.
Exan-Iple 9.1 . I. The Weyl group of type A, is isomorphic to the sym-
metric group S,+1. It is best described operating on the subspace of an
(I + I)-dimcnsional Euclidean space with orthonormal basis eo, el, . . . , e,
whose elements satisfy Xo +Xl + . . . +Xl =0. The elements of Woperate
on 1) by permuting the coordinates xo, Xl, . . . , Xl in all possible ways.
Thus the polynomial invariants are the symmetric polynomials in
xo, Xl, . . . , Xl. These are all polynomials in the elementary symmetric
polynomials. Since XO+Xl+ ... +XI=O we have 3f=[R,[II, 12,..., lz],
where It is the elementary symmetric polynomial of degree i + 1.
Example 9.1.2. The Weyl group of type B, operates on a Euclidean
space with orthonormal basis el, . . . , e, by prmuting .the coor.dina.tes
Xl, . . . , Xl in all possible ways and by changmg the sIgns arbitranly.
Thus the polynomial invariants are the symmetric polynomials i.n
xi, x, . . . , x;' These are all polynomials in the elementary symronetnc
polynomials in xi, . . . , xf. Thus ] = n[h, /z,. . ., Iz], where It IS the
ith elementary symmetric polynomial in xi, . . . , xf.
In general, a polynomial of form Ax11X2... xm, A E [R" is caled a
monomial of degree kl +k2 + . . . +km. The degree of an arbitrary
polynomial P is the greatest degree of any monomial const!tuen and will
be denoted by deg P. A polynomial is called homogeneous If all ItS mono-
mial constituents have the same degree.
For each PE$>, the average of P under W is defined by
1
Av P= ---- L: w(P).
I WIWEJ-V
LEMMA 9.1.3. If PE  then Av PE].
POLYNOMIAL INVARIANTS OF THE WEYL GROUP
J25
PROOF. Let WE W. Then we have
-"\,
w(Av P)= -, 1 I 1: JVw'(P)= -II 1: w'(P)=Av P.
W eW W eW
.
9.2 A Theorem of Chevalley
The fact that .1f is a polynomial ring was originally proved by Chevalley
[5]. In order to prove Chevalley's theorem we first need a preliminary
lemma. Let + be the set of polynomials in  with constant term 0,
and let 3f-I-=] n+. Let ]-I-- be the ideal of £-> generated by 3J+. The
elements of ]+ therefore have form Plh + . . . +PkJk, where Pt E,
J, E] 1- for each i.
LEMMA 9.2.1. Suppose Jl, J2,. . ., Jk are elements of .1f such that
Jl is /lot in the ideal of 3f generated by lz, . . . , Jk. Let PI, P2, . . . , Pk be
homogeneous polynomials in $ such that Pl1r + . . . +PkJk=O. Then
PI E #:v.1f+.
..-.....
PROOF. We show first that Jl is not in the ideal of $ generated by
J2, . . . , Jk. Suppose this were false. Then
1r = Q2J2 + . . . + QkJk,
Now for each WE W we have
Qt E #:v.
.
W( QiJi) = w( Qd w( Jt) = w( Qt)Jt.
Thus Av (QtJt)=(Av Qt)Jt. It follows that
1r=AVJl=(Av Q2)J2+... +(Av Qk)Jk.
Since A v Qt E 3f, this means that 1r is in the ideal of 3J generated by
J2, . . . , Jk, a contradiction.
We prove that PI E#:v]+ by induction on deg Pl. If deg PI =0, PI is-
constant. Since Jl is not in the ideal of fb generated by J2, . . . ,Jk, we
mllst have PI =0. Thus PI E]+ in this case.
We now assume deg PI > O. The root system <D is a finite su bset of Y
and, for each rE <D, the hyperplane orthogonal to r is given by an equation
Hr = 0, where Hr is a homogeneous polynomial of degree 1. Consider
the polynomial H'r(Pt)- Pt. This polynomial vanishes at all points for
which [-Ir is zero. Since Hr is an irreducible polynomial, .fIr divides
..:.:ro,..

"-'
r
f---<O
LJ
[
[- "
.;;;
";-
l,.O'
r'
i
t,.-..,
[
f .
L_.,
L.
t
,
i
I
t...
r
L
126
SIMPLE GROUPS OF LIE TYPE
POLYNOM[AL INVARIANTS OF THE WEYL GROUP
127
Wr(Pt) - Pt, thus
PROOF. Suppose P(h,. . ., In) =0 wjth P=I= O. We may assume, by
comparing terms of a given degree, that all monomials in h,..., In
which Occur in P have the same degree d in Xl, . . . ,Xl. Let PI. = iJP/iJft.
Then Pt(h, . . . ,In), i = I, . . . , n, arc elements of 31 and not alI the Pi
are zero. Let 1B. be the ideal of 3S generated by PI, P2, . . . , Pn. We may
choose the notation so that PI, . . . , Pm but no proper subset generate
1{ as an ideal of 31. Then there exist polynomials Qt, 1 E 3f such that
Wr(Pt)-Pt = Hr. Pt,
Pi E i:>.
Now Pi is homogeneous, hence wr(Pt) is homogeneous of the same degree.
Thus Wr(Pt)-Pi is also homogeneous, and it follows that PI. is homo-
geneous. Moreover deg Pi < deg Pi. Now we have
PIJI + . . . +PkJk=O.
Thus
Wr(PI) 1r + . . . +Wr(Pk) Jk=O
m
PI. =  Qi, jPj,
j=l
i>m.
and so
Now each PI. is homogeneous in Xl, . . . ,Xl of degree d-deg ft. Thus,
by comparing terms of the same degree in Xl, . . . , Xl on both sides, we
may assume each Qt. j is homogeneous of degree deg Pi - deg Pj. Tlms
deg Qt.j=deg Ij-deg ft.
Now P(fr, . .. , In) =0, thus OPjOXk=O for k= 1,. . . I. Hence
 o _! =0,
i=10Il OXk
 PI. _?!! =0.
i=l OXk
Hr(PIJI + . . . +PkJk)=O.
Since Hr is not identically zero we have
PIJI + . . . +PkJk=O.
But deg PI < deg PI and so PI E3J+ by induction. Thus
Wr(PI)-PIE3J+.
Now w(3J+)=3J+ for each WEW, hence w(3J+)=i:>3I+. Thus each
WE W operates naturally on the quotient ring {f:pj3J+. We have seen that
Wr operates trivially on PI in this quotient ring for each rE<D. Since the
Wr generate W, each WE W operates trivially on PI in the quotient ring.
Thus
W(PI)-PI E 3J+.
It follows that Av PI-PI E3J+. But PI is homogeneous with deg PI >0,
hence Av PI E3J+. In particular Av PI E3J+ and so PI E3J+ as re-
quired. .
It follows that
m oj n (m oJ )
L: Pi ._ +  L: Qi, jPj .--.!- =0,
i=l OXk i=1/I-j-l j=l OXk
m ( Oli n oJ )
L: PI. --+ L: Qj. f _t =0.
i=I OXk j=7II+1 OXk
We now apply 9.2.1. PI,. ", PrJ are in 3J and PI is not in the ideal
of Jf generated by Pz, . . . , Pm. Each of the polynomials
oIl. n oIj
- + L: Qj. 1. .-- --.,
OXk j=m-I-l OXk
Now the ideal ]+ of fb is generated by the homogeneous elements
of.] of positive degree. By Hilbert's basis theorem there is a finite subset
of this generating set which generates 3J+. Thus there is a set It, h, . . . ,I1J
of homogeneous polynomials in 3J such that /r, . . ., In generates ]+
but no proper subset has this property.
i = 1, . . . , n1,
is homogeneous m Xl, . . . , Xl of degree deg It -]. Thus by 9.2.1 we
have
THEOREM 9.2.2. There is no polynomial P#-O such that
P(lt,... , In) =0.
Thus /r, . . . ,In are algebraically independent.
oIr tl oJ
. --- + 2: Qj. 1 __.J E3I+.
aXk ;-m+1 OXk
We now multiply this polynomial by Xk and sum over k. For a homo-
geneous polynomial!J in Xl, . . . , Xl we have, by Euler's formuJa,
, a Ij 1
L... Xk -- = deg Ij. j.
1=1 aXle

128
SIMPLE GROUPS OF LIE TYPE
POLYNOMIAL INVARIANTS OF THE WEYL GROUP
129
Therefore
PROOF. In the above notation we must show that n = I. Let
n n
drg h. h + L:; deg Ij. Qj, IIf = 2: ltRi,
j=111-/-1 i=l
K= fR(XI, . . . , Xl)
be the field of rational functions in Xl,. . . ,Xl over fR. Similarly let
k = fR(h, . . . , In) be the field of rational functions in II, . . . , In over fR.
Then we have
where each Rt E +. All the terms on the left-hand side are homogeneous
of degree deg h. Comparing terms of this degree on both sides we obtain
n
deg h.II + 2: deg Ij. Qj, dj= 2: ItSt,
j=m-/-l i
fR eke K.
where the sum on the right extends over some subset of 1, . . . , n not
including i = 1 (since the monomials in hRI have too large a degree).
It follows now that h is in the ideal of fb generated by h" . . . , In and
we have a contradiction. This completes the proof. II
Since Xl, . . . , Xl are algebraically independent over fR, the transcendence
degree of Kover fR is given by
tf. deg. K/rR = I.
Also by 9.2.2 we have
THEOREM 9.2.3. Every element of 3J is a polynomial in II, . . . , In
tf. deg. k/rR. = 11.
PROOF. It is sufficient to prove this for homogeneous polynomials
in 3J. Let J E3J be homogeneous. We use induction on deg J, the result
being clear if deg J =0. Suppose deg J> O. Then J E3J+ and in particular
J E 3J+. Thus we have
Since
J=Plh + . . . +P1Jn
tf. deg. K/fR = tf. deg. k/fR + tf. deg. K/ k
by Galois theory (see, for example, Jacobson [2]), we consider tf. deg. K/k.
Now K is generated over k by Xl, . . . , Xl. However, each Xt is an algebraic
element over k. For the polynomial
.......
for certain PI, . . . , Pn Efv. Since J, II, . . . ,In are all homogeneous we
may assume each Pi is homogeneous also, with deg Pi = deg J - deg It.
Then
n (t - H{Xt))
wEIY
J=Av J=Av Pl. II + . .. +Av Pn.In.
has Xi as a root, and its coefficients are the elementary symmetric poly-
nomials in W(Xi) for all WE W. These coefficients are invariant under
each element of W so are in 3J and therefore in k. Thus K is generated
over k by a finite number of algebraic elements over k, and so
tr. deg. K/k=O.
Av PI, . . . , Av P1J are homogeneous polynomials in 3J of degree less
than deg J. Thus they are polynomials in II, . . . ,In by induction, and
so J is also. II
COROLLARY 9.2.4. 3J=[R[h"..., In] is isomorphic to the polynomial
ring inn generators over IR.
It follows that 11 = I.
.
PROOF. This follows from 9.2.2 and 9.2.3.
.
9.3 The Degrees of the Basic Invariants
II, . . . , In is called a set of basic polynomial invariants of V. We now
determine the number of invariants in a basic set.
Now the set II, . . . , /z of basic polynomial invariants of W is not uniquely
determined. It is not difficult to show, however, that the degrees of the
polynomials in a basic set are uniquely determined.
THEOREM 9.2.5. The number of invariants in a basic set is equal to
the rank of the JYeyl group.
..-...
PROPOSITION 9.3. I. Let II" . . , Ii; I,..., 1; be two sets of basic

",,-
[
[
["'
--
, ,
["

i
t___
i "
L.
[4
r-
i

\_,,
r--
C"
--
L.
L
130
POLYNOMIAL INVARIANTS OF THE WEYL GROUP
131
SIMPLE GROUPS OF LIE TYPE
polynomial invariants. Then we may arrange the numbering so that
deg Ii = deg (for i = I, . . . , I.
Then T2=T. Let Mo be the set of XEAf with T(x)=O and J\-h be the set
of XEM with T(x)=x. Then AI = lvlo EB Ml since T is idempotent. The
trace of T is the dimension of lvlI. Also the J--V-invariants of lvl are just
the elements which lie in A-h. For if x is 1--V-invariant,
PROOF. Each of l, . . . , 1; is expressible as a polynomial in I l' . . . , I l
and conversely. Consider the matrices
1
T(x)=---- :2: x=x,
I W I WE JV
whereas conversely if T(x) = x we have
u-{x) = wT(x) = T(x) = x
for aJI IV E f-V. This completes the proof.
II
(aI)
aJ. .
J
( aJ.)
af; ,
These are inverse matrices since
l aJ. al; aJ. ( 1 if i = j,
1;1 all aIj = aj; = 0 if i =F j.
Thus the determinant I aldal; I is non-zero. It follows that for some
permutation p of I, 2, . . . , I,
We now return to the natural representation of Jl1 on V. Let HI E 1'V
be an element with eigenvalues 1'1, . . . , Ai on V. The At are in the complex
field. Let Vc be the complexification or V. Then IV may be represented
by the diagonal matrix diag (AI, . . . , Ai) with respect to a suitable basis
of VI[;' Thus, for each t EC,
det (1- tlV) = (1- Alt)(I - A2t) . . . (I - Alt).
IT =FO.
i=1 aI p(i)
By renumbering I,..., I; we may assume p is the identity. Thus
aldal=FO for each i. This means that it, as a polynomial in J, . . . , J;,
involves I and so deg Ii  deg J;. In particular
l l
:2: deg Ii  :2: deg J.
i-I i=1
If we consider t now as an indeterminate, l/det (1 - tl-l') may be expressed
as a power series in t. In fact
1 ( \ \2 2 ( \ \2 2 .)
-----= 1 +/\}t+/\}t +...) 1 +/\2t+/\2t + . . .
det (I - tw)
. . . (1 + Alt + A;t2 + . . . )
By symmetry we must have equality. Thus deg Ii=deg I for each i. .
We shall denote the degrees of the basic invariants by dl, dz, . . . , {ht
and shall derive some properties of this set of integers.
 ( " A]A2 . . . 1..;1) tn.
nO k,+...-I-kln
LEMMA 9.3.2. Let M be any finite-dimensional W-module over a field
of characteristic O. Then the dimension of the subs-pace formed by the
W-invariants of M (i.e. the elements XEM such that Iv(x)=xfor all WE J.-V)
is equal to the trace of the linear transformation
We write A v l/det (1- tw) to denote the power series
1 1
rr-vl wv (ret (T-tw)
in f. The ncxt resuJt shows that this power series can be expressed simply
in terms of the degrees dl, . . . , dz.
1
-}: w
I W Iwe W
PROPOSITION 9.3.3.
of M.
PROOF. Let
Av 1 - IT 
det (1- tw) i=} (1- tal)
1
T=- }: w.
I Wlwew
as power series in t.

132
SIMPLE GROUPS OF LIE TYPE
POL YNOMIAL INVARIANTS OF THE WEYL GROUP
133
PROOF. We have earlier defined the action of W on the dual space j
of 1) and on the ring i;J of polynomial functions on V. In the present
proof we take V, V,  over the complex field instead of the real field
as before.
,\, . . . , l are the eigenvalues of w on 19, thus XlI, . . . 'l-l are the
eigenvalues of w on fJ. However these eigenvalues are roots of unity
and w is a real transformation, hence the eigenvalues of HI on f} are
1, . . . , l. Let Y1, . . . , Yl be corresponding eigenvectors spanning V.
Yl, . . . , Yl are linear combinations (possibly complex) of the original
basis Xl, . . . , Xl of 'iJ. The polynomial functions yily2 . . . yl for all
sets of non-negative integers k1,.. . ,kl with k1 + . . . +kl = n form a
basis for {bn, the space of polynomial functions which are homogeneous
of degree n. Since W(Yt) = tYt we have
H'(yl . . . y;l) = Al . . . flyl . . . yfl.
Thus the eigenvalues of w on {bn are the complex numbers tl... ;l.
Hence the cocfllcient of tn in the power series lldet (1- tl1') is the sum
of the eigenvalues of w on {bn, i.e. the trace of Hi on fiz>n. It follows that
the coefficient of t n in the power series A v lldet (1- tw) is the trace of
the linear transformation
-,
THEOREM 9.3.4. (i) d1dz. . . dl= I WI,
(i i) (h + d2 + . . . + dl = N + I.
1
--  W
VIWEJ17
on fiz>n. By 9.3.2 this is the dimension of the space ivn n]' = ifn. Now
ifn has as a basis the set of all polynomials 1{1/:J.2. . . 1ft of degree n,
where h, . . . , It are a set of basic invariants. Since deg it = di the number
of sllch polynomials is the number of solutions of the equation
dle1 +dzez + . . . +dlel=n,
which is in turn the coefficient of tn in the power series
PROOF. Consider det (1-tw). This is (1-t)1 if w= 1, (1-t)l-l(1 +t)
if HI is a reflection, and a polynomial not divisible by (1- t)I-1 otherwise.
Now the only elements of W which are reflections are the elements
Wr with rE <D. For suppose W contains a reflection Wll in a hyperplane
II not orthogonal to any root. H contains a vector v not orthogonal to
any root, and wl:l(v)=v. v lies in a chamber C (see section 2.3), and
since w}J(C) is also a chamber containing v we have W/l(C)=C. Hence
H'll = 1 by 2.3.2 and we have a contradiction. Therefore W contains
exactly N reflections, where N = I $+ I .
We now apply 9.3.3. :M ultiplying both sides by (1- t)1 we have
IT 1 - 1 (1 + N(l-t) +(1- t)2 F(t))
i=1 1 +t+ . . . Jttdj-1 IWI (1 +t) ,
where F(t) is some rational function whose denominator is not divisible
by 1- t. (The first term on the right comes from the unit element, the
second from the N reilections, and the third from the remaining elements
of JY.)
Putting t = 1 we obtain
-.--
1 1
d-;J;l - fWl '
hence I WI =dldz. . . dl.
To obtain the second result we first differentiate and then put t = 1.
We have
/1 1 ( _1+2t+ ... +(dt-l)td'-)
i=1 1 +t+ . . . +td,-1 i=1 1 +t+ . . . +td,-l
(1 + tdl + tZdl + . . . )(1 + td2 + t2dz + . . . ) . . . (I + tdl + tZd, + . . . ).
This is the coefficient of t n in
N 1
= ----- .--+G(t)
I W /1+1 '
and so the result is proved.
.
where G(t) is a rational function whose numerator is divisible by 1- t.
Putting t = 1 we obtain
1 1 1 N
-. .  (dt-l)=--.
2 d1d2 . . . dl i=l 2 I WI
I 1
n
;=1 (1- tdt)
By applying this result we can determine the sum and the product
of the degrees dl, . . . , dl.
It follows that

dl +d2+ . .. +dl=N+l.
.

134
SIMPLE GROUPS OF LIE TYPE
POLYNOMIAL INVARIANTS OP nIP. W£YL GROUP
135
... "'-
Let h, . . . , lz be a set of basic polynomial invariants and
divides J. But
o(h, . . . , lz)/o(xl, . . . , Xl)
deg J=N=deg ( I1 Hr).
re+
be the Jacobian of the map
Hence
[
(Xl, . . . , xI)(h(XI, . . . , Xl), . . . , h(XI, . . . , Xl)).
The Jacobian is the matrix whose (i, j)-coefficient is oIl/oxj. Let
J= 1_I)__1
0(X1, . . . , Xl)
be the determinant of this matrix. J is a homogeneous polynomial in
Xl, . . . , Xl of degree
J=/I.. n Hr
r e (1)+
for some /I. E fR.
.
f
,
9.4 A Theorem of Solomon
I
 (dt-I).
i=l
We now turn to the proof of the identity
 tl(w)= III (tdt_2).
wEW i=1 t-1
This factorization of the polynomial tl(w) was first proved by Bott
[I], with a suitable interpretation of the integers £11, . . . , £It, by considera-
tions involving the topology of Lie groups. We give here a proof due to
Solomon [4] in a slightly l110diIJed form due to Steinberg [16], where
£11, . . . , £ll are interpreted as the degrees of the basic polynomial invariants
of v.
We define
.,,-
i
to
By 9 .3.4Jhas degree N. We now show that J has a factorization into linear
factors.
 ,-
THEOREM 9.3.5. For each root rE <1)+ let Hr=O be the equation of the
hyperplane orthogonal to r. Then
..,-'V""'...
s
.
,
,
., \. ..-'
J=>... n Hr
,E+
I
!
i
t ".
(or some >.. E IR.
PROOF. Consider the map T of t) into itself given by
T(XI, . . . ,xl)=(h(XI, . . . , Xl), . . . ,h(XI, . . . ,Xl)).
For each point X=(XI, . . . ,Xl) of t) at which J=I=O there exist open
neighbourhoods of X, T(x) in 1-1 correspondence under T. (See, for
example Loomis and Sternberg [1 ].) Now suppose X lies in a reflecting
hyperplane Hr, rE <1>. Then any open neighbourhood of X contains points
a, b such that a=l=b but wr(a)=b. Then we have
It(b)=h(wr(a))= Wr(lj(a))=h(a)
for i= 1, . . . , 1. Thus T(b)=T(a) and so J=O at x. Hence the polynomial
J vanishes at each point at which the linear polynomial Hr vanishes.
Since Hr is an irreducible polynomial, Hr divides J. This is true for all
r E <1>+, thus
Pw(t)= 2: tl(w),
wE TV
.,.........
n Hr
re <11+
1 tdt - 1
Pw(t) = 11 -,
i=l t-l
and Pwit), PwAt) denote the corresponding polynomials for the Weyl
groups WJ, where J is any subset of fI. The idea of Solomon's proof is
to show that Pw(t), Pw(t) satisfy the following identities:
 (-1)1 JI Pw(tl=tN
J PWJ(t)'
 (-I)IJI .!!=tN.
J pWJ(t)
Assuming by induction that PWJ(t)=Pwit) whenever J is a proper
subset of fT, it follows from these identities that Pw(t)=Pw(t).
We concentrate first on the polynomial Pw(t).
,
 -.'
L_

136
SIMPLE GROUPS OF Lffi TYPE
POLYNOMIAL INVARIANTS OF THE WEYL GROUP
137
LEMMA 9.4.1. Lef IV be an element of WJ. Then few) is the same whether
IV is regarded as an elernent of the JYey! group W or the Weyl group WJ.
summed over the elements xEWfor which XWX-1EWJ. Now XWX-1EWJ
if and only if xwx-1(CJ)=CJ by 2.6.1, and this holds if and only if
w fixes x-1(CJ)' Now there are I1J(I-V) such elements x-1(CJ) fixed by
wand for each one of them the element x may be chosen in I WJ [ ways.
Thus
PROOF. Suppose r is a positive root not in <DJ. Then r is a positive
combination of roots in 11 involving some root in n - J. If ri E J, wr.(r)
still involves this root in 11 - J with a positive coefficient. Thus H'r/r)
is still a positive root not in (!)J. Repeating this argument we see that
w(r) is a positive root not in (DJ for all WE WJ. Thus aU positive roos
transformed by w into negative roots are in (PJ. The result follows by
2.2.2. .
I
q (w) =---,.1 WJ I 11J(W) = 11J(W).
J I WJ
.
PROPOSITION 9.4.3.
Lemma 9.4.1 shows that we may write l(w) for WE WJ without
ambiguity.
We now consider the operation of rV on the Coxeter complex, as
described in section 2.6. By 2.6.3 each element of the Coxeter complex
can be transformed under W into just one element of the form
 (-I)1J1 nJ(w)=det w.
J
(v, 1')=0 for rEJ )
CJ= V. .
, (v,r»O for rEl1-J
Consider the orbit of the Coxeter complex under W containing CJ.
For }II E W we define I1J(W) to be the number of elements in this orbit
which are fixed by IV. By 2.6.1 I1J(W) is eq ual to the number of left cosets
x W J fixed by w under left multiplication.
Now I1J( w) has a useful interpretation in terms of the theory of charac-
ters. If G is any finite group and H a subgroup of G we define the opera-
tions of restriction and induction of class functions in the lIsual way.
If X is a class function on G we denote by XII the restriction of X to H,
and if 1> is a class function on H we denote by 1>c the induced class function
on G. (pc is defined by
To prove this we again consider the operation of w on the Coxeter
complex. However, we first require a preliminary lemma. In this we
consider not the Coxeter complex defined in terms of the reflecting hyper-
planes, but a complex ::f defined similarly by any finite set of hyperplanes
of t}. The dimension of an element 1k.E::f is defined as the dimension of
the smallest subspace of 'P containing it.
"-h."
LEMMA 9.4.4. Let n( be the number of elements of .Yt of dimension i.
Then
 (-1)111t=( _l)dlm 11.
i
. 1
1>C(g)=--  4>(xgx-1)
I HI XEG
summed over those elements XE G for which xgx-1 EH.
PROOF. We use induction on the number of hyperplanes. Suppose the
result proved for a system of 11 hyperplanes and that an additional hyper-
plane II is then added. Each element of ::f of dimension i which is cut in
two by H has corresponding to it an element of dimension i-I in H
separating the two parts. Thus the sum
 ( - l)f 11,
i
remains unchanged.
.
- 1
lr.;/w) = , WJ I  1
PROOF of 9.4.3. Let U be the subspace of 'P of elements fixed by
w. The clements of the Coxeter complex which are fixed by IV are just
the ones which lie in U, by 2.6.2. We now apply 9.4.4 to U. 11( is the
number of elements of the Coxeter complex which have dimension i
and lie in U. Since dim CJ=/-I J I we have
n,=  nJ(w).
IJI =l-i

LEMMA 9.4.2. Let lWJ be the ullit character of WJ. Thus lw/w)=1
for ail WE WJ. Then q:J(w)=I1J(w) for all IV E W.
PROOF.


'-
138
SIMPLE GROUPS OF LIE TYPE
POLYNOMIAL INVARIANfS OF THE WEYL GROUP
139
Thus, by 9.4.4,
L (-1)1 nl=( - 1)' L (- 1)/JI nJ(w)=( _l)dfm U.
i J
We now turn to consider the polynomial
L (- I)IJI nJ(w)=det w.
J
.
Pw(t) = }!r ( - 1_).
i=1 t - 1
In order to show that FIV(t) satisfies the required identity it is necessary
to look more closely at the operation of Won the ring fi;J of polynomial
functions on lJ. As before, ] denotes the set of invariant polynomials
uncler 111. A polynomial P E  is said to be alternating if w(P) = det IV. P
for all 1\1 E W, and the set of alternating polynomials is denoted by Jr.
As before, Hr denotes the linear form representing the hyperplane ortho-
gonal to the root r, defined by H,.(x) = (r, x).
f'

tJ
Now w, being an orthogonal transformation of 11, has eigenvalues which
are I, -lor pairs of complex conjugates of modulus 1. Thus
det w=( _l)'-dlm U.
"
t
It follows that
l
THEOREM 9.4.5.
L: (-l)IJI Pw(:l. = tN.
J Pwit)
LEMMA 9.4.6. A polynomial P E fip is al/ernating (! and only if it is
the product of an invariant polynomialll'ith
t- -,
f
L,
PROOF. Let DJ be the set of distinguished coset representatives of
WJ in W defined in 2.5.8. Define
PDit)= L: t'(w).
WEDJ
n Hr.
r E \1,+
PROOF. We show first that
"-
l_-.,
Then, by 2.5.8, we have
n H,
r E 'J)+
'd_
J
i
I.. ,"
PW(t) ><=PWJ(t) .PDAt).
is alternating. If s is any root we have IVs(lf,)= Hw.(r). For
Therefore
Ws( l-l,{X)) = llr( ws(x)) = (r, wix))
= (ws(r), x)=Hw.(r)(x).
Let rt E IT be any fundamental root. Then
;
}
t__
 (-l)/JI Pw(t) =  (-l)IJIPnit)
J Pwit) J
= 7 (- 1)\J1 (  tZ(W»
w{J) S +
=  (  (-I)IJI) t'(w).
weW J
w(J) s +
Let J w be the set of roots r E n such that w(r) E q>+. Then the coefficient
of tl(w) in the above sum is
wri( n Hr) = - (IT Hr),
rE4D+ rE([)+
f
:
i
't ___
(
I
\ ---
since wri transforms rt into -rt and permutes the remaining positive roots
amongst themselves. Hence
11'(. n Hr) =det w. IT Hr
rE (D-f- re (D-f-
and so
L (-1)IJI=(I-1)IJwl.
JsJw
n Hr
r e (D+
L
This is 0 unless Jw is the empty set, when it is 1. However if Jw is empty
w transforms every positive root into a negative root, so w = Wo by 2.2.6.
Hence the above sum is tl(wo) = t N and the result is proved. .
is alternating. Furthermore if Q E ]' we have
W( n Hr. Q) =det w. IT Hr. Q
r E (D-f- rE (D-f-

140
SIMPLE GROUPS OF LIE TYPE
POLYNOMIAL INVARIANTS OF THE WEYL GROUP
141
-.,
and so
n Hr.Q
r E Qj+
PROOF. Let X(w)=tr bnw be the trace of w on n. Let XWJ be the
rcstnctlOn of X to WJ and xJ be the induced character of W. Then
X :: J = X. 1  J'
For
is also alternating.
Now let P be any alternating polynomial. Then Wr(P) = -P for each
rE (1). Let x be an element in the hyperplane orthogonal to r. Then
1
x1f. J(w) = rw7j
1
 xwixWX-l)=_, W I
XEW J
xwx-1e WJ
 X(w)
xe W
xwx-'E WJ
-P(x)= wr(P(x))=P(Wr(x))=P(x).
Hence P(x)=O. Thus P vanishes at all points for which Iir vanishes,
therefore Hr divides P. In fact
1
=x(w)'1 WJ I
 1 =x(w).Ir.;J(w),
xE W
a:wx-' E W J
IT lIr
r e <1>+
Now by 9.4.2 liw)=nJ(w) and by 9.4.3 we have
 (-1)IJ1nJ(w)=det w.
J
divides P. Thus
P = IT Hr. Q
r e <1>+
for some Q E. But now
det }-I?P=H-{P)=W( IT Hr).H'(Q)=det w. IT Hr.w(Q)
rE<1>+ re<1>+
Therefore w(Q)= Q for all WE Wand so QE 3L Thus every alternating
element is the product of
It follows that
with an invariant element.
.
 (-l)IJI xr.;iw)=det w. X(w).
J
We now average over Wand obtain
(-I)IJI xn;J(  W)=_11 I  detw.x(w).
J I wi WEW W wE W
The left-hand side can be simplified since
x1tC 12iv w)XWJC J I w,f"J w).
..-..,
fI Hr
r E <1>+
Let fipn be the set of homogeneous polynomial functions on lJ of
degree 11, let 3Jn=3J n1ivn and ]n=3J nn. Let 3JJ be the set of poly-
nomial functions in  invariant under VJ and (3JJh = 3rJ n n. ]n,
(3JJ)-n and jr n are all finite dimensional vector spaces over fR. Also we
have
For
X:J(I  Iwww)=IJ IT w Xwixwr1)
XWZ-1E W J
dim ]n=dim ]n-N
for all n  N by 9.4.6, since
1
=-- :b XwJ(y),
I WJIIIEIVJ
IT Hr
, E <1>+
Thus we obtain
:b (_I)IJI XWJ(-  W) =-1 1 I :b det w.x(w).
J IWJ\wewJ W weW
\Ve now apply 9.3.2.
( 1 )
Xw --- :b w
J I WJ I we WJ

is homogeneous of degree N. If n < N then dim jrn=O.
PROPOSITION 9.4. 7.
2: (-I)IJI dim (]J)n=dim 1!n.
J

142
POLYNOMIAL INVARIANTS OF THE WEYL GROUP
143
SIMPLE GROUPS OF LIE TYPE
1
-, _I  det w.xCw)
W WEW
where d, . . ., dl.11 are the degrees of the basic invariants of W.1 on
1).1, the subs pace of 1) spanneu by roots in J. Since 11/., operates as the
identity on the subspace of 1) orthogonal to 1).1, the degrees of lhe basic
polynomial invariants of WJ on l> are d, . . . , d,'J I' 1, . . . , 1. Thus the
cocllicient of tn in
'-'
is the dimension of the subspace of invariants of n under WJ, viz.,
dim (3h )7&' Also
[
is the dimension of the subspace of invariants of n under the W-action
w
p  det w.w(P)
However det w. w(P)=P if and only if v(P)=de[ w.P; thus we require
the dimension of the space of alternating elements of T" viz., dim ]11'
Hence
( 1.11 1) 1 1
i1 (I=- tdi) . [r=-il-1J1) = (T - t)IPwJ [t)
is dim (3JJ)n, also as in 9.3.3. However
2: (-I)IJI dim (].I)n=dim ]n
.1
[
2: (-I)IJI dim (3b)n=dim in.
.1
.
by 9.4.7. It follows that the rational funCtions
[
We are now able to show that the polynomial Pw(t) satisfies the analogue
of9.4.5.
-, (-1)1.11
 _______ m._ _ _
J (l_-t)l Pwit)'
tN
(i-=-t)T F wet)
PROPOSITION 9.4.8.
have the same power series expansion, and so are equal.
.
"1-'
L
L (-1)1.11 !wQ2 =tN.
.1 Pwit)
We now complete the proof that Pw(t) =Pw(t).
THEOREM 9.4.9.
L.
PROOF. We prove this result by comparing the coefficients of tn in
the power series expansion of the rational function!)
(-1)1.11 tN
L( - and ---
.1 1- t)'Pwit) (1- t)lPW(t) .
I ( t dt - 1)
2: . tHw) = IT - -.. ---- .
WEW i=1 t-I
f'
L"
Now
PROOF. We use induction on 1= rank W. We may assume inductively
that Pwit)=Pwit) for aJl proper subsets J of n. By 9.4.5 and 9.4.8
we have
r '
t
1 I 1
(1- t)' Pw{t) - EI (1- tdj)
and the coefficient of tn in the expansion of this function is dim 3f,J as
in 9.3.3. Thus the coefficient of tn in
t N -( -I)ITII (-1)1.11
--- = L ----
Pw(t) Jell Pwit)'
tN-(-I)ITII (-1)1J1
----- =  ----
Fw(t) Jell Fwit)'
{U'
}
L_
tN
(1- t)-TPw(t)
is dim 3Jn-N, and this is equal to dim :if n by 9.4.6.
We now consider the operation of WJ on lJ. We have
IJI (tdl-I)
Pwit)= n - ,
il t- 1
Since the right-hand sides are equal by induction it foHows that
Pw(t) =Pw(t).
II
J
. l.."
We have now been able to derive a multiplicative formula for the
orders of the finite Chevalley groups.
;.....----
I -
I
L

144
SIMPLE GROUPS OF LIE TYPE
-,
THEOREM 9.4. 10. Let G= JL( q) be a finite Chevalley group. Then
I
I G I = d q N (q d 1 - 1)( q d. - 1) . . . (q d, - 1),
CHAPTER 10
where d is defined as in section 8.6, N is the number of positive roots of
1[, alld dl, . . . , {ft are the degrees of lire basic polynomial invariants of W.
The Exponents of the Weyl Group
PROOF. This follows from 8.6.1 and 9.4.9.
.
In the present chapter we shall show that the set of integers d1, . . . , dl
occurring in the multiplicative formula for the order of the finite Chevalley
groups can be obtained in three essentially difrerent ways. They have
been defined as the degrees of the basic polynomial invariants of the
Weyl group, but we shall show that they can also be defined in terms
of the eigenvalues of the Coxeter elements of the Weyl group and also
in terms of the partition of the positive roots into roots of a given height.
This latter definition gives a particularly simple way of calculating the
i ntcgers in the individ ual cases.
The present chapter may be regarded as a digression from the main
theme of this book, and the reader primarily interested in the properties
of the Chevalley groups and twisted groups may prefer to omit it at a
first reading, referring to 10.2.4 and 10.2.5 for a knowledge of the
numbers (ft, . . . ,dl in the individual groups. The equivalent definitions
of these integers are nevertheless of considerable interest, and it is probable
that their significance is not yet fully understood.
.-..........
10.1 A Theorem of WeyI
\Vc prove first a well known factorization theorem of Wey1. In order to
do this we derive some properties of the vector
S=l  r.
rEI!>+
LEMfA 1O.J.1. If pol E IT then WPt(S) = S - Pt.
PROOF. This is evident from the fact that wPt transforms Pi into -Pi
and permutes the other positive roots. .
.-.."
LEMMA 10. I .2. S lies in the fundamental chamber.
145

146
SIMPLE GROUPS OF LIE TYPE
THE EXPONENTS OF THE WEYL GROUP
147
--
PROOF. It follows from 10. 1 . 1 that
PROOF. Let
j
2(Pt, S2 = 1
(Pt, Pt)
for a11 pj En. Thus (Pt, S) > 0 for all Pt.
x= L: r
rEO
and Pi En. Then
II
( s - x (p;)t5) = 1 - j > o.
LEMMA 10.1.3. Let ql,..., ql be the fundamental weights. Then
S = ql + . .. +ql.
[:
PROOF. S is certainly a linear combination
l
2: Ajqj
i-l
Since 2(x Pi)/(Pt, Pi) is an integer, this means that 2(x, Pt)/(Pt, Pt):(O.
Therefore (x, Pi):(O for all Pi Err. However, x is a sum of positive roots
and so
I
X = L: A-tP't,
i=l
of ql, . . . , ql. However
with each Aj  O. Hence we have
f--
l
( 2pj ) {I if i = j,
(Pt, Pi)' qj = 0 if i =1= j,
I
(x, x) = L: At(X, Pi):( O.
i=l
as in section 7. 1. Thus
It follows that (x, x)=O, whence x=O and Q is empty.
.
[...
( S) =Aj
(pi, Pt)'
and therefore Aj= 1 for i= I, . . . , I.
.
LEMMA 10.1.6. Let n be a subset of <1>+. Then
S- L: r
reO
. "'-"
L
LEMMA 10.1.4. Let WE W. Then
is either in one of the reflecting hyperplanes or is a transform w(S) of S
by sotne element of the JVeyl group.
f-
i
t"".
W(S)=s-  r,
rEO
PROOF. Suppose
where n is a subset of <1>+ with I Q I =I(w).
S-  r
reO
[
PROOF. Let n be the set of roots in <1>+ which are not in w(<1>+). Then
!
I
'- -
S- W(S) = -!- }-:; r- t  w(r)=  !".
reO -w(r)eO reO
is not in any reflecting hyperplane. Then
S- L: r
,-£ n
Moreover, I Q I =/(w-1)=/(w), using 2.2.2.
.
lies in some chamber. By 2.3.2 there exists WEWsuch that
r
I
t
LEMMA 10.1.5. Let n be any subset of <1>+. Then if
W(s- 2: r)
rE 0
""'-
S-  r
reO
is in the fundamental chamber. However
L.
lies in the fundamental chamber, .a is empty.
l1'(s-  r) =s-
reO
 r
reOI

148
SIMPLE GROUPS OF LIE TYPE
for some other subset Q1 of (1)+. (This follows from the definition of'
S and the fact that w permutes the roots.) Thus by 10.1.5 Ql is empty
and so
S-  r
reO
is a transform of S.
As usual we denote by Q the set of integral combinations
1
2.: niqt
i=l
of the fundamental weights. Q is an additive abelian group. It will be
more cO.'1Yenient in the following discussion to regard Q as a multi-
plicative group instead, anel so we define e( Q) to be a multiplicative
group isomorphic to Q. The elements of e(Q) have form
1
I1 e( ql)nl
i=l
and the map e from Q to e(Q) satisfies
e(a +b) = e(a). e(b) , a, bE Q,
e( -a)=e(a)-l.
Let A be the rational group algebra of e(Q). The elements of A are
finite sums
2.: Axe(X),
x
where XEQ and AxEQ. The natural operation of v on Q can be trans-
ferred in the obvious way to an operation on e(Q) and then extended by
linearity to give an operation of v on A, thus making A into a W-module.
We consider the alternating elements of A, viz., those which satisfy
w(a)=det w.a
for all WE W. We denote by 8 the lincar map of A into itself given by
0= 2.: det w.w.
tVeW
LEMMA 10.1.7. The image of A under 8 is the set of all alternating
elements oj A.
THE EXPONENTS OF THE WEYL GROUP
149
-\.
PROOF. Let WE Wand a EA. Then we have
w. 8(a)=w.  det w'. w'(a)
w'eW
=detw.  det(ww').ww'(a)
w'eW
.
=det w. 8(a).
Thus 8(a) is alternating.
Now let a be any alternating element of A. Then
8(a)=  det w.w(a)= 2: a=\ W! a.
weW weW
Thus
8C  a)=a
and so a lies in 8(A).
.
"-''"
theorem of Weyl giving a factorization of the expres-
We now prove a
sion 8(e(S)).
THEOREM 10.1.8.
8(e(S»=e(-S). IT (e(r)-l).
re c1>+
. thO tl1eorem we give an example to illustrate it. Suppose
Bcfore provmg IS I t Then
.y) .s of type Az with fundamenta roO s PI, pz.
the root system '-! I'd ( ) - Y Then
S=!(Pl+]J2+(Pl+PZ»)=Pl+PZ. Let e(Pl) = X an e pz - .
8(e(S)=XY-X + y-l_X-ly-l+X-l- Y
and we have the factorization
Xy _ X + y-l_ X-I y-l + X-l- y = X-I y-l(XY -l)(X -1)( y -1).
PROOF. Let
--.
a=e(-S). D (e(r)-l).
r e <lJ+
f

150
SIMPLE GROUPS OF LIE TYPE
--
[
We show that a is an alternating element of A. For each Pi En we have
wpla)=e( -Wp,(S)). n (e(lVp,(r))-l)
rE4;1I+
f"
iii!
I
Il. ",,;
=e(Pi-S). n (e(r)-I). e( - P-l)-1
rE4;1I+ e(pi)-1
=e( -S). n (e(r)-I).e(Pt).L-=' P-l)-=
rE\l>+ e(pt)-l
= -a.


t
L,
It follows that w(a) = det w. a for all WE Wand so a is alternating.
We now multiply out the product in a. We have
a=e(S). n (l-e( -r))
rE 4;11+
r
f
iJ
=e(S). L: (-1)101 e(- L: r)
Os4;1l rE!)
= L: (-1)101 e(s-  r).
Os4;1l+ rEO
Now a is alternating, and so e(a) = I W I a. Thus
;'-
t
\--.,
a= 2: (-1)101 e(e(s-  r)).
I W I Os<D rEO
Now if
S- 2: r
rEO
lies in some reflecting hyperplane Hs we have
e(e( S-ror)) =0.
For wand wWs give contributions to the sum
2: det W e(w(s - L: r))
w rEn
which are equal and opposite. Thus we need only consider subsets .0
of cI>+ such that
s- r
rEO
'-
(
\
\ -
does not lie in any reflecting hyperplane, and so has the form w'(S)
for some w' E Wby 10.1.6. In such a case we have I n I =/(w') by 10.1.4.
n'IE EXPONENTS OF THE WEYL GROUP 151
Thus
1 (- l)l(w') 8(e(w'(S)))
a =---- -- 2:
I V I w' E If'
I det w' . 81V'(e(S))
= .----- 2:
I W I w' Elf
SInce
(-l)l(w') =det w'.
Hence 1 e(e(S))
a=----- 
'$i.it I f'V I w'JV
= e(e(S)),
SInce
ew' =det w'. e.
This completes the proof of Wey1's theorem. ..
10.2 A Theorem of Macdonald
Now the elements of e( Q) are uniquely expressible in the form
l
n e(qt)nt,
i=l
where I1j ElL. We may introduce a total ordering on e(Q) by means of the
first difference in the exponents Ill,..., nl. Let x, y be two non-zero
elements of A. Then
x=Ae(qI)ml . . . e(ql)ml+lower terms in e(Q),
Y = f-Le( ql)n1 . . . e( ql)nl + lower terms in e( Q),
where A#O, f-L#O. Hence
XY=Af-L e(ql)m1+nl .. . e(ql)1nI+nl+lower terms in e(Q)
and so xy# O. This shows that A is an intcgral domain.
Let F bc the field of [ractions of A and F[t] be tile polynomial ring
over F in the indeterminate t. The following remarkab1e identity in F[t]
is due to I. G. Macdonald.
THEOREM 10.2.1.
w.crl, :-J:i ;;:}D =w/(W).

152
SIMPLE GROUPS OF LIE TYPE
THE EXPONENTS OF THE WEYL GROUP
153
-,
We again illustrate this theorem by an example before proving it.
Suppose <P has type Al and that r is the single positive root. Then Mac-
donald's identity states that
I-tee -r) + 1-te(r) -1 +t.
1-e( -r) 1-e(r)
lies in any reflecting hyperplane. Moreover if
S- l: r
rEO
does not lie in any reflecting hyperplane, it has the form w'(S), where
w' is an element of W with l(w') = I Q I (see 10.1.6 and 10.1.4). Using
this information we have
PROOF. We again consider the element
a=e( -S). 11 (e(r)-l).
r E \D+
l: (11 I-t e( -W22) =! l: (_t)l(w') 8(e(w'(S)))
WE W r E \D+ 1 - e( - w(r )) a w' E W
=! l: tl (10') . det w' . Bw'(e(S))
a (O'E IV
Since a is an alternating element we have
J.v(a)=det w.a=e(-Jv(S)). n (e(w(r))-l)
r E (1)+
=e(w(S)). 11 (I-e( -w(r))).
r E \D+
=! l: tl(w'). 8(e(S)),
a w' E IV
Thus we have
since (-l)l(W')=det w' and Bw'=det w'.B. But a=B(e(S)) by 10.1.8
and so the result follows. .
rr (l-e( - w(r))) =e( -w(S)). det w.a.
r E \1>+
Now the polynomial
...-.."....
It follows that
( I-t e( -w(r)))
wv r!L l-e-C -w(:;:5)
= l: (det we(w(S)). n (I-te(-w(r))))
a WE JV rE\D+
= l: det we(w(S)). l: (-t)IClI.e(w(- 2: r))
aWEJV Oc\D+ rEf}
=! l: (-t)IOI l: det w.e(w(s- 2: r))
a 0 c \D+ WE JV rEO
2: tl(w)
WEW
has been considered in Macdonald's identity as an element of F[t],
where F is the field of fractions of the rational group algebra of e(Q), and
a factorization of this polynomial has been obtained in F[t]. However,
one can make use of this factorization in F[t] to obtain a factorization
of
l: tl (to)
WEW
in the very much smaller domain O[t].
= 2: (-t)IOI e(e(s- l: r)).
a 0 c (1)+ rEO
THEOREM 10.2.2.
As in the proof of 10.1.8 we have
th(r)+l_1
 tl(W)= 11 --
weW rel1>+ th(r)-l '
e( e( s- ro r)) =0
where her) is the height of the root r.
S-  r
rE f}
PROOF. Let P be the additive group generated by the fundamental
roots PI, . . . ,Pl and e(P) be the corresponding multiplicative group.
Let h be the homomorphism from Pinto 71.. taking value 1 at each
--.
if


154
SIMPLE GROUPS OF LIE TYPE
THE EXPONENTS OF TI IE WEYL GROUP
155
"=='
fundamental root. h is called the height function. Then there is a
homomorphism
will cancel, and in order to see which terms remain after canceIlation
we consider the number of positive roots of a given height. Let kt be
the number of roots of height i. Then inspection of the root systems
(see section 306) shows that
1= k1 ?; k2 ?; k3?;; . . . .
t
e(x)t-h(X)
from e(P) into the infinite cyclic group generated by an element t.
Let B be the rational group algebra of e(P). Since P is a subgroup
of Q (see section 7.1), B is a subalgebra of A. The above homomorphism
from e(P) into <t) extends to an algebra homomorphism from B into
Oft, I/t], the set of rational combinations of the powers of t. (Q[t, 1ft]
is the rational group algebra of the infinite cyclic group <t).) This algebra
homomorphism may itself be extended to an algebra homomorphism
from the polynomial ring B[t] into Q[t, l/t] under which t is mapped
into t. For since t is an indeterminate over B, its image may be chosen
arbitrarily in 0 [t, l/t] and an algebra homomorphism is then uniquely
defined. We denote this homomorphism by r/;. Then r/;: B[t]O[t, I/t]
satisfies
Now (kl, k2, . . . ) is a partition of N. The dual partition of N has I parts
and will be denoted by (m1, 1-r12, . . . , ml).
THEOREM 10.2.3.
[-''''

I (enj+1 - I )
L: tl (w) = IT ' ----.----- .
WETV i=l t-l I
PROOF. All the terms in the expression
r
L-
r/;(e(r))=t-h(r), rE,
r/;(t)=t.
Now Macdonald's identity
L: (IT I-t-=(-w(r))= L: tl(w)
WE TV r E + 1 - e( - w( r )) WE W
may be interpreted as an identity in B[t]. For we may remove the dl.
nominators to obtain an identity in A[t], and then observe that all the
elements of Q appearing in this identity are in P. We now apply the
algebra homomorphism r/; to both sides, and obtain an identity
( 1- tI+h(W(r»)
h n ---- ---- = L: tl (w).
WE W rE+ 1- th(w(r» wE W
[h(r)+1_1
IT --- -- - ---
rEID+ tlL(r) - 1
......
; LJ
cancel with the exception of terms t11l1+1_ I, t11l2+1- I, . . . , tmrl-1-1
in the numerator and (t -I)l in the denominator. .
COROLLARY 10.2.4. The degrees of the basic polynomial invariants
of}Vare given by dl=mt+I, i=I,..., I.
PROOF. This follows from 9.4.9 and 10.2.3.
.
f
e

'\. _0
It is easy to determine the integers 1111, . . . , ml in the individual cases
by inspecting the root systems.
 I
. t
r--
L
However if wl=I there is some rE(f)+ such that h(w(r))=-l. Thus the
contributions to the left-hand side from all non-identity elements of W
are zero. Hence
PROPOSITION 10.2.5. The integers m1,...  ml are asfo/lows:
/111, ///2, . . . , ///1
and the theorem is proved.
.
All, 2, . . . , /
Bt 1, 3, 5, . . . , 2/-1
Ct 1, 3, 5, . . . , 2/-1
Dt 1, 3, 5, . . . , 2/- 3, /-1
G2 1,5
F4 1,5,7,11
E6 1, 4, 5, 7, 8, 11
E7 1,5, 7, 9,11, 13, 17
E8 1,7,11,13,17,19,23,29
1- t1+1t(r)
IT ------ = L: tl(w)
r E + 1 - t h (r) WE W
f
i
t_
Now it is clear that in general a large number of terms in the product
t1t(r)+1-1
n -----
rE \11+ th(r) -1
----

156
SIMPLE GROUPS OF LIE TYPE
PROOF. Calculate the numbers kl, k2, . . . from section 3.6 and form
t11e dual partition. .
It will be observed that the integers ml, . . . ,ml determined in 10.2.5
satisfy a condition of duality. If the Ini are arranged in increasing order
we have
ml+ml=m2+ml-I=... =l+h(R),
where R is the (unique) root of ITiaximum height. We have not yet seen
any reason why this duality should exist. However, this becomes clear
by giving an entirely dilTerent definition of the mt, based on properties
of the 'Coxeter elements' of the Weyl group. The approach via Coxeter
elements will also explain another property of the integers Ini observable
from 10.2.5, the fact that they exhibit a definite tendency to be prime.
10.3 The Class of Coxeter Elements
A Coxeter element of the Weyl group W is an element of form
WTlWT2' . . Wrp
where rl, r2, . . . , rl is a fundamental system in <1>.
THEOREM 10.3. 1. The Coxeter elements of W form a conjugacy class
in V.
PROOF. It is clear that any conjuga te of a Coxeter element is a Coxeter
element. For
WWrl . . . 'WTI W-I = Ww(rl) . . . Ww(rl)
and if rl, . . . , rl is a fundamental system so is w(rl), . . . , w(rl).
In showing that any two Coxeter elements are conjugate it is sufficient
to consider Coxeter elements corresponding to a fixed fundamental
system, since any two fundamental systems are equivalent under the
Weyl group (2.2.4). We take the fundamental system n = {PI, . . . ,pI}
and consider the Coxeter elements corresponding to this. WPlWp2, . . W1Jl
is one such element and the others are obtained by changing the order
of the factors. The [act that they are all conjugate follows from the
following lemma. (\
THE EXPONENTS OF THE WEYL GROUP
157
LEMMA 10.3.2. Suppose the fundamental roots PI, . . . ,PI are written
rOlind a circle, as shoum in Figure 3. Then any permutation of PI, . . . , PI
may be obtained by performing a succession of interchanges (PiPi), where
PI, Pi are not joined in the Dynkin diagram and are adjacent on the circle,
and then reading clockwise around the circle beginning from a suitable
point.
Ii
Figure 3
.-..,\
PROOF. We use induction on I. The resuIt is clear if 1= 1 or 1=2,
since no interchanges have to be carried out at all. Thus we assume
l 3. Now the Dynkin diagram contains some node joined to at most
one other node. Let PI correspond to such a node. On omitting PI from
the circle we may by induction obtain any permutation of PI, . . . , pl-I
by interchanging adjacent pairs not joined in the Dynkin diagram and
then reading round from a suitable point. We show that all these inter-
changes can be carried out even when PI is present. It is sufficient to
show that Pi, Pi can be interchanged if Pt, Pi are not joined in the Dynkin
diagram and if Pl is adjacent to both. (See Figure 4.)


Figure 4
I'
Now PI is not joined in the Dynkin diagram to at least one of pt, pj-
assume without Joss of generality that Pl is not joined to Pt. Then by
interchanging PI., PI and then Pl., Pi we have succeeded in interchanging
the order of Pt, Pi when PI is removed. Thus we may make the required
arrangement of all the roots other than Pl. But then PI, since it is joined
-.

158
SIMPLE GROUPS OF LIE TYPE
THE EXPONENTS OF TIlE WEYL GROllP
159
'-
,.-
to at most one other node in the Dynkin diagram, may be moved around
the circle by a succession of steps in either a clockwise or an anticlockwise
direction into its required position. .
Let /1, . . .,fi, be vectors of unit length in the directions of PI, . . . ,pl
so that Pi= I Pi I it. Then It, . . . ,ji is a basis for 1), aJthough not an
orthonormal basis. However, there is a uniquely determined dual basis
JI, . . . ,1£ satisfying
f

LJ
Now given any arrangement of the roots PI, . . . , Pl around the circle
and any starting point on the circle, a Coxeter element 11"1..' W'l is
determined. rI is the starting point, and rl, r2, . . . , rl appear clockwise
around the circle starting from rl. If we take the same arrangement
around the circle, but a different sta;ting point, we obtain a conjugate
Coxeter element. If we change the arrangement around the circle by
interchanging adjacent roots not linked in the Dynkin diagram, the
Coxeter element is unchanged since the corresponding reflections com-
mute. (If we have interchanged the first and last terms, a conjugate
Coxeter element is obtained.) Thus 10.3.2 shows that any Coxeter element
of the form W'l . . . wr, can be obtained from WPI . . . WPl by a succession
of operations which either leave the element unchanged or give a con-
jugate element. This completes the proof of 10.3.1. .
f (1 ifi=j,
(li, Jl) = 0 if i# j.
Let (jt,jj) =mij and ]VI be the I x I matrix !vI = (mi1). Then (It,jj) = (M-I)lj.
Now !vI is a symmetric matrix whose diagonal coc/licients are 1 and
whose non-diagonal coefficients arc non-positive (2.1.4), Thus 1- M is
a symmetric matrix whose coefficients are all non-negative. This matrix
plays a useful role in deriving the properties of the Coxeter elements.
'"
,
,
t
L"
. r.
f
I
L",
Let A be a non-zero eigenvalue of 1- M and
'-
The conjugacy class containing the Coxeter elements of W is caI1ed
the Coxeter cJass.
,, [ ::J
f
!
, ..
be a corresponding eigenvector. Then
(1- M) u=AU
; .
10.4 A Dihedral Subgroup of the Weyl Group
and, since 1- M is symmetric, A is real and
real. We define two elements a, b of V by
k
a= L: tilt,
i=l
6, . . . , l may be chosen
We shaH obtain further information about the Coxeter elements by
showing that each Coxeter element can be embedded in a dihedral sub-
group of "V which operates faithfully on a certain 2-dimensional subspace
oft}.
1
b = L: td;,.
i=k+l
\ I
i
LEMMA 10.4.1. The set offundamental roots PI, . . . ,Pl may be divided
into two disjoint subsets, each of which contains roots l-vhich are all orthogonal
to one another.
LEMMA 10.4.2. (i) (a, a)=(b, b).
(ii) The angle 8 between a, b is given by cos 8 = A.
PROOF. M may be written in the form of a block matrix
PROOF. Remove a node from the Dynkin diagram which is joined to
at most one other node, and use induction on the number of nodes. .
M=( h A )
A' h-k'
where A is a certain k x (l-k) matrix and A' is the transpose of A. M-l,
which is also symmetric, may be written in the form
'''---
We suppose that the fundamental system n = {PI, . . . ,pI} is decomposed
as in 10.4.1 into two subsets
! i
I
I
PI, . . . ,Pk; Pk+1, . . . , Pl.
M-l=(;, ).

160
SIMPLE GROUPS OF LIE TYPE
THE EXPONENTS OF THE WEYL GROUP
161
If we write
U(:: )
subgroup of W, since it is generated by two involutions. This dihedral
group operates in a particularly simple way on the 2-dimensional sub-
space of 1) containing the points a and b.
--...
-Au2=AUI,
- A'ul =AU2.
PROPOSITION 10.4.3. Let A be a non-zero eigenvalue of I-M and u
a corresponding eigenvector. Let a, b be the vectors defined above and r
be the circle with centre the origin passing through a and b (see Figure 5).
Then the group (WI, W2) operates on the points of r. In particular WI is
the reflection in the line Ob, W2 is the reflection in the line Oa and W is a
rotation around r through an angle 26, where cos 6 = A.
the equation (I-A1)u=Au gives
It follows that
- BAu2 = ABu 1 ,
Now since M-IM=h we have
I-DA'U1=ADu2.
(*) Cu2= ABu I,
C'U1=ADu2.
BA+C=O,
c' +DA'=O.
Therefore
However
k
(a, a)=  tttj(M-I)tl=(ul)'Bu1,
i.j=1
l
(b, b)=  tttj(M-l)tj=(u2)' Du2.
. i,j=k+l
The equations (*) now show
(ul)'Bul = A-I. (u 1)' Cu 2 = A-I. (u2)'C 'ul = (u2)' Du2.
Figure 5
=(ul)' CU2=A(UI)' Bul=A.(a, a).
Since (a, a)=(b, b) we have cos 6=A.
.
PROOF. It was shown in 10.4.2 that ABu1- Cu2 = 0 and this implies
that (It, Aa-b)=O for i= 1, . . . , k. Thus Aa-b is a linear combination
of !k+l, . . . ,fz, so also a linear combination of Pk+1, . . . , pl. Hence
)'2(Aa-b)= -(Aa-b). However, w2(a)=a and, since cos 6=A, Aa is the
projection of b on Oa. Thus W2 leaves r invariant and operates on it as
the rel1eclion in the line Oa.
SlIniJarly it was shown that C'UI-ADu2=0 and this implies that
(j;, a-Ab)=O for i=k+1,...,1. Thus a-Ab is a linear combination
of PI,... ,Pk and so WI(a-Ab)= -(a-Ab). Also wI(b)=b and Ab is the
projection of a on Ob. Thus WI leaves r invariant and operates on it as
the reflection in the line Ob.
Finally, since Oa and Ob are inclined at an angle 6, w= WIW2 is a
rotation through 26. The whole dihedral group (WI, W2) therefore operates
on r in an obvious way. .
Therefore
(a, a) = (b, b).
The angle between a, b is given by
k 1
I a ]./ b I.cos 6=(a, b)=   tttj(M-I)tj
i=lj=k+l
We now define elements WI, W2 of W by
WI = Hipl . . . WPk,
W2 = WPk+1 . . . 'WPI.
10.5 Eigenvalues of the Coxeter Elements
Then WI, Wz are involutions, being products of reflections with respect
to mutually orthogonal roots. Their product W = WIW2 is a Coxeter element.
We consider the group <WI, W2) generated by WI, W2. This is a dihedral
Bc:fore applying the results of section 10.4 to give further information
. about the Coxeter eIeents, we state and prove a cJassical.result on real
........

"---
I'Xl
['"
. ';i
r--
-.
l.o<-'
, -
I
t,...
L
[. ."

f'
i
,,_c,'
[
f
I
L_.
t_--
\

L
r._
162
THE EXPONENTS OF THE WEYL GROUP
163
SIMPLE GROUPS OF LIE TYPE
symmetric matrices wbich we shall need, known as the Frobenius-Perron
theorem.
A real symmetric matrix M = (mu) is called positive semi-definite if
xMx'  0 for all x ElRl, and indecomposable if it is impossible to split
up the set 1, 2, . . ., I into two non-empty complementary subsets I, J
such that mtj=O whenever iEI, jE J.
matrix TMT'. M - f-Ll is singular, so its nuB-space has dimension 1 and
contains a vector whose coclTicients are all positive. Thus f-L occurs as
eigenvalue of M with multiplicity 1, and has an eigenvector whose coeHi-
cients are all positive. ·
We apply the Frobcnius-Perron theorcm to the situation discussed in
10.4.3. Let (I.,f=(mij) be the matrix defined by I71lj=(jt, fj). Then Iyf
is positive definite, since the Killing form is positive definite (section 3.3).
Moreover mij  0 when i =I- j by 2. I ,4.
PROPOSITION 10.5.1. Let M be a real symmetric matrix such that
mij  0 for all i #- j, and suppose that !vI is positive semi-definite and indecom-
posable. Then the eigenvalues of !vI are all real and nOll-negative. The
smallest eigenvalue has multiplicity 1 and has all eigenvector whose coeffi-
cients are all positive.
COROLLARY 10.5.2. If the vectors a, b arc chosen aJ' ill scctiol1 10.4
I\'ith respect to the largest eigcl1value of 1- iV!, then each point oj the circle
I' lying strictly betll'eell a and b is ill the fundamental chamber.
PROOF. The eigenvalues of M are real since M is symmetric, and
non-negative since 11-1 is positive semi-definite. There is an orthogonal
matrix T such that TlI-fT' is a diagonal matrix whose coellicients are the
eigenvalues of !vI, by a well-known theorem of linear algebra.
Let Va be the null-space of M, i.e. the set of XEl sllch that xM =0.
Yo is also the set of XEz such that xli-Ix' =0, as is easily seen by con-
sidering the diagonal matrix TMT'. We shall show that dim Vo  l.
Suppose dim Vo> 0 and let x=(cq, a2, . . . ,al) be a non-zero vector
in Vo. Let y=( I al I, I a21 , . . . , I all ). Then we have
PROOF. The eigenvector aI,..., l) corresponding to the smallest
eigenvalue of !vi corresponds to the largest eigenvalue of 1- !vI. Thus
we may assume each i > O.
Now we have
k
a=  i.h.
i=l
l
b=  di;
i=k-j-l
thus every point c 011 r strictly bet ween a and b has the form
l
c = 2.:: Adi,
i=l
OyMy'xMx'=O,
since m11O if;#- j. It follows thatyMy' =0 and so YEf'o. Thus f'o contains
a non-zero vector whose coordinates are all non-negative, and we have
where At> 0 for i = 1, . . . , I. This means that (c, fi) > 0 and so (c, Pi) > 0
for each i. Thus c lies in the fundamental chamber. ·
I
 I at I mtj=O.
i=l
We are now in a position to acquire further information about the
Coxeter elements.
Let I be the set of i with O:t#-O and J be the set of i with ai=O. Suppose
jE J. Then all the terms I O:t I mtj in the above sum are non-positive, and
so I O:t I mtj=O for all i. If iEI we have I O:t I #-0 and hence 111'£j=O. Thus
111lj=O for all iEI,jE J. However, !v! is indecomposable and lis non-empty,
therefore J must be empty. Thus each coefficient of x is non-zero. Since
this holds for each non-zero vector in Vo we must have dim 1)0  I.
Furthermore if dim Vo= 1, then l>o contains a vector whose coefficients
are all positi ve.
Let f--L be the smallest eigenvalue of M. Then M - f-LI satisfies the hypo-
theses of the proposition, as can be seen by considering again the diagonal
THEOREM 10.5.3. The order of the Coxeter elements is 2N/ I.
PROOF. Let w be a Coxeter element and Iz be the order of w. Let r be
the circle defined in section 10.4 with respect to the largest eigenvalue
of 1- lvi. Then Hi operates as a rotation 011 r. Now the cyc1ic group (w)
generated by w operates faithfu11y on r. For if an element or Woperates
trivially on r it fixes some point in the fundamental chamber by 10.5.2,
so is the identity by 2.3.2. Thus w has order II on I'.

164
SIMPLE GROUPS OF LIE TYPE
THE EXPONENTS OF THE WEYL GROUP
165
a
is a fundamental system. Since Pk+l, . . . ,pz are mutually orthogonal,
the only such roots are i:.Pi. Thus a lies in exactly /-k reflecting hyper-
planes.
Since a lies in / - k reflecting hyperplanes the same is true of all trans-
forms wt(a), i = 1, 2, . . . . Similarly band alJ its transforms wt(b) lie in
exactly k reflecting hyperplanes. However, each reflecting hyperplane
intersects the plane of r in a line meeting r at one of the transforms
wf(a) or wf(b). For otherwise there would be a reflecting hyperplane
intersecting r at a point strictly between a and b, contrary to the fact
that such points lie in the fundamental chamber. Now there are h trans-
forms of a and Jz transforms of b alternating round r. Thus the total
number of reflecting hyperplanes is
""..
.. ""-"''''',
Figure 6
Let c be the point on r which is the reflection of b in Oa (see Figure 6).
Then the angle between Ob and Oc is 28. Since w is a rotation through
28, there exists an integer i> 0 slIch that wt(a) lies on the arc bc. Let;
be the least such positive integer. If wi(a) lies between a and b, both a
and wf(a) lie in C, the closure of the fundamental chamber (10.5.2).
Thus wi(a)=a by 2.3.5. On the other hand, jf wf(a) lies between a and
c it is evident that w1(b) lies between a and b. Thus both band ».,i(b)
lie in C, and so wf(b)=b. In either case we have wi= 1 since (w) operates
faithfulJy on r. Hence i=h and 8= TIjh.
Now
![(/-k) Jz+klz]=-llh
since each hyperplane meets r in two points. Thus N = -lIlt, and so h = 2Nj I,
as required. .
COROLLARY 10.5.4. A Coxeter element w has an eigenvalue e27Tt!h.
.-
k
a=  gilt
i=l
PROOF. As w operates on r as a rotation through 2TIjh, w has an
eigenvalue e27Tt!h in the plane of r.
 
 t""
and so a is orthogonal to the roots Pk+l, . . . , PI and their negatives.
We show that these are the only roots orthogonal to a. Suppose
I
 TjtJi
i=l
COROLLARY 10.5.5. A Coxeter element w has an eigenvector v with
eigenvalue e2Tri!h sllch that v is not orthogonal to allY root.

r;

;:: ...
k
2: gfTji=O.
i=l
PROOF. Let v be an eigenvector with eigenvalue e27Tt!h lying in the
olane of r. Since HI operates as a rotation in this plane v is not real, so
i:D. Suppose (v, r)=O, where rE <D. Then (D, r)=O since r is rea.l, and t
follows that r is orthogonal to every vector in the plane of r. Smce thIS
plane contains points in the fundamental chamber we have a contradiction.
Thus v is not orthogonal to any root. .
is a root orthogonal to a. Then
Since each i > 0 and aU Tji have the same sign, this implies that
Tjl=Tj2= . . . =Tjk=O.
PROPOSITION 10.5.6. A Coxeter element has no eigenvalue 1.
Thus the root
2: Tj lJi
i=l
PROOF. This is much easier to prove than the preceding results.
Suppose a Coxeter element w fixes a vector v. Then
WP1WP2' . . wplv)=v
is a linear combination of Pk+l, . . . , pl. By 2.5.1 the roots which are
linear combinations of Pk+l, . . . ,Pl form a system in which Pk+1, . . . ,pl
and so
wPS' . . wp.(v)=wP1(v).
--...

",- '-'
--,
[
l---'
''-a
1'-
i,,,,,,
L"
r-
L
L..
L_
I r
"-
[N
- .
f..__
166
THE EXPONENTS OF THE WEYL GROUP
167
SIMPLE GROUPS OF LIE TYPE
Now WP2" . Wpz{v)-v is a linear combination of P2, .. ., pz and WPl(V)-V
is a scalar multiple of Pl. Since PI, P2, . . . , pz are linearly independent we
have
of linear factors representing the reflecti ng hyperplanes. Since /I is not
in any reflecting hyperplane, J=lO at (Yl, 0, . . . , 0). Thus we may choose
the numbering of the invariants 11, . . . , lz so that ait/aYt=lO at
Wp2" . wp(v)=wplv)=v.
Thus v is orthogonal to PI and wP2 . . . wPI fixes v. Repeating the argument
we see that v is orthogonal to PI, P2, . . . , pz, hence v=o. Thus w fixes
no non-zero vector and so has no eigenvalue 1. .
(YI, 0, . . . , 0).
Thus we have
aft '\ A. 1 . 1. . I . 1
-a-- = I\tYI "'t- + terms mvo vll1g Yt WIt 1 z> .
Yi
Hence
10.6 A Theorem of Coleman
It = '\iYI d,-IYt + terms involving different monomials,
where .\t#O.
We now apply wand use the fact that it is an invariant. Let
We now prove a theorem of Coleman relating the eigenvalues of the
Coxeter elements to the degrees of the basic invariants.
Y= Ylfl + . . . +Ydi.
THEOREM 10.6.1. Let the eigenvalues of a Coxeter element be
Then
,mI, m2, . . . , ml,
w-1(y)= YI-mIjl + . . . + Yl-m1l.
Since for any polynomial P(y) we have II'(P(y» =P(w-l(y» it follows
that
lvhere  = e21Ti!h and ml, m2,..., mz are positive integers less than h.
Theil the degrees of the basic invariants It, 12, . . . , lz of Ware
ml + 1, m2 + 1, . . . mz + I .
W(Yt) = Yi. -ml.
Hence
welt) = Atp-d,-m'Yldi-IYt +terms involving different monomials.
But w(lt)=lt and, since .\t#O, we obtain
PROOF. The eigenvalues of a Coxeter element ware hth roots of
unity, where h is the order of w, so are powers of ,. By 10.5.6 '0 is not
an eigenvalue, so the eigenvalues are of form ,rn" where 0 < mt < h. Let
fl, . . . ,/z be a basis for the complexification {; of  such that
w(jt) = ,mift.
P-dt-mi = I.
Now h-mi occurs as an eigenvalue of w whenever mi does, since
w is a real transformation. We now renumber the basis vectors /I, . . . ,fz
so that the eigenvalue ,mi is replaced by ,h-mi. With this new numbering
we have
Let
Xlel + . . . +xzez = Ylfl + . . . + yzh,
where el,..., ez is an orthonormal basis of 'V. Then Yl, . . . , Yz are
linear functions in Xl, . . . , Xz and vice versa. Now we may assume that
ml = I by 10.5.4 and that /I is not orthogonal to any root by 10.5.5.
Let
l-clt+mt- It = 1
and therefore
'd,-l = ,m,.
"It follows that (It - 1 == mt mod h.
Now the numbers h-m'l, i= 1,. . . , I, are a permutation of the numbers
1tI1. ;=1,..., I. (We have again used 10.5.6 here.) Thus
l z
2:: (h - mt) = 2: mi,
i=] i= I
J = I i! I
acyl, . . . , Yl)
be the Jacobian determinant of a set of basic polynomial invariants
of Wexpressed in terms of YI, . . . , Yz. By 9.3.5 J factorizes into a product

168
SIMPLE GROUPS OF LIE TYPE
THE EXPONENTS OF THE WEYL GROUP
169
which gives
I
L: mt=N.
i=l
Note. The equivalence of (i) and (iii) can be shown directly by an
argument of Steinberg [3].
The integers nIl. . . . , ml are caIled the exponents of the Weyl group.
We have now obtained three equivalent definitions for these exponents.
They can also be defined in terms of the topology of the corresponding
Lie group, i.e. in terms of the Betti numbers of the compact Lie group
with Weyl group W. The Betti numbers are the coefficients of the Poincare
polynomial of this Lie group, which factorizes as
I
II (l + t2m,+!)
il
I
2:: mt = 1/h.
i=l
But 1Jh=N by 10.5.3, and so
Also we have
I
 (dt-1)=N
i=l
by 9.3.4. Since dt - 1 == nI, mod hand 0 < m, < h, these equations imply
that d, - 1 = nIt for i = 1, . . . , I. .
(cr. Bott [1]).
G. Lusztig has pointed out that Macdonald's identity 10.2.1 can be
derived analytically by using a Lefschetz fixed point formula due to
Atiyah and Botl.
Coleman's theorem shows that one can determine the degrees of the
basic invariants of J--V merely by looking at the operation of a Coxeter
element on V. This is one of several indications that the class of Coxeter
elements is of particular significance in the Weyl group. It appears to
be the conjugacy class which is 'as far removed as possible' from the
unit class.
-'\
COROLLARY 10.6.2. Let k, be the number of positive roots of height
i and let (ml, 1112,..., nIL) be the partition of N dual to the partition
(kl, k2, . . . ). Then the eigenvalues of a Coxeter element are
ml, m2, . . . , ml,
where  = e21Ti/ h.
PROOF. This follows from 10.2.4 and 10.6.1.
.
The duality
nIl +nIl =m2 +ml-l = . . .
now follows immediately from the fact that w is a real transformation.
Since thc common valuc of these sums is I +h(R), wherc R is the highest
root, we also obtain the following result.
COROLLARY 10.6.3. The folio wing three integers are equal:
(i) The order h of the Coxeter elements,
(ii) 2N/ I,
(iii) 1 +1z(R), where R is the highest root.
-\

'-
CHAPTER 11
Further Properties of the Chevalley Groups
........
\Ve now return to prove some further properties' of the Chevalley groups.
We shall show that these groups are all_simple, apart from a few exceptions
when the underlying field is very small. We shall also prove that the
Chevalley groups of type AI, Bl, Cl, DI are isomorphic to certain classical
groups.
11.1 The Simplicity of the Chevalley Groups
We first prove a criterion for simplicity valid for any group with a
(B, N)-pair. We recall from section 8.2 that in such a group G the quotient
group W=N/B nN is generated by a set o[ elements Wi, iE/, such that
wf = 1. Let G' denote the commutator subgroup of G.
\..-
THEOREM 11. 1 . 1. Let G be a group with a (B, N)-pair satisfying the
following conditions:
-;",
.'
(a) G= G',
(b) B is soluble,
(c) n gBg-l = I,
{JEG
(d) the set I cannot be deconzposed into two /lon-empty complementary
subsets J, K such that Wj commutes with Wk for all j E J, k E K.
Then G is simple.
'...w_:..
PROOF. Let Gl be a normal'subgroup of G. Then GIB is a subgroup
of G containing B, thus GIB=P J for some subset J of 1, by 8.3.2. Let
K be the subset of I complementary to J. Let j E J and k E K, and let I1j, l1k
be elements of N corresponding to Wj, Wk E W under the natural homo-
morphism. If we define the length function l(w) as in the proof of 8.2.3
we have I(Wjl'Vk»I(Wk). By 8.2.4 we see that
'- '\....-
;.: -.
Bl1jB . BnkB s Bnj11kB.
170
"'==- .'
.;
:1 :
FUIU I Il:R 1'IZUl'i:lz m:s OF -11 JE ClII:V..\LLE'I liIUJl;['S
171
. .
-' )
..;:]
:.; ..1
,;
t. .
',\
;\:\1i
.:t{
'f
:. J;.
:,:,*
.1'"
;f!J
-:(!
;: IJ
.-\f-
.I\ .
!;1
'itj. <
. :11
:'
'1
"" !....
Now we have
G1B=P J= BNJB
and therefore
BnjB n Gl =I- cpo
Since Gl is a normal subgroup of G this implies
nlcBnjBl11 n Gl =I- cpo
HO'Never
llkBl1jBl11c s; I1kBl1jT1kB s; BI1j11kB n BI1J.;l1jl1lcB
by axiom BN 4. Thus we have
BI1j11lcB n Gl =I- cp or Bnk/1jl1kB n GFI=-.
The former condition implies that I1jnkENJ, whence l1kENJ and W],;E ftVJ.
This contradicts the fact, shown in the proof of 8.3.4, that the elements
Wi form a minimal set of generators for W. Hence we have
BI1/cl1jnkB n G1 =I- 4).
This means thalllkl1j/IlcENJ and \Vlc\VjWkE J-VJ. Thus
1\'J,;1\'jWlc E I'V J n J..v{j I k} = J..vU}
by 8.3.4. It follows that \Vi.;JI.'jWk is either I or \Vj. \Vk\Vjll'k = 1 implies
that Wj = 1, which is impossible; thus we have IVkJl.'jI-Vlc = IVj. Hence for
each jE J, k EK we have Wj\Vk = W/cH'j. Since I does not decompose into
two non-empty complementary subsets with this property, either J or
K must be empty.
Suppose K is cmpty. Then J=/ and G1B=G. Thus we have
GIGl=GlBIG1BIG1 nB.
Now B is soluble and so GIGl is soluble also. However, the [act that
G coincides with its commutator subgroup G' means that G has no non-
trivial soluble factor group. Hence Gl = G.
Suppose J is empty. Then G1B=B and so Gl is contained in B. Since
Gl is normal in G we have
Gl s; n gBg-l = 1.
(fEG
: ",1,
::, J.
i'l!
J
. .
...
i, !
.it
.1.1 L.
'::'...:(:.
. "
- . 
'l"
:. .if
Therefore Gl = 1.
Thus Gl is either G or 1, and so the group G is simple.
III
......
wI.
Theorem 11.1.1 shows that groups with a (B, N)-pair have a definite
tendency to be simple. Now the Chevalley groups G=JL(K) all have
. 1I1.
;- t;.'
,.:101
;;tl ':
:;r
:"- -,;;
OJ-

 .
:. .. ..
t · ,-:
 ...
(" .
)
172
SIMPLE GROUPS OF LIE TYPE
(B, N)-pairs by 8.2.1. Thus we may use 11.1.1 to try to prove the
simplicity of the Chevally groups. The proof goes through in all but a
few cases.

t !I!
; vii
I .
...
. t. ":"
:
THEOREM 11.1.2. (i) Let JL be a simple Lie algebra over C and K be
an arbitrary field. Then the Chevalley group G = JL(K) is simple, except
Jor Al(2), Al(3), B2(2), G2(2).
(ii) Each Chevalley group (even a non-simple one) has trivial centre.
I"
T:,
I
 i
. <.
: < ';
£.. j'
. f J'
)W; -,
{:. I
..)t .
\: to
.. "(l
.; .
 f 'L L
.. -,-
PROOF. By 11.1 .1 it will be sufficient to prove that
G=G' and n gBg-l=1.
oeG
For the Borel subgroup B is certainly soluble, being the semi-direct
product of the nilpotent group U with the abelian group H. Morever,
the set Wi of distinguished generators of Ware the fundamental reflectlOns,
and these cannot be decomposed into two non-empty complementary
commuting subsets since JL is simple.
We show first that G has n9 non-trivial normal subgroup contained
in B. Let Gl be such a normal subgroup. Now by 2.2.6 the Weyl group
W contains an element Wo which transforms every positive root into a
negative root. Let no be a corresponding element of N.
Then we have TT -] V
nOVJ10 =
by 7.2. 1, and also 110 UIIn"f;I = VII.
Now Gl is contained in UH, so also in VH since it is normal. Thus
Gl  UH n VII=I-I
by 7.1 .3. However, H normalizes U and so we have
[U, II]  U
 .
;- .:-.
, 1 .. :;
. -t:'
t L-' :
'\., ,.,.,
t. : ",. 'j
: ".... . I
t
}:
and
[U, G1]  U n Gl  U n H = 1.
 ,;'
Thus every element of Gl commutes with every element of U. Let
hex) E Gl. Then
hex) Xr(J) h(X)-l = xr(x(r )) = xr(1)
for all r E CP+. It follows that x(r) = 1 for all r E <1)+. Hence X = 1 and hex) = 1.
Thus Gl = 1. We have therefore shown that
n gBg-l=1.
oeG
/:-'
:  J
{: . !
.'
... -,':
. .U
';'1t
. .
I..... ..
,t
t ':'
.... :; 
..... "" -- ; t :
¥';
\If -

FURTHER PROPERTIES OF THE CHEV ALLEY GROUPS
173
-......
We show next that G has trivial centre. Let Z be the centre of G. The
argument used in the proof of 11 . 1 . 1 shows that either
G=ZB or Zs n gBg-l.
oeG
."'5 .:
If G =ZB then B is normal in G. Since by 8.3.3 B is its own normalizer
in G we have ZB=B. Therefore Z  B, and since Z is normal we have
Z  n gBg-l.
(leG
Hence Z= 1.
In order to show that G is simple it remains to prove that G = G'. We
shall require the following lemma.
LEMMA 11.1.3. (i) Let r E cD and t be a nOll-zero element oj K. Let Q
be the additive group oj weights. Then there is a K-character X oj Q such
that x(r) = t2.
(ii) There is a K-character X oj Q such that x(r) = t, unless J[ =Al or
JL= Cl and r is a long root.
-...,
PROOF. (i) Let Xr, t be the K-character of Q defined, as in 7.1.1,
by
Xr, tea) = t2(1', a)/(r, r).
'.. " \
Then Xr, t(r)=t2.
(ii) Let P be the additive group generated by the roots. We recall
from section 7. I that P, Q are free abeIian groups of rank I and that
P is a subgroup of Q of finite index. Let m be the greatest integer such
that (1/m)rEQ. (m is the highest common factor of the coefficients of
r when expressed as an integral combination of ql, . . . , qz.) Let
1
1'1 =- r.
m
Then
1
- 1'1 f/= Q
11
for any n> I, and therefore rl forms part of some basis for Q. Hence
a K-character X of Q can be chosen so that x(rl) is an arbitrary non-zero
element of K. Let
h-
r-(r,r)
.....--.,.,

.
;
:.'
!br
, i

t
'"W'
"
"f.-,.
t
r\

;;;L
,.
t
f
L
{
--
:L
L
,
f
-
\......
L
f
L
174
SIMPLE GROUPS OF LIE TYPE
be the co-root correspvnding to r. Then (hr, r]) £ g: since 1'1 E Q, by 7.1. 1..0:
Thus
2=
(I', 1') m
is an integer, hence In = 1 or 2.
If In = 1 we can certainly choose X so that X(r) = t, and this is so unless
trE Q. We first consider what happens when I' is a fundamental root.
Let I' = Pi. Then
t
I' = L: Ajiqj
j=]
by 7. I, and tr E Q only if each Aji for j = I, , . . , I is divisible by 2. A
glance at the Cartan matrices of the simple Lie algebras (listed in section
3.6) shows that this can happen only if JL = A 1 or if JL = Ct, (l  2), and
I' is the long fundamental root. If r is not a fundamental root it can be
transformed into one by an element of Wand the property tr E Q is
preserved. Thus tr E Q holds only for the roots of A I and the long roots
of Ct. II
We may now complete the proof of 1 1 . 1.2 and to do so we must show
that G=G'.
Suppose first that K has at least four elements. Then there is a non-zero
element t E K sllch that t2 i= I. By J 1 . I .3 there exists for each I' £ (I) a
K-cbaracter X of Q such that x(r) t= I. Then h(X) E H by 7. I . I and we
have
hex) X?{/) IrCd-1 = xr(X(r) t).
It follows that Xr«X(r)-I) I)£G' for a11l£ K. Let UE K and choose
u
. I = - --- .
x(r) - I
Then Xr(U) E G'. Since the elements Xr(U) for all I' E!'Q), U E K generate G, we
have G' = G.
Next suppose that f( = GP(3). If J1.. is not of type A I or Ct. J 1.1.3
shows that there exists for each I' E (I) a K-character X of Q with xCI') t- 1.
Then G' = G as above. Thus suppose that G = Ct(3), where I 2. Now
G is genrated by its root subgroups Xr for all I' E (P. Since Xr, XtDCr)
are conjugate subgroups of G by 7.2.1, and so equivalent modulo G',
the factor group GIG' is generated by the images of the root subgroups
Xr and it is sufficient to take one root from each orbit of Q) under W,
-" .
-,--
.. J 

"
.. .
  .
I ';1 ,
t1'
: 
11 o.
\ , 'r
U!i <I.
t
['I
rll! ·
'\ "
J, ','
.." 't
FURTHER PROPERTIES OF THE CHEV ALLEY GROUPS
175
I .

 i i.e. one root of each length. If I' is a short root of Cl there is a K-character
X of Q such that xCr) -II, by 11.1.3. Thus Xr is contained in G' in this
,. case. Let s be the long fundamental root of Ct and I' the fundamental
root joined to it in the Dynkin diagram. Then we have
".
[Xs(l), xr(I)]=xr+s(:t 1)x2r+s(:t 1)

by the commutator formula. Now I' +s is a short root and 21' +s is a
long root. Thus Xr+s  G' and Xr+sC:t I) X2r+s( i I) E G'. It follows that
XZr+s s G' and that G' = G.
Suppose finalIy that K = GF(2). If all the roots of 'If. have the same
length, GIG' is generated by the image of xr(1) for any single root r.
[f JL.:;6 A 1 we can choose 1', S E cJ:> such that I' + S E cJ:>. Then we have
.t', .
, .'
I'
.
I,
i ;
if  (
\'."-
i' .
1a. ,J..
II '
- 'I .
I, '
I .
t
.I 01 .
., .'
[Xs(l), Xr(1)] =xr+sCl)
by the commutator formula. Thus Xr+sCO EG' and so G' = G.
Now suppose there are roots of two different lengths. Then GIG' is
generated by Xr( 1), xs( I), where r, s are fundamental roots of different
lengths. We may assume I' is a short root, s a long root and that r+sE<{).
We assume also that the nodes corresponding to 1', s are joined by a
double bond in the Dynkin diagram, i.e. that 1L -:J G2. Then the commutator
formula gives
"
I. 
01.
,
1
[x.ll ), X7'( 1)] = xr+s(l ) x2HsCl )
r
J
, .'
,r-
, .
and so xr+sC 1) X2NS( 1) E G'. Here I' +s is a short root and 21' +s a long
one. Suppose 11. t- 82. Then the Dynkin diagram of 11.. has two nodes
joined by a single bond. Let 1'1, 1'2 be the corresponding fundamental
roots. Then
'u' -
l..- ,,:
"r ·
, .
 J
[Xr2(1), xr,(Ol = Xrr1-r2( 1)
and so Xrt+r2( 1) E G '. Lf r} + 1'2 is a short root this implies that all the
root subgroups corresponding to short roots are in G'. Thus Xrt-s(1)EG'.
But then x2r+-s(I) EG' and G' contains all the root subgroups corresponding
to !0ng roots also. Hence 0' = G. A similar argument clearly applies if
1'1 + 1'2 is a long root.
The only Chevalley groups not covered in the above argument are
AI(2), A 1(3), B2(2), G2(2) and so the theorem is proved.
. ,
.
R 1 .
f
: l
'.
The four groups which have not been proved to be simple are al] in
fact not simple. A t(2) has order 6 and is isomorphic to the symmetric
group 53. A 1(3) has order 12 and is isomorphic to the alternating group
I
1-
<-
1 r .

.1
d
.4 j..  4
176
SIMPLE GROUPS OF LIE TYPE

A4. B2(2) has order 720 and is isomorphic to the symmetric group 56.
Finally G2(2) has order 12096 and has a simple subgroup of index 2
isomorphic to the unitary group PSU3(32).
. l
11 .2 Classical Lie Algebras in Matrix Form
q, .
We wish to show now that the Chevalley groups Az(K), Bz(K), Cz(K),
Dz(K) are isomorphic to certain classical groups. In order to do this
we first describe a matrix representation of each of the simple Lie algebras
Az, Bz, Cz, Dz. We refer to Jacobson's book [1 J for proofs of the statements
about the simple Lie algebras which we require in this connection.
f -
\
.?' .
11.2.1 The algebra Az
The Lie algebra Jf. of all (I + 1) x (I + 1) matrices of trace 0 over C is
isomorphic to Az. Let  be the subalgebra of all diagonal matrices in
JL and eij be the elementary matrix with (i, j)-coefficient 1 and other
coefficients O. Then  is a Cart an subalgebra of JL and
JL= EB  Ceij
i'i=j

t
is a Cartan decomposition. Let h E  be defined by
h = diag P,o, AI, . . . , Ai).
[heij] =(Ai - Aj) eij.
. .
:
Then we have
 t

'l
..
Thus we evidently have a root system of type Az (cf. section 3.6) and the
root space negative to Ceij is Ceji. Let
hij = [eijejd = eu - ejj.
Then hij is the co-root corresponding to the root space Ceij. For h'i
is certainly a scalar multiple of this co-root, however we have
[hij, eij] =2eij
and so hij must be the co-root itself.
Let 8 be the map of JL into itself given by
6(x) = -x',

.
1 .
, I
j :
-I
. ' .1
1 \
I l
. .,
f
where x' is the transpose of x. Then 6 is an automorphism of JL. For
[-x', - y'] =x'y' -y'x' = (yx)' -(xy)' = - [xy]'.
t
FURTHER PROPERTIES OF THE CHEV ALLEY GROUPS
l
177
Also we have
8(eij)= -eji.
Thus if r is the root whose root space is Ceij and if we define the root
vector er by er = eij, then
8(er) = - e-r
for all rE <1:>. Now we have
[eres]=Nr,ser+s
and applying 8 this gives
[e-re-s] = - Nr, Se-r-8'
Hence N-r. -8= -Nr. 8. Since by 4.1.2 we have
Nr, sN-r. -13 = - (p + 1)2
'.
it follows that
Nr, s = :t (p + I).
Thus the elements hij corresponding to the fundamental roots and the
elements eij corresponding to all roots form a Chevalley basis of 1L.
--.
LEMMA 11.2.2. Let A be an n x n matrix over C. Then the n x n matrices
T satisfying
T'A +AT=O
form a Lie algebra. If T is a nilpotent matrix satisfying this condition
then
(exp T)' A( exp T) = A.
. P?OF. The set of matrices T satisfying T' A + AT=O is closed under
addIlOn and scalar multiplication. Let TI, T2 satisfy this condition and
'\ consIder the Lie product [TIT2] = TIT2-T2Tl. We have
TT; A - T;TA = - TATI + T{ AT2,
ATIT2 - AT2TI = - TIAT2 + T2ATl.
I
1

" ...
. Thus
rTtT2]' A +A[TIT2] =0.
..-: Now Suppose that T is nilpotent. Thus Tk=O for some k and
r.
.. -
...f
I.. ..
:
on

k-l I
exp T =  - Ti
£..J., .
i=O I.
..-.


[
r
L",
-""r'
,-"",
l_.
f
L.

L-
L
[
"r._
\
"--'
L
f'-
1 l/j
SIMPLE GROUPS OF LIE TYPE
FURTHER PROPERTIES OF TilE CIIEVALLEY GROUPS
179
Then we have
where
Is
(exp T)'A =exp (T').A = z: - (T')iA
i l!
1
=  -:-,- (-I)iATi=A.exp (-T).
\ ,.
0< i <j,
..
It follows that
( etj-e-j. -i,
- e-t, -j + ejt,
er= ,
et, -j-ej, -t,
-e-i, j +e-j, i.
(exp T)'A(exp T)=A.
.
h transforms these root vectors according to the folIowing formulae:
11.2.3 The algebra D,
We now take
[h, eij-e-j, -t]=(A!-A})(etj-e-j, -i),
[11, -e-£. -j +eji] =(Aj-Ai)(-e-l, -j+ej£),
[11, ei, -j-ej, -t]=(Al+Aj)(et, -j-ej, -l),
[It, - e-i, j +e-j. i] = ( - Ai - A})( - e-t, j +e-j, i).
A=(O
I,
' )
in 11.2.2 and consider 2/x21 matrices
Thus J[. is a simple algebra of type Dl (compare section 3.6) and the
above decomposition is a Cartan decomposition. The co-roots of J[. are
the clemen ts
T= (Tn
T2I
T12)
T22
[eij-e-j, -i, -e-t. -j+ej, i]=eu-ejj-e-t, -i+e-j, -}.
rei, -j-ej. -i, -e-i, j+e_j, i]=eii+ejj-e-i, -i-e-j, -j.
satisfying T' A + AT=O. T satisfies this condition if and only if T22 = - TII
and T12, T2I are skew-symmetric. Let JL be the Lie algebra of aU such
matrices and j[) be the set of diagonal matrices in JL. The elements of JL)
have form
For in each case we have [hrer]=2er. As in 11.2.1 the map 8(x) = -x'
is an automorphism of JL, but this time it satisfies O(er) = e--r for each
root r. To show that the fundamental co-roots and the root vectors er
form a ChevaJlcy basis we observe that D(ier)=-(-ie-r) and that
[ier, -ie-r]=hr. Thus the fundamental co-roots together with ier (r
positive) and - ie-r (r positive) form a Chevalley basis, as in 11.2.1.
H follows that the fundamental co-roots togcther with er, e-r (r positive)
also form a Chevalley basis, using the general ideas of section 4.2.
'\1
'\2
h=
'\,
-'\1
-'\2
11 .2.4 The algebra B,
This time we take
r  0 0
0 Il
A=
l  Il 0 i
I
f
I
I
-'\1
Nring the rows and columns ( 2, . . . , I, -1, -2 ... -I we have
_' -..._ ____________.____.____.__.__.___._ ..____.....-.__-__" __.--.--..---.........-----.-1--.--.---.-L ____..,_
J(, = jQ ED  teer,

180
SIMPLE GROUPS OF LIE TYPE
and cons.ider (2/+1)x(2/+1) matrices T satisfying T'A +AT=O. These
form a LIe algebra as in 11.2.2.
Let
( Too TOI T02 )
T= TIO Tu T12 1
Tzo T21 T22 1
1 1
?e the expression of T as a block matrix. Then T satisfies T' A + AT=O
If and only if T22 = - TIl, T12 and TZ1 are skew-symmetric T = _ 27"
T " ,10 1 02,
2U = - 21 01 anJ Too = o. Let JL be the Lie algebra of all such matrices
and ;t[) be the set of diagonal matrices in JL. The clements of ji) have
form
o
Al
Iz=
AZ
-AI
-,\Z
Numbering tle_ ows and columns 0, 1, .. . , I, -1, ..., -I we have
JL=  EB 2: Cer,
where
etj-e-j, -t,
-e-t, -j+ejt,
er=
et, -j - ej, -t,
- e-t, j +e-j, t,
2ew - eo, -I,
- 2e-to + eO{.
o < i < j,
_ i
I
f
FURTHER PROPERTIES OF THE CHEVALLEY GROUPS
181
Ti
I t }
n.
ill
 i ---,
j i
III
 r !
i J i "-
ill
"I!
U1 . -......
T
II i
j f ........
'I
!I
, i
'I
L
II
Ii
1 i
II
II
II ,
--"'\
I' -"'"
j
I'
!
II
II
II ? -'""!
II
r
i - "'""'\
J j
I )
fl - ..........
II
II
"
h transforms these root vectors according to the following formulae:
[Iz, etj-e-j, -d=(At-Aj)(e'j-e-j, -t),
[h, -e-I, -j+ejt]=(Aj-A{)( -e-t, -j+ejt),
[h, et, -j-ej, -{] =(A{ + Aj)(e" -j-ej, -t),
[h, -e-I, j+e-j, t]=( -A{-Aj)( -e-t, j+e-j, t),
[h, 2e'lO-eo, -t]=At(2em-eo, -I),
[h, - 2e-'lO + eod = - A{( - 2e-'lO + eo{).
JL is a simple Lie algebra and the above decomposition is a Cartan de-
composition giving a root system of type B (cf. section 3.6). The co-roots
of Jr. are the elements
[etj-e-j, -t, -e-t, -j+ejd=eu-ejj-e-t, -t+e-j, -j.
[e" -j-ej, -{, -e-I, j+e-j, {]=eu+ejj-e-I, -t-e-j, -j,
[2ew-eo, -t, - 2e-to +eot] =2eu - 2e-{, -to
For in each case we have [hrer] = 2er.
Let S be the matrix diag (2, 1, . . . , 1). Then the map 8 defined by
8(x) = - S-lX'S
is an automorphism of JL It has the effect of transposing the matrix,
changing its sign, halving the first row and doubling the first column.
Thus it can be seen that
e(er)=e-r
for all roots r. Hence the fundamental co-roots and root vectors er defined
above form a Chevalley basis of Jr., as in 11.2.3.
11 .2.5 The algebra Cl
This time we take
( 0
A-
-h
)
and consider the Lie algebra of all 21 x 21 matrices satisfying T' A + AT=O.
Let
T= (Tn
T21
TI2).
T22
G


'-
l:
[
[
'-.-.
\....""
[
[
f'
l..,
f
L___
.,

.' t..,.,..
..
. \
, \.
,, .....;.......j,' 
182
SIMPLE GROUPS OF LIE TYPE
FURTHER PROPERTIES OF THE OIEVALLEY GROUPS
183
Then T satisfies T'A+AT=O if and only if T22= -T{l and T12, T21 are
symmetric. Let J(. be the Lie algebra of all such matrices and J[) be the
set of diagonal matrices in J(.. The elements of  have form
, Al
r
I
I
h='
r
f
I
,
elements
[eil-e-f. -i, -e-i. -1 +eli]=ctt-cjJ-e-i. -l+C-f.-j,
[ei. -j+ej. -t, e-t, j+e-j, t]=eu+ejj-e-i, -i-e-j, -j,
[ei, -i, e-i, i] = eu - e-i, -t.
AZ
-AI
For [hrer] = 2er in each case. The map 6(x) = -x' is an automorphism
of JL such that 6(er)=e-r for all roots r. Thus the fundamental co-roots
and vectors er again form a Chevalley basis.
11.3 Identifications with some Classical Groups
- AZ
We make two useful observations about the matrix representations of the
simple Lie algebras AI, Bl, CI, DI which have just been described. J n
the first place it is evident that for all the matrices Cr in these representa-
tions we have c=O. In fact e;=O in all cases except for
( 2ew - eo, -i,
Cr=
-2e-i.o+eOi
Numbering the rows and columns 1, 2, ..., I, -I, -2, ..., -I we
have
J(. = 1L) EB 2: Ce-r,
where
etj-e-l. -i,
- e-l. -1 + efl,
in type BI. In these cases we have
et. -1 + el. -i,
o < i < j,
2 ( -2et, -i,
e =
r
- 2e-t, i.
er=
e-I, j +e-l. I,
et, -t,
Thus
e-I, I.
exp (ter) = 1 +ter+tf2e;.
\.
h transforms these root vectors according to the following formulae:
[h, eiJ- e-J, -tJ = (Ai - Al)(e'J - e-J, -I),
[h, -e-I, -J+ejt] = (AJ-At)( -e-t, -J+elL),
[11, e" -l+eJ, _tJ=(AI+Al)(ef, -J+eJ. -i),
[11, e-t,l+e-l, i]=( -At-Al)(e-I. f+e-j,,),
[11, et, -I] =2Atet. -I,
[11, e-I. tJ = - 2Aie-i, I.
J[ is a simple Lie algebra and the above decomposition is a Cartan de-
composition giving a root system of type Cl- The co-roots of J(. are the
Secondly we note that the coefficients of exp (ter) are all of the form
n, nt or nt2, where n ElL. This is because the coefficients of e; (when this
is not zero) are all divisible by 2. This fact enables us to transfer to an
arbitrary field. For each matrix er in one of the above representations
and each element t in an arbitrary field K, exp (ter) is a well-defined non-
singular matrix over K.
. Let JL be a simple Lie algebra of type AI, Bl, CI, Dl and G=JL(K)
be the ChevalJey group of type JL over K. Let G be the group of matrices
generated by the elements exp (ter) for all rEel) and all tEK. By 4.5.1 we
have
exp (t ad er).x=exp (ter).x.exp (ter)-1
. for all XEJ1.K. Thus there is a homomorphism a of G onto G=Jf.(K) such

184
SIMPLE GROUPS OF LIE TYPE
that
(J
exp (ter)-+exp (t ad er).
We determine the kernel of a.
LEMMA 11.3.1. The kernel of the homomorphism a:G-+G is the centre
ZofC.
PROOF. Let YEG be in the kernel of a. Then yxy-l=X for all xE'lLJ{.
In particular y commutes with ter for alt r E (1). Thus y commutes with
exp (ler) and so y is in the centre Z of G.
Conversely, suppose y is an element of Z. Then y commutes with
exp er for all r E (I>. If'lL is not of type Bl we have
exp er= 1 +er;
thus y commutes with er. Since
[ere-r]=hr,
y also commutes with hr for all r E cI>. Thus y is in the kernel of a. Now
suppose that JL is of type Bl. Then y commutes with er whenever e;=O,
and if e;i=O y commutes with (exp (er)-I)2=e;. Thus provided K does
not have characteristic 2, y commutes with
exp (er)-l-!(exp (er)-l)2=er
and it follows as above that y is in the kernel of a. Finally, if K has charac-
teristic 2 an easy matrix calculation shows that y is a scalar multiple of the
identity, and so is in the kernel of a. II
We are now able to prove a theorem of Ree identifying Chevalley
groups of type Al, Bl, Cl, Dl with classical groups.
THEOREM 11.3.2. (i) Ai(K) is isomorphic to the linear group PSLl+1(K).
(ii) Bl(K) is isomorphic to the orthogonal group PD.2l+1(K, fll), where fn
is the -quadratic form
X+XIX_l +XzX_z+ ... +XlX-l.
(iii) CI(K) is isomorphic to the symplectic group P Spzl(K).
(iv) Di(K) is isomorphic to t/Ze orthogonal group PD.21(K, fD), where fn
is the quadratic fonn
XIX-l +XZX-2+ .. . +XlX-l.
FURTHER PROPERTIES OF THE CHEVALLEY GROUPS
185
j:
Hi
jtil
, I
, I :
,I.
pi
!Ii
ill:
II
L\
n,
Iii
IIi
t
i;
I'
i;
If.
I;
i!
I j
n-
,\
J'
! II J
. ,
J
i
II
"
I {
II
If
II
II
!t
P
! t
II
Ii
i;
II
II
IIi
II
Ii
; 
If
II
IJ
Ii
Ii
ii:
. t,
o
II
.1
i I
I.
d
II
!'
I !
I
I:
I-
I-
f
\
Note that In and In are both quadratic forms of maximal index in the
spaces concerned (cf. section 1.4).
PROOF. (i) Let G=Al(K). Then C is the group of (I + 1) x (I + 1)
matrices generated by
1+ teij
(i # j)
for all t EK. These matrices generate the group SLl+1(K). Thus, by
11.3.1,
G GjZ PSLl+l(K).
(iv) Let G= Dl(K). Then, by 11.2.2 and 11.2.3, G is a group of
21 x 21 matrices generated by clements T satisfying
T'AT=A,
I .......
where
A = (OIL ).
lz 0 .
If the characteristic of K is not 2 all such matrices represent isometries
of the quadratic form
--"
, XIX-l +X2X-2 + . . . +XlX-l
J
f F"'''
I
\
and so G is a ubgroup of 02l(K, fn). Now G is generated by matrices
t
\
(I+t(eii-e-i, -i),
I-t(e-t. -j-ejt),
I +t(et, -j-ei, -i),
I-t(e-1,.j-e-i, i).
 orI'Ii
o < i < j,
i
i
It is easily seen that these generators leave invariant the above quadratic
form aJso when K has characteristic 2. Thus G is a subgroup of 02l(K,fn)
in all cases. Moreover it is shown in Ree [1] that the above mat_i
gC!lf;rate D.2l(K, fn), the commutator subgroup ofo;;ri(,-lrTIius--
Pc
t _
i-
f;
f
G fJ.2l(K,fn)
and it follows by 11.3.1 that
G  GjZ  PfJ.2l(K,fn).
\
. (iii) Let G = Cl(K). Then G is a group of 2/ x 21 matrices generated by

,-J.
l'
f-'-'

t
!€C '-


'.;;.1
1
t;J
,
o-C"
:
 }
!

L_.
(
I'

\_---
,
i
(. .-
'",,-
i __'"
*
186
SIMPLE GROUPS OF Lm TYPE
FURTHER PROPERTIES OF THE CHEV ALLEY GROUPS
187
elements T satisfying T'AT=A, where
and it is shown in Ree [1] that these matrices generate D.2z+1(K,IB), the
commutator subgroup of 02l+1(K, In). Thus C,;;; D2Z+1(K, In) and
G G/ZPQ2l+l(K,llJ).
-'
A =Cl ;).
Such matrics are elements of the symplectic group Sp2z(K). C is generated
by the matrIces
I +t(etj-e-j, -c),
I-t(e_I, -j-ejl),
I +t(et, -j +ej, -c),
o < i < j,
I-t( -e-I, j-e-j, i),
I + te" -i,
1+ te-j, I.
But these matrices generate the symplectic group Sp2z(K) (cf. Ree [I]).
Thus CSp21(K) and
G  GjZ  PSp21(K).
ii) Let G = Bt(K) and suppose the characteristic of K is not 2. Then
G IS a group of (2/ + 1) x (2/ + 1) matrices generated by elements T satisfying
T'AT=A, where
2 0
0 0
A=
0 Iz
o
Now suppose K has characteristic 2. In this case G can he considered
as a group of linear transformations of a vector spacc )) with basis
[10, VI, . . . , Vl, V--l, . . . , V-Z, where
eap. Va = Vp.
Now in the set of generators for G described in 11.3.2 (ii), the terms
involving ew and e-t. 0 vanish when' K has characteristic 2. Hence the
subspace W of lJ with basis VI, . . . , Vz, V-I, . . . , V-Z is invariant under G.
We show that C operates faithfully on ill.
We remarked in j I .3. I that the ccntre Z of G consists only of scalar
multiples of the identity. Howevcr, the elements of G satisfy T'AT=A,
where
o
Iz
2 0 0 (----,
0 '....- ' )
0 I, /
/
A= I
! 1 ll
I
\
and AI can only satisfy this if A 2 = 1. In characteristic 2 this implies that
A= 1. Hence Z = 1 and C is isomorphic to G. It follows that G is simple.
(We exclude the exceptional case B2.(2).) Thus C acts faithfully on W.
Now the quadratic form In in characteristic 2 is non-degenerate but
has defect 1 (see section 1.6). The vector space 11 on which In is defined
therefore has non-singular symplectic subspaces of codimcnsion 1, and W
is sllch a subspace. We therefore compare the action of C on W with the
action of the symplectic group Sp2l(K).
Now the group C acting on l}) is generated by the folJowing matrices
(using the fact that -1 = 1 in K):
I-+t(eij-e-j, -t),
1- t(e-f, -j- ejl),
I-+t(et. -j-+ej, -'l),
1- t(e-t, j - l:'-j, t),
1-+ t2et, -t,
1-+ t2e_f, f.
(/
(I I,
o I, 0 -<I-
Such matrices represent isometries of the quadratic form
2
Xo + XIX-l +X2X-2 + . . . +XIX-1
and so C is a subgroup of 021+1(K,!B). Now G is generated by matrices
I +t(etj-e-J. _I),
I-t(e-I, -j-eJI),
I +t(el, -:-j-eJ, _I),
I-t(e-i. j-e-j, I),
l+t(2elO-eo, -1)-t2eC. _I,
1-/(2e-t, o-eo, f)-t2e_t. t

188
SIMPLE GROUPS OF LIE TYPE
If these generators are compared with the ones in 11. 3.2 (iii) it is clear
that (J is a subgroup of Sp2l(K). Moreover if K is a perfect field each
element of K is a square and so G=Spzl(K). Thus
(J = Sp2l(K) = 02l+1(K, I B)
as in section 1.6 and so
Bl(K) ';;; PDZl+1(K, In),
since (J is simple.
Now suppose that K is not perfect. The situation is now more com-
plicated since the orthogonal group 02l+1(K, In) is a proper subgroup
of Sp'll(K) (cr. section 1.6). However it is easily checked that the given
generators of the group (J lie in 02l+1(K, In). The commutator subgroup
D.zl+1(K, In) is generated by 'orthogonal transvections' corresponding to
elements x E'V with IlJ(x) EK2, except possibly when /=2 (cf. Dieudonne
[1], p. 59). The generators 1+ f2et, -t and 1+ t2e-t, i are orthogonal
transvections of this kind and it is shown by Dieudonne [4] that the
group C contains all such orthogonal transvections, so must contain
D.2l+1(K, In). Thus C lies between D.2l+1(K, In) and OZl+1(K, In). Since
G is simple we have G=D.zl+1(K,IB) and so
Bl(K)  PD.2l+1(K, In)
in this case also.
.
--....
CHAPTER 12
Generators, Relations and Automorphisms in
Chevalley Groups
] 2. 1 A Theorem of Steinberg
The Chevalley group G = 1L(K) is generated by a set of elements Xr(/)
for all n:c: (I>, f EK. We consider the problem of finding relations involving
these generators which are sufficient to define G as an abstract group.
Such a system of relations has been discovered by Steinberg.
Let lir(f), 11,{f) be the elements of G defined as in chapter 6 by
h,(t)=.p, (
I1r(t)=4>r ( 0
-t-1
-l ).
).
,,.
By 6.4.4 we have
nr(f) =Xr(f) X-r( - f-l) Xr(t),
hr(t) = I1r(f ) I1r( - 1)
and these equations show how to expn:ss I1r(f) and hr(t) in terms of the
generalors of G.
Now the following relations have been shown to hold in G:
X,(fl) x,-(f2) = Xr(/1 + (2),
[xs(u), Xr(f)]= IT Xir+js(Cijrs( -t)£uj),
i,j>O
I1r(/l) hr(f2) = I1r(flf2),
lIt2#O,
.
where Cijrs are integers defined as in 5.2.2 for linearly independent
roots r, s. Steinberg's theorem shows that these relations are almost
suflicicnt to define G abslractly, the only additional relations required
being ones to ensure that the group so defined has a trivial centre.
189
---.

Ii..
190
 '-'
1_,
_J
[I
.
'",-q
I:
t
t '""-
L",
f"
L
[.
r-
Ld
td
L.

r- ..
f
L."
l
SIMPLE GROUPS OF LIE TYPE
GENERATORS, RELATIONS AND AUTOMORPHISMS IN CI1£VAllEV CROUPS 191
1
. 
if
!I
,;
;f
THEOREM 12.1.1. Let 1[ be a simple Lie algebra with JLAI and
let K be a field. For each root r of JL and each element t of K introduce a
symbol Xr(t). Let G be the abstract group generated by the elements x,.(t)
subject to relations
where
\Ve shall show that the corresponding reJation
iir(t) xo(u) i'ir(t)-l= Xwr(g)(7]r, st-ArBU)
holds in C. Suppose r, s are linearly independent. Then
ji,-(t) xs(u) iir(t)-I
=x,-(t) x-r( - t-I) Xr(t) X8(U) Xr(t)-I X-r( - t-I)-I Xr(t)-I.
The commutator relation shows that this can be expressed as a product
of terms of the form Xir+1s(V), where j> O. Now there is a positive system
(1) I- in l1) containing all rools of form ir + js, where i, j are integers and
j>O. With respect to SLJch a positive system we have
flr(t) xs(u) izr(t )-1 E 0,
XWr(8)( 7]r, 8t-ArBll) E O.
But 0 is an isomorphism from fj onto U and we have
O{iir(t) xs(u) iir(t )-1) = O(XWr(S)( 7]r, I5t-ArBU)).
iir(t) Xs(ll) iir(t )-1 = XWr(S)( 7]r, st-Ar8U)
I
i'
i:
i
I
Xr(tI) Xr(t2)=Xr(ti +t2),
[Xs(U), Xr(t)] = n X!r+1s(Ct1rs( - t )tu1),
i,j>O
"r(tI) I1r( t2) = I1r(1It2),
11,-(t)=flr(t) iir( -1)
jjr(t) =Xr(t) X-r( - t-I) Xr(t).
Let Z be the centre of G. Then GjZ is isomorphic to the Chevalley group
G=JL(K).
tlt2 i= 0,
and
PROOF. (a) There is a homomorphism e of G onto G such that
O(Xr(t))=Xr(t).
We must show that the kernel of (J is Z.
As usual, let <D be the set of roots of JL and <D+ be a positive system
in <D. Let a be the subgroup of G generated by the elements Xr(t) for
rE<D+. Then the commutator relation shows that each element of -0 can
be expressed in the form
xrl(h) Xrlt2) . . . XrN(tN),
Hence
as required.
If r, S are not linearly independent we argue differently. Suppose r=s.
Since <}) is a simple root system of rank greater than 1 (we have assumed
lL =/:- AI), there exists rl E <1>, rl  r sllch that (r, rl) > O. Then r does not
begin the ri-chain of roots through it. Let r.2 begin the 'I-chain of roots
through r. Then r=irl +r2 for some i> o. Also
Cnrlr2 = Mrlr'},! = :i: 1
where rI, . . . , rN is the set of positive roots. Thus
O(Xrl(tI) . . . XrN(tN)) =Xrl(tI) . . . XrN(tN).
However, by 5.3.3 each element of U is uniquely expressible in the form
Xrl(tl) . . . XrN(tN). Thus the map 0 : D U is an isomorphism.
(b) We now consider the expression
nr(t) X8(U) llr(t)-1
in G. Since nr(t)=hr(t) flr we have
nr(t) xs(u) llr(t)-l=hr(t) nrX6(U) n;1/Zr(t)-1
=hr(t) XWr(6)(1]r, sU) hr(t)-1, by 7.2.1,
=xWr(s)(7]r,8tArpU), where p= Wr(S), by section 7.1,
=XWr(8)(7jr, t-ATBU).
by 5.2.2. Now the commutator relation shows that
[Xh{t2), xrl(h)] = II Xtrl+1riCi1rlr2( - tI)1 t).
i,j>O
We now tranform each term in this relation by iir(t). Let <D+ be a positive
system in <I) containing all roots of form
iWr(r1) + jWr(r2),
iO, jO.
Since
fir(t) xlu) iir(t)-1=xwr(8)(7]r, st-Arsu)
whenever s :i: r, it is evident that all the transformed terms lie in a with
the possible exception of
'-lr(/) Xr(Cnrlr2( -tI)i t2) iir(t)-1.

192
SIMPLE GROUPS OF LIE TYPE
GENERATORS, RELATIONS AND AUTOMORPHISMS IN CHEVALLEY GROUPS 193
Hence this term lies in 0- also. Since Cilrlra t:= 0 it follows that
iir(t) Xr(U) iir(t )-1 E 0
for all UEK. Also we have
But
XWr(r)( Tjr, rt-Arru) E D.
Using the relation just proved we have
ii,(t) h:;(u) -iir(t)-1=i1r(t) iisCu) iisC -I) iir(t)-l
= iiwr(s)(-fJr, st-Anu) iiwr(s)( - r;r, st-An)
=hwr(s)(Tjr, st-AnU) iiwr(s)(1) nWr(S)(1)-1 hWr(s)(r;r\ IItAr.)
=hwr(s)(Tjr, st-Ar.U) hwr(s)(r;r1, sIAr.)
= hWr(S)(U).
--"\
B(iir(t) Xr(U) iir(t )-1) = B(xwr(r)( Tjr, rt-Arru))
and B : D U is an isomorphism. Thus
iir(t) xr(U) iir(t )-1 = X-r( - t-2u),
since Tjr, r= -I, Arr=2.
Finally suppose s= - r. Then, from the above formula we have
iir(t)-l X-r(U) iir(t)=xr( -t2u).
I-fence
and so
ilr(IYl=ii,{ -I)
iir(t) hs(u) iirU)-l=hwr(s)(u).
(d) Let iir=iir(l) for each rE<D. Let Il be the subgroup of G generated
by the elements h,{I) for all rE<D, t=lOEK, and N be the subgroup
genera ted by Il and iir for all r E (D. Since i1r{t) = hr(/) iir it is dear that
"r( I) E IV for aU 1 =I O. Since
iirhs{u) n;:l =hWr(s)(u)
it is evident that Il is a normal subgroup of IV.
Now B(N) = Nand B(Il) = H. Thus e induces a homomorphism from
JVI17 onto NIH, which is isomorphic to W by 7.2.2. This homomorphism
maps iirO to Wr. However it follows from relations proved in (c) that
iirnsiT;I=iiwr(s)(Tjr, s)=lzwr(s)(Tjr, s) iiwr(s).
! .
j.
I
!
But
Thus
iir(l) X-r(U) nr(t )-1 =Xr( - 12u).
iir(t) X-T(U) iir(t)-1= XWr(-T)(Tjr, -rt-Ar.-ru),
as required, since Tjr, -r= -I, Ar, -r= -2. The relation
iir(t) xs(u) iir(t)-I=xWr(s)(Tjr, st-Ar.u)
is therefore valid in G in aU cases.
(c) We now derive some consequences of the relation proved in (b).
Let r, S E (I) and t, u be non-zero elements of K. Then
iir(t) iiiu) iir(t)-1
=iir(t) xlu) X-s( -u-1) j:o{U) ilr(1)-1
=XWr(lJ)(Tjr. st-Ar3U) j'Wr(-S)( -Tjr, _sC-Ar. -3U-1) XWr(s)(Tjr, sl-Ar.u)
='twr(S}(rJr. st-Ar3U) 't-Wr(S)( - Tj;:-l, stAr.zrl) Xwr(s)(Tjr, BI-Anu)
=iiwr(B)(Tjr. st-Arau)
by 6.4.3. Thus
This implies that
iirH. fisH. (iirH)-l =fiWr(B)il.
Also
ii;:1 =nr( -I)=hr( -1) fir
and so
n;=lzr( -1)
and
(iirH) 2 = H.
Now the Weyl group W is defined as an abstract group by generators
U'r, rE <I), subject to relations
iir(t) HsCu) nr(t)-I=iiwr(8)(Tjr, Bt-AroU).
Now consider the expression
;ir(t) hsCll) iir(t)-l.
w; = I,
WrwsW;:-l = Wwr(s),
by 2.4.3. Also we have a homomorphism from IVI n onto W mapping
"rn to Wr, where
.....-...
(nrfi)2=R
iirH. iisfi. (iirB)-l =iiWr(B)n.

\.
----
r
[
[ .
i!:r>

fI- ,
I
",,,",I
J."
t- .
t"-",,
---"I

-t<.,,-,
, -
!
t...
(
L,
.........
-
)
t,,-
t
194
GENERATORS, RELATIONS AND AUT01-,1ORPIIlSMS IN CHEVALLEY GROUPS 195
SIMPLE GROUPS OF LIE TYPE
!t follow that this homomorphism is an isomorphism, and so RI R is
IsomorphIc to W.
. (e) Let 13 be the sugroup of G generated by 0 and n. We show His
III the normalizer of U, so that B= DEi. Now
hr(t) xs(u) hr(t)-I=i1r(t) fir( -1) xs(u) iir( -1)-111r(t)-1
=iir(t) XWr(8)( 1]r, s( -1)ATBu) ii1{t)-1
=Xs(1]r, s7]r, wr(S)t-Arp(-I)Ar.u), where p=wr(s),
=xs(tAr.u),
since 1]r, s1]r, wr(s) =( -1)Ar. by 6.4.3 and Ar, wr(s) = -Ars. Thus
hr(t) xsCu) hr(t)-I=Xs(tAr.u)
for all SE <D, and so n is in the normalizer of O.
Let Xr be the subgroup of G generated by xr(t) for all teK. Let n be
the fundamental system contained in <D+. For each root ren let Or be
the subgroup of 0 generated by Xs for all SE<})+ with s-:Fr. The com-
mutator relations show that
For each IV E J-V let
'y 1 = {I' E Q.>+; 11'(1') E <l)-I-},
'F 2 = {r E ())+; 11'(1') E (I)-}
o= n Xr,
T E 'i'l
and let
0-;;;= fI Xr.
rE'}" .
Then, since the map 8 : 0---4 U is an isomorphism, 0= 0:; 0;; by 8.4.1.
A]so, using the relations proved in (b), we have
i1wO:;if;;/ s:; 0,
where iiw is an clement of R correspondihg to WE JV under the natural
homomorphism. Hence
BilwB=BiiwHD:;O;;= BHJlw 0;% 0;;
= BnwO;;.
Thus each element gEG can be written in the form
g=br1-wu,
where 5EE and flEO;;.
(h) For each WE W we choose a fixed coset representaive irwEfiI nd
assume that ih = 1. We show that each element of G has a Unique expresSIOn
of the form
Or= n Xs,
.e'"
'if:.r
where the product can be taken in any order. Moreover, Xr and X-r
normalize Or as in 8. 1 .1. Thus iir normalizes Or also.
(f) We now show that B uBiirB is a subgroup of C for aU rED. As
in 8.1.4 it is sufficient to show that
firBiir £ 11 u BiirB.
g = it'llnwu
with fI'eD, hER, WEW, uED;;.
Suppose that
Now we have
firBiir= iirBjz-;l =firOfifi;l
=firXrOrHIi";:l = X-rOrH s:; X-rl1.
Moreover X-r £ E UEflrB; for X-r(t) EB if t=O, and if ti=O
X-r(t) =X,.(t-l) fir( - t-1) Xr(t-1)
=Xr{t-l) hr( -t-1) 11,-xr(r1)EBilrB.
Hence E u EiirE is a subgroup of G.
(g) It now follows exactly as in 8.1. 5 that
EfiB. BiirB s:; BniirB u BiiB
for all fiER, reD. Arguing as in 8.2.2 we then obtain RfJR£BfUl and
finally G=BRB.
fih/iwliil = u;h'.!,ilw./iz.
Then
8(l/) 8(h1) 8(i1wJ O(u1) = 8(ii;) 8(hz) 8(iiw.;) 8(ii.J.
By the uniqueness of expression in G proved in 8.4.4 we have
8(ii;) = 8(D;),
8(h1) = 0(n2),
Wl=W2,
8(iir) = 8(ii2).
Thus ;IWl = l1w2 and, since 8 : D---4 U is an isomorphism, we 1ave. ii = ii
and ih = liZ. It follows that hI = T12 and so the expression for g IS uIlIque.

1
n ti'= 1
i=l
,II
, f
j  I
!. t ..--.....
[1 i
i ,
.{ ;
! I
!' I
! ! -"""
i ;
L I
I-I
II
1- ;
I !
I I:
II:
I \:
II;
i I i
I ; t
I ! i 
r
II
I.
,if
I ! I
I I
! I
i I i
I
!
t
I
I
1 II
!
i 1\
I
I
""-"'"
196
SIMPLE GROUPS OF LIE TYPE
GENERATORS, RELATIONS AND AUfOMORPHISMS IN CHEVALLEY GROUPS 197
(i) We now consider the sugroup 1/. Let fir be the subgroup of G
generated by hr(t) [or all liQ In K. We show that B is abelian and that
E=Ep1J7P2" . BpI,
where IT = {PI, P2, . . . , pt}. Now
hr(t) hs(u) hr(t)-I=iir(t) i1r( -1) hs(u) fir( - I)-I fir(t)-l
= iir(t) h1O,(S)(ll) i1,{t)-I
=hs(ll)
by the relations proved in (c). Thus His abelian.
In order to show tha t
also. It follows that
1
H= IT lip,.
i=l
(j) We consider finally the kernel of 8. Let u'niiwu be in this kernel.
Then
8(u') 8(h) 8(n1O) 8(u) = 1.
By uniqueness of expression in G we have
we need the relation
1
H= n Hpl
i= 1
8(u') = 1,
8(h) = 1,
8(fiw) = 1,
8(u) = 1.
Hence w= 1 and so 1110= 1. Also i' = 1 and u= 1 since 8 : O U is an
isomorphism. Thus u'llllwii=h, and so the kernel of 8 is contained in H.
Let
hr(t) i1s(u) hr(t)-l =iis(tA'8U).
This can be derived from the relations proved in (e) as follows:
hr(t) ii;;(ll) hr(t)-l=hr(t) .fs(u) -\:-s( -u-1) xs(u) hr(t)-l
= xs(tA'8 u) x-s( - tAr. -8 u-1) XitA'8U)
=I1s(tAr8u).
Let hr(t) be one of the generators of E, If r E IT then certainly
1
hr(t)E IT Epl"
i=l
Ot.herwise r=H'(pt) for some PtEI1 and we use induction on I(w). There
 SED such that w=lVsw', where 1(1V')=/(w)-1. Let P=W'(PI).
hr(t)=hw8(p)(t)=11s!lp(t) ii;-l
=11s/1s(tApB)-1 hp(t)
=hs(t-ApB) hit).
Both hs(t-ApB) and hit) may be assumed to lie in
I
n = [I np,(tt)
i=l
be in the kernel of 6. Then
hx,(t) h-l=xr(t .IT ti')' where nt=Ap,r,
t=l
by the relations proved in (e). Applying 8 we have
xr(t) =xr(t p ti').
t=l
Hence
and so h commutes with .ir(t). Since the elements x,{t) generate G, h
must bc in the centre Z of C. Thus the kernel of 8 is contained in Z.
Ilowcvcr the ChevaJIey group G has trivial centre by 11.1.2. Thus Z is
the krncI of 8 and the proof is complete. .
1
n BpI
i=l
The group G defined in 12.1. 1 is called the universal Chevalley group
of type JL over K. It can be shown that in the universal ChevaIJey group
each clcmcnt of 11 has a unique expression of the form
npl(t1) np2(t2) . . . np,(tl)
; I
, I
! III
: Jt
i'
by induction, thus hr(t) is in
1
n H Pt
i=l
!;
l

198
SIMPLE GROUPS OF LIE TYPE
GENERATORS) RELATIONS AND AUTOMORI'II'SMS IN CIIEVALLEY GROUPS 199
\......J
'l
and that H is therefore isomorphic to a direct product of copies of K*.
the multiplicative group of K. A system of relations defining the Chevallcy
group G may be obtained by adding to the relations defining the universal
group () further relations defining the subgroup H in terms of the generators
hplt ).
IfE=Al, when the argument of 12.1.1 breaks down, it has been shown
by Steinberg that the result remains valid if the commutator relations are
replaced by the relations
Fir(t) Xr(U) ii,{t)-I=X-r( -t-2u),
where t, UEK and t#=O.
Let 0 be a universal Chevalley group and N be a subgroup of its centre
2. Then N is normal in G and the factor group G/N is a group generated
by elements x,{t) in which the relations of theorem 12.1.1 are valid.
Such groups 0/ N are also often called Chevalley groups. Thus all the
Chevalley groups for a given JL and K are factor groups of the universal
Chevalley group G' and they all contain as factor groups the 'ordinary'
Chevalley group G  0/2. The group G is often called the adjoint Chevalley
group of type E over K.
It is not difficult to determine which elements
if und only if
l
n ti1i1=1
il
for j= 1, . . . , 1. Now it was shown in section 7.1 that if P is the additive
group generated by the fundamental roots and Q is the additive group
generated by the fundamental weights then Q/P is generated by elements
lb. . . . . ijz subject to relations
\r;!j,"_ 
l
 Aij{!i=O.
il
"-I,,- 
If we compare this with the relation
l
[[ t;IJ= 1
i=l
we see that
l
n hplf.t) E2
i=l
if and only if tl, . . . , tl determines a K-character of Q/P. In fact 2 is
isomorphic to the group of K-characters of Q/P.
L,
l
n hpt(tt)
i=l
12.2 Diagonal, Field and Graph Automorphisms
of fj are in the centre 2. We know that these are precisely the elements
for which
l
n hpt(tt) = 1
i=1
\Ve turn now to a discussion of automorphisms of ChevaJIey groups.
An automorphism is uniquely determined by its effect on the generators
Xr{t ).
L,.
,n
l".,_
i1 hptCtt)= il h(xPt, t,)=h (i1 Xp" t,)
as in section 7. 1. Th us
PROPOSITION 12.2.1. Let G=JL(K) be an adjoint Cheva/ley group.
A bijective map of G onto itself is all isomorphism provided it preserves the
relations
in G. Now
[
l 1
n hpt(tt) = 1 if and only if n Xp" t, = 1.
i=1 i=1
EXI. X,.(lI) Xr(t2) =Xr(tl +t2),
fA2. [Xb'(U),Xr(t)]= n Xt,.+1s(Cijrs(-t)fuj),
i,jr,60
fA3. hr(tl) hr(t2) = hr(11/2), h, t2 #= 0,
. where hr(t) = llr(t) l1r(-I) and n,{/)=xr(t) X-r( -t-1) x,-(t).
This is so if and only if this character takes value 1 at each fundamental
root. Now
\..-
r
-
l
n XPt, t,= 1
i=l
PROOF. Such a map determines an automorphism of the universal
ChcvalJey group G by 12.1.1. Such an automorphism leaves Z invariant
ince this is a characteristic subgroup of G. lt therefore induces an auto-
morphism of G/Z, which is isomorphic to G. .
"JoOi.-
X t (pj) = t(Ptl PJ)/(p"p,) = t;1u
p" 'J J .
Thus
{
I
L__.

200
SIMPLE GROUPS OF LIE TYPE
We next describe some particular kinds of automorphism.
DIAGONAL AVTOMORPHlSMS. It was shown in section 7. I that G is
normalized by II in the group of all automorphisms of JLK. Thus if
heX) EfJ the map
gh(X) gh(X)-l
is an automorphism of G. If hex) is in fI but not in H the automorpl1ism
is called a diagonal automorphism. If the automorphisms of 1[/( are
represented by matrices with respect to the Chevalley basis the diagonal
automorphisms of G are obtained by transforming by suitable diagonal
matrices.
FJELD AUTOMORPHISMS.
Then the map
Let f be an automorphism of the field K.
Xl{t)xT(f(t)),
rEW, tEK,
can be extended to an automorphism of G. To show this we must verify
that the relations !Jf1, !Jf2, !Jf3 are preserved. This is clear for !Jf1 and !Hz
and, since hr(t) is mapped into hr(f(t», for!Jf3 also. The automorphisms
obtained in this way are called Held automorphisms of G.
GRAPH AUTOMORPHISMS. Automorphisms of this type arise from
symmetries of the Dynkin diagram. A symmetry of the Dynkin diagram
of JL is a permutation p of the nodes of the diagram such that the number
of bonds joining nodes i, j is the same as the number of bonds joining
nodes p(i), p(j) for all i i=- j. The non-trivial symmetries of the connected
Dynkin diagrams are indicated in the figure.
Al
fj
E6

0-----0-=---.0-- -----
<>-0-0-----_ -<)
o
)
-
a==:::D
D4
82
G2

....--......

0"---'1 o
GENERATORS, RELATIONS AND AUTOMORPHISMS IN CHEVALLEY GROUPS
201
...--.,-."
PROPOSITION 12.2.2. Suppose all the roots of J[ have the same length
(ll/{} let p be a symmetry of the Dynkin diagram of lL. Then p determines
(l permutatioll of the corresponding fundamen!al. syster:z D.. Let T be he
iiI/ear tram/ormation of  into itself which cOlllczdes wzth tIllS permutatzoll
0/1 II. Theil T is an isomelry of 13 and T(<P) =(1).
PROOF. Let ntj be the number of bonds joining the nodes corresponding
to PI, pjED. Then 4(pt, pj)2
ntj=---
(Pi, Pi)(PJ, Pj)
by section 3.4. Now nij=np(i)pUJ and, since all the roots have the sa
length, this implies (pi, pj)2 = (p p(t), P p(j»)2. Since (pi, Pj)  0 for all Z,)
wllh i -# j we have (Pi, Pj) = (p p(t), p pU»),
i.e. (Pi, Pj) = (T(pi), T(pj)).
Thus T extends by linearity to an isometry of t>. Now T(Pi) = P p(t) .and
I (" -l-W It foHows that TWT-1EW for aU WEW, SInce
t 1crc!ore TWptT - Pp(i)' )
H' is generated by its fundamental reflections. Let rEeD. Then r=w(Pi
for some IVE W, PiED. Thus
T(r) = TW(Pi) = TWT-1. T(Pi) E wen) =<.1>.
-'''''>
Hence T{<l)) = <P.
If T is as in 12.2.2 we write T(r)=f for rE:cD.
PROPOSITION I . 2.3. Suppose J[ is a simple Lie algbr wl:oe roots
aJ/ have the same length, and let rf be a map of <I> into llself arzsrng from
a .!J)'mllletry of the Dynkin diagram of'lL. Then there exiJ't numbers Yr= :t 1
slIch tllat the map xr{t)Xi'{Yrt)
can be extended to an automorphism of G. The Yr can be chosen so that
')'r= 1 if rE n or -rEn.
.
PROOF. We use the isomorphism theorem for simple .Lie algebras
i\'cn in 3.5.2. Take JL'=J[ and P=Pp(t). Then there eXists an auto-
morphism of JL such that
ere,.,

e-r---+- e-r,
hrhi'
/

202
SIMPLE GROUPS OF LIE TYPE
GENERATORS, RELATIONS AND AUTOMORI'HISMS IN CIIEVALLEY GROUPS 203
1. ..
, I
--
,- Thus
-..
and so
r -
l
erl -+ Yrle,l'
erz-+YrZe;:2'
Then the transformation maps a into b, b into a and 2a +b into 2b +0,
\\ hil:h is not a root. Thus the symmetries of the Dynkin diagran:s of
Iyp Bz, Gz and F4 do not in general give rise to graph autom?rphlsms.
However, it is in fact possible to find such graph automo.rphlSl:1S over
artain fields K, although the form of these automorphlsms IS more
complicated than that given in 12.2.3. We shall dscribe graph, auto-
morphisms of the groups B2(K) and E1(K) when K s  prect held of
characteristic 2, and a graph automorphism of Gz(K) If K IS a perfect
field of characteristic 3.
for rEn. We show that, for each rEeD, er-+y,.ei' under this automorphism,
where Yr=:!: 1. This is true for the fundamental roots. If r is positive
but not fundamental we use induction on h(r). r can be written as
r = r1 + r2 where r1, r2 E <1>+ and h(r1) < h(r), l1(r2) < h(r). By induction we
have
Nrl. r2er-+YrIYr2Ni'1' i'zei'
Yr= YrIYr2Ni'I' '2_ + 1.
Nrl. r2 -
, .
er-+ Yrei',
hr-+ hi'
LEMMA 12.3. 1. Suppose <1) is all indecomposable self-dual root syste,."
of type Bz, Gz or F4. Let p be the non-tn',vial symmetry of the DYIl/Wl
diagram. Theil if n = {Ph. . . , PI} is a fundamental system of <I),
{pp(l), . . . ,Pp(l)}
;s a fimdamental system of <1) dual to I1 (cr. 3.6. 1).
PROOF. The angle between P p(t), P (J) is the same as tle angle betwee
P. P since Pis a synlmetry of the d1t\gram. Also P p(l) IS a long root 1
I, J .. . .
and only If pc IS a short one.
For negative roots we use the fact that [ere-r] =hr. This shows that
YrY-,..= 1, so that Y-r= :!: 1 also.
We now transfer to an arbitrary field K. Since we have an automorphism
of JL under which
.
L_,
jf

for all rE (I>, the same map determines an automorphism of J!.K for any
field K. Let this automorphism of JLK be denoted by (). Then
()xr(t) ()-l=() exp ad (ter) 8-1
=exp ad ()(ter)= exp ad (Yrtef)=Xf(Yrt)
by 5.1.1. Since the elements Xr(t) generate G it follows that 8 normalizes
G in the automorphism group of JLK. Hence the map
g-+8g8-1
is an automorphism of G which transforms Xr(t) into Xf(yrt).
We obtain in this way automorphisms of order 2 of Al(K), l 2; DI(K),
14; and E6(K). We also obtain automorphisms of order 3 of D4(K). .
12 3 2 Let r E (I), where (I> is as in 12.3. 1. Suppose the co-root
LEMMA .,.
Irr satisfies
1
hr= L, Ilihp,.
i ,=1
....\;!-,iIp'"
t,,,
Thm the element
;= = L: l1iP p(i)
i=l
is ill (1).
12.3 Graph Automorphisms of BlK) and FlK)
PROOF. <I> is self-dual and {p1,...,pl}; {pp(lh...,Pp(l)} are dual
fundamental systems. Thus
1
L 11ip p{i)
i=1
'--
If the roots of 1.. are not all of the same length the situation is more com-
plicated, since the argument of 122.2 and 12.2.3 no longer applies.
The linear transformation of Y which extends the permutation of the
fundamental roots is no longer an isometry, nor does it map the root
system into itself. For example, consider the case JL=B2 and
cI>+={a, b, a+b, 2a+b}.
is in <1> if and only if
1
1: nthpt
i=1
has the form hr for some r E <D.
.

204
SIMPLE GROUPS OF LIE TYPE
Since <1> * * - <1> the - fIt..
d -, map r-+r 0 '*' mto itself defined by 12 3 2 lIas
or er 2. We observe that if . .
l
r=  ntp1.
i=l
then
-_  . (Pt, Pt)
r - £..J 11£ --.--- P p(£)'
i=l (r,r)
For example, supps 1L=B2 and Pl=a, P2=b as shown in Figure 7.
T!len fo eac1 rE.<l>, r IS the root in the direction obtained by reflecting
r 111 the Ime blsect1l1g a, b.
b
o+b
20+b
-0
o
-20-b
-b
Figure 7
, PROPOSTION 12.3.3. Suppose G is a Chevalley group B2(K) or F4(K),
H Iere K IS  .pc:/ect field 0/ characteristic 2. For each root r E <I) define
;\(/ ) to be 1 if r IS a short root and 2 if r is a long roof. Theil the map
Xr(t)xi{tA(P»), rE<I>, tEK,
can be extended to an automorphism of G.
PROOF. We show that this map preserves the relations l, 2, Pl3.
To prove that f!Il1 is preserved we must have
xp(t lA(f») Xp(l.} (i'») = Xp«t1 -/- t2)A(f»).
Th!s .is c1er if ;\f)= 1, and if ;\(f)=2 it is valid over a field of charac-
tenstlc 2 since t1+1;=(l1 +t2)2.
Consider now tbe relation f!Il3. Under the above map we have
nr(t )Xf(ti\(f») X-f« - t-1)i\(f») Xp(tA(i'») =np(ti\(f»),
I1r( t )/1r( ti\(r») /lp( -1) = 11;o(tA(r»),
GENERATORS, RELATIONS AND AUTOMORPHISMS IN CHEVALLEY GROUPS 205
....-.."
again using the fact that K has characteristic 2. It is now clear that 3
is preserved.
Finally consider relation 2. If r, s are roots such that r +s is a root,
the roots in the subspace spanned by r, s form a system of type A2 or
B2. (The former possibility cannot occur if 1L=Bz.) Thus it- is sufficient
to show that the following relations are preserved:
(a) [Xill), xr(t)] =xr+s( - Nr, BtU),
where r, s are of equal length and inclined at 2-n,/3.
(b) [xs(u), xr(t)]=Xr+B( -Nr, 8tU),
where r, s are short roots inclined at TTJ2.
(c) [xs(u), Xr(t)] =xr+s( - Nr, IJtu) X2r+s(Czlrst2U),
where r is short, s is long, and r, s are inclined at 37TJ4.
Now K has characteristic 2, so (a) becomes
[Xs(u), Xr(t)] =xr+B(tu).
-....
Applying the_given map we obtain
[XsCuA(l»), Xf(tA(i'»)] = Xr+s«tU)A (r+s»).
However
[XS(llA(S»), X;{tAU'»)] =Xf+S(tA(f)uA(S»)
and so we have a valid relation, since r+s=f+s and A(i)= ;\(S)= A(r+s).
(b) becomes [xs(u), Xr(t)]= 1 since Nr, 8= ::t2=O. Thus (b) is pre-
servcd.
(c) becomes
[Xs(u), Xr(t)] =xr+s(tU) X2r+B(tzU).
Applying the given map we obtain
[xs(u), Xf(tZ)] = .xr:tB(tZu2) );2r+s(t2U).
However
[Xs(u), xp(t2)] = Xf+s(t2U) Xf+2S(tZUZ).
Also r+s=f+2&, 2r+s=f+s and so (c) is preserved.
Since K is perfect, every element of K is a square. Hence the mono-
morphism pf G into itself which extends the above map is bijective, so
is an automorphism of G. .


\..",,-j
..-
[
f"
U
LJ
r'
f
t."..
'-'-
\U>J
t-
t
L.
.
!

,-,.,,
\
"--
r
!
L.
r
t
.
r=
r
\
{ .
206
SIMPLE GROUPS OF LIE TYPE
12.4 A Graph Automorphism of GlK)
We now consider the group Gz(K) and show that this has a graph auto-
morphism when K is a perfect field of characteristic 3. Let PI = a, pz = b
be fundamental roots of Gz. Then, for each r E <1\ r is the root obtai ned
by reflecting r in the line bisecting a, b.
Figure 8
It is considerably more difficult to prove the existence of the graph
automorphism for Gz(K) than for B2(K) and F4(K). There are two reasons
for this difficulty. Firstly, since K has characteristic 3 instead of 2, care
must be taken over the signs of the structure constants Nr. s. Secondly,
the graph automorphism of Gz(K) has a 'nice form' only if the structure
constants are chosen in a special way, whereas for B2(K) and F4(K) they
could be chosen arbitrarily.
PROPOSITION 12.4.1. Let G = Gz(K), where K is a perfect field of
characteristic 3. For each r E <!> define ,\(r) to be 1 if r is short and 3 if r
is long. Then the structure constants N,..8 of G2 can be chosen in such a
lvay that the map
Xr(t )Xy(t'\(f»),
extends to an automorphism of G.
r E cD, t E K,
PROOF. It is sufficient to show that the given map transforms all
relations l, 2, 9f3 into valid relations, since the perfectness of K then
impJies that the map induced on G is surjective.
GENERATORS, RELATIONS AND AUTOMORPHISMS IN CHEVALLEY GROUPS
207
Now the relations 9fl are certainly preserved. Since
nr(t )lli(tA(f»),
Izr(t )hf(tA(r)),
.  I ed The non-trivial relations £J1!2 to be
the relations ;;n3 are a so preserv .
considered have the following forms:
(a) [xs(u), x,{t)] =Xr+s( - Nrstu),
where r, s are short rots inclined at 1T/3.
(b) [Xill), Xr(t)]=Xr-ts( -Nn;tu),
where r, s have the same length and are inclined at 21T/3.
(c) [.\''s(u), Xr(t)] =xns( - CllTstU) X2r+s(C21T8t2u 3 2
X X3r+s( - C3Irst:JU) X3r+2s( - C32rst 1I ),
where r is short, s is long, and r, s are inclined at 571'/6.
Now (a) becomes [xs(u), Xr(t)] = I,
since Nr. 8= :t 3 =0. Hence (a) is preserved.
Applying the given map to (b) we obtain
[xlu'\(S»), Xp(t'\(f»)] =xnlJ( - NT8t'\(r+s)u'\(r+8»).
But [xs(u'\(!}, Xp(t"-(f»)] = Xi'+( - NT'. stA(r)llA(S»).
Since r+s=r+s and I\(r)=.-\(s)=.-\(r+s) we obtain a valid relation by
transforming (b) if and only if
Ni. s=Nr. 8.
We now apply the given map to (c). We obtain
[x,(u), Xt(t3)] =Xr+B( - Cllr8t3u3) XZT+s(CZIrstUU3)._ 3 Z
X X3r+s( - C31Tst3U) X3r+Z8( - C32rst U ).
We wish to know when this is a valid relation. However
[x,{u), Xr(t3)] = [Xy(t3), Xs(u)]-l
= {X.Hf( - CllSft3U) XUH( CZHrt3u2)
X X3S+i( - C31Sft3U3) X3H-2f( - C32Ut6113)}-1
= X3H2P( C328pt 6 u3) X3Hf( C3 Ui(3113)
x X2,Hf( - C2Upt3u2) XHP(CUSft3U)
=X3,Ht(C3 Uit3U3) X3HZy(C32Sft6u3)
x Xs-t-i(Cllsrt3U) X28+f( - CZHft3uZ),

208
SIMPLE GROUPS OF LIE TYPE
the terms involving s+f, 2.f+f commuting since
NHf, 28+;'= :t3=O.
CHEV ALLEY GROUPS
GENERATORS, RELATIONS AND AUTOMORPHISMS IN
209
Now we have
They then a/so satisfy the condition
C32rs::>:: C211f
. l nd r s are inclined at 57T/6.
whenever r is short, s IS ong, a ,
d condition follows from the other
PROOF. We show that the thir
two. We have

r+s=3s+f,
2r+s=3.f+2f,
3r +s=s +f,
3r+2s=2s+f.
Thus the transform of (c) is a valid relation provided
- CUrs = C311f,
CZ.Uf = iN!, fN9, HI'
-1 -l-N Nr r+sNr, 2r+s. -2Ns, 3r+s
-"2"' 11 r,8 ,
usi ng relations of the first and second types. However the relation
N Nr 2r+8 = - NrH, 2r+8Ns, 3r+s
r, 8 .
 _II""
C3lrs= - CllU,
C32rs = C21lf,
C21rs = C32U'
The first two sets of equations are clearly equivalent, since whenever
r, s are a pair of roots related as in (c) J";, f are also. Similarly tIle last
two sets of equations are equivalent. Thus the given map preserves aU
the relations provided
Nr, 8=Nf, I whenever r, s are as in (b).
( - C3lrs = CllBf whenever_r, s are as in (c).
C32rs = C2lU
is easily deduced from the fact that
(r+s) +(2r+s) +( -s)+( -3r-s)=O,
as in 5.2.1. Therefore we have
C211F= - iNr, r+sNrH, 2r+sN8 3r+8
= !Nr+8, rNr+8. 2r+8
---.
=tMr+8, r, 2
It is more convenient to write the first of these conditions in the form
- iNr, 8 = H,,, I whenever r, s are short roots inclined at 27T/3,
since both sides of this equation, when interpreted as rational integers,
now have absolute value]. (Note that -t= 1 in K!)
In order to complete the proof of 12.4.1 it is sufficient to prove the
following lemma.
= C32rs.
.. "bIe to choose the signs of the structure
\Ve now show that It IS pOSSI " . t. tied The result of
h t the iven conditIOns are sa IS .
constants Nr,8 so tag N, N. 2 +b and Nb 3a+b can
4.2.2 shows that the signs of Na. b, a, a+b, a, a ,
3a+2b
LEMMA 12.4.2. The structure constants of the Lie algebra Gz over
the complex field can be chosen so that
- Nr. s=Nf, 8
>
-
i" .....11.)01
!;;
whenever r, s are short roots inclined at 27T/3,
- C3lrs = CllSi'
'-j
whenever r is short, s is long, and r, s are inclined at 57T/6.
-30-2b
Figure 9
i
(

'-
.f
-
.
["'.'.
-.-
-
[w
r
.\.--
t,,,,,
f
t"
'-
L
f .
i
\.--
#
I
l.__
r-
--
,"-.
r-
t..,
.....lV
SIMPLE GROUPS OF LIE TYPE
be chosen arbitrarily and that the remaining N" , are then determined.
Define I = i 1 by
Na. b= £I,
Na, a+b = 2 2,
Na. 2a+b = 3 3,
Then
Nb. 3a+b = 4.
l 3
Na+b, 2a+b= -3 ___
4
by 4.1.2 (iv) and the remaining N" , may be calculated using 4.1.2 (i),
(ii), (iii).
Now if r, s are short roots inclined at 2",/3 so are -r, -s and the
reJation - IN,, ,= Nt. 1 is equivalent to - l N--<. _, = N-t, _I. Thus it is
sufficient to check this relation when (r, s) is (a, a+b), (2a+b, -a) or
(a+b, -2a-b). In all these three cases the relation holds if and only
if 2 = - q.
SimilarJy, if r is short, s is long, and r, s are inclined at 5",/6 the same
is true of - r, - s; a nd the relation - C31,. = C "Ii is equ i valen t to
- C3, I, -r. -,= C" I, -I, -,. Thus it is sufficient to check this relation
when (r, s) is (a, b) (2a+b, -3a-b) (a, -3a-b) (2a+b, -3a-2b)
(a+b, -3a-2b) or (a+b, -b). In the first four cases the relation holds
if and only if <2 = - <3 and in the last two cases if and only if <2 = _ <4.
Thus the structure constants satisfy the required conditions if and only
if - <2= <3= <4. This completes the proof. .
It may be useful to give an explicit cboice of the structure COnstant,
for which the graph automorphism has the form given in 12.4.1. Thi,
is done in the table below. The structure Constant N" , appears in the
row indexed by r and the column indexed by s. In this presentation We
have chosen l= -1, 2=-1, 3=], E"4=1.
12.5 Automorphisms of Finite Chevalley Groups
Tbe automorphisms so far described, the diagonal, fieJd and graph
automorphisms, together with the inner automorphisms, are sumeient
to generate the whole group of automorphisms of G= 1f.(K) if the field
K is finite. We shaJJ now prove this result, wbich is due to Steinberg.
;UTOMORPHIsrviS IN CIIEVALLEY GROUI'S
GENERATORS, RELATIONS AND
211
b, 2a b, 3a b, 3a 2b
b a+b,2a+b.3a+b,3a+2b, a, b. a
a, , 2
3
-I -2 3 -1 1
tl -1
b 1 3 -] -2
3 I
'H.b 2 2 -2 1
+b -3 -3 -I 1
311 1- b -1 -1 -1 1
3CJ t. 2b 1 2 -3
-3 -2 -1
.-- Cl 1 1 -1
--b 1 -2 -3
2
-ll-b -3 -1 -I 3 3
-2l1--b -2 2 -1
- 3,1 - b 1 -1
1 -1
-.3a 2b
I JL is simple and K = G F( q).
1 LtG - JL(K) lV len I I
THEOREM ] 2.5.. . - Then there exist inner, diagona, grap 1
L ,{ 0 be an aulomorp/usm of G. () _ . / 'i'
t . . d g f such fhat - J( 2 .
lIlid field aufomorp/usl7ls " , ,
. Then U V are Sylow p-sub-
PROOF Let q be a power of the P}nme p. any tw Sylow p-subgroups
. 4 10 By Sylow's t leorem, . .. "te
groups of G, by 9. . . . .1 a Sylow p-subgroup of G, so IS conJug
arc conjugate. Now 8(f!) IS a s . automorphism h of G such that
lo U. Thus there eXlsts1 an mlleI () U) = U. Now Ole V) is a Sylow
O(U)=i (U). Let ()l =i1 ().. Then l( Tl () (V) =x-IUx for some
1 0 onJug'lte to U. IUS I B W
p-subgroup of G so IS C . t  .n the form x=hnwZl, where bE , WE ,
can be wnt en I
XEG. Ho:vever, x 'So b normalizes U we have
liE U- as m 8.4.3. mce
w -1
()1(V)=U-Jnw Unwu.
V -1 Hence U nu-1n-;;}UJ1-wu= 1
Now Un V = 1 and so ()1(U) n ()l )  2. 1 this implies that W trans-
. o.tfoIlowsthatnwUn;}nU=1. y... t thusw=woby2.2.6.
an I 0 . oot into a negative roo,
fnrms every posItive r
Therefore nl Unw=J1--;o1Ullwo = V.
I 'sm Y---Hr1yu
L t . be the inner automorp 11
. I t()(V)=lrIVu. e 12 V
II follows t 13 I '-1 8.>(U)U and O,(V) . _
of G and let 82=12 01, Then ("') d similarly we have G(V)- VH.
Now 1(;(U)=Ul-I by 8.5.2. Ill, n t Also UHnVl[=J[ by 7.1.3
Ul-I d VII mVdnan 0 t s the
Thus O2 leaves an UH = B invariant, it permue
. d '0 8'J(H)=H. Since 82 leaves. . B 8 3.2 these have the
an s - 0, b -, ps contalI1lI1g B. y .
millimal parabohl; su grou

-uou...c. .....uuJ:'s OF LIE TYPE
fq.rm
P{r)=B UBllrB, rEII.
We shall show now that P{) n V-x
" l' - -r. We bave
P{r}=B UBl1rB
= (B U BllrB) 11;:1
= BIl;:l U B/lrBn;:l
=BIl;:l UBX_r.
Consider tbe subset Bn-l n V W 1
r . e lave
BIl;:l n V = Bn-l nn TT-l
l' wounwo
c (B -}
/1, nwo nnwoU) 11;
c (B -]
However I1r nwoB n BnwoB) nw.
Bn;:lnwoB nBnwoB=rp
by 8.2.3, tbus Bn;:l n V-,I. also H
-'f' c . ence
. p{r) n V = BX-r n V =(B n V) X-1'= X-r
SInce B n V = 1 by 7.1.2.
Now 8z maps V into itself and
fundamental roots r E II. Thus 8 permutes the subgroups Per} for the
One can show in a similar way ZtJ::;utes the subgroups X-r for rEfI.
rE II. However if r d. . Z permutes tl1e subgroups X for
Ii " s are Istmct fundament'l r
ormula implies that [X X J _ 1 b a roots, the commutator f
that [Xr, X-r] 1. Thus"if r-Ssri 7xers the rsu1t of chapter 6 implies ,.
this relation is preserved by 82 we hav:' -8] = 1 If and only if rs. Since .
f
i
I
i

1
f
}
2
r
f
t'
02(Xr) = Xp(r),
82 (X-1' ) = X-p{1'),
where p is a permutation of II.
Let 8z.xr(l) -x (I) L b
- per) 1'. et X e tbe K-cbaracter of P defined by
X(p(r)) = 11', r E IT
Let d be the diagonal automorphism .
r E II,
of G. Then we bave
Yh(x) yh(X)-l
82. xr(I) = d. Xp(1")(1).
-f
i
I
f Let 83 =d-182. Then

g
4
I We shall show that

i 83.x-r(I)=X-p{r)(I).
. Let 8a.x-1'(1)=X-p(1')(A). We prove that A=l by using the homomorphism
I from SL2(K) into <Xr, X-r). We have
1
f
I
GENERATORS, RELATIONS AND AUTOMORPHISMS IN CHEVALLEY GROUPS
213
83. X1'(1) = Xp(1') (1 ).
( :)(-: )( :)(-: )( :)(-: )
and it follows that
Xr(1) x-r( -1) xr(1)=X-r( -1) x1'(l) X-1'( -1).
Applying 83 we obtain
Xp{1")(l) x-p(,')( - A) Xp{r)(I)=x-p{r)( - A) Xp{r)(1) X-p(1')( -- A).
However
( :)L )(
L )( :)(-
l)=(l-A
1 -A
0) (I-A
I = A2_2A
2-A),
I-A
lA)
and, since the kernel of the homomorphism SL2(K)<Xp{rh X-p{r»
contains only lz and possibly -/z, we have A= 1 by comparing the above
matrices. Thus
83 .xr(1) =Xp{r)(l),
83 .x-r(l) =X-p(r)(1)
and it foJIows that
83 (nr) = 83(x1.(1) x-r( -1) xr(1))=np(r).
We now show that the permutation p of n induces a symmetry of the
Dynkin diagram. Since 83(lI)=1I and 83(nrJ1s)=l1p(r)l1p{8), the order of
he coset I1rllslI is the same as the order of the coset flp(r)l1p(8)1l in the
group N/ll. Using the isomorphism betwecn N/ fI and V, the order of
H'rll's is the same as the order of wp(r)Wp(s) in W. Thus the number of
bonds joining the nodes r, s in the Dynkin diagram is the same as the
number of bonds joining p(r), p(s). Hence p is a symmetry of the Dynkin
diagram.
II
----...,
; -.....
: 
.
-,f--
1;'
f
y

r;",;..,:,;:
f
......" ::
)
.1


214
SIMPLE GROUPS OF LIE TYPE

-...... ?'
We prove next that there is a graph automorphism g of G such that
g(Xr)= Xp(r) for all rEf I. By 12.3.3 and 12.4.1 this is so proviud
K has characteristic 2 if 1£=B2 or F4 and p is not the identity, and K
has characteristic 3 if 1£ = G2 and p is not the identity. Suppose p is the
non-trivial symmetry of the Dynkin diagram of 1£, where 1£=B2 or F.I.
Then there exist roots a, bE IT interchanged by p such that the roots
which are linear combinations of a, bare
l
:ta, :tb, :t(a+b), :t(2a+b).
f'''''''
L
Now nbXan'b1 = Xa+b. Applying 03 we have 11aXbn-;;,l = 83(Xa+b). I-Tenee
03(Xa+b)= X2a+b. However [X2a+b, Xb]= 1. Applying 8"31 we obtain
[Xa+b, Xa]=].
But
['
--.

[Xa+b(l), xa(l)]=x2a+b(-Na, a+b).
Thus Na. a+b=O. But Na, a+b= :t2 and so K has characteristic 2.
Now suppose 1£-= G2 and p is the non-trivial symmetry of the Dynkin
diagram of 1£. Then p interchanges the fundamental roots a, b of 1L and
the other roots are
[
t'
:ta, :tb, :t(a+b), :t(2a+b), :t(3a+b), :t(3a+2b)
Now nbXall'b1 = Xa--l-b and so, applying 03, we have naX b11;;1 = 03(Xa+b).
Thus 03( Xa+b) = X3a+b. Also naXa+bn1 = X2a+b and, applying 03.
nbX3a+bn'b1= 03(X2a+b). Thus 83(X2a+b)= X3a+2b. Now [X3a+2b, XbJ= I.
Applying 831 gives [X2a+b, Xa] = 1. However
[X2a+b(1), xa(I)] =X3a+b( - Nu. 2a+b).
Thus Na, 2a+b=O. But Na. 2a+b= :t 3 and so K has characteristic 3.
Thus in each case there is a graph automorphism g of G such that
g(Xr)=Xp(r) for rEn. Let 84=g-183. Then 0iXr)=Xr and 8-1(X-r)=X-r
for aU rEn. Also 84 fixes Xr(1) and x-r(1), thus 84 fixes 11r for aU rE If.
It foHows that 04 fixes each Xr, rE<D. For r=w(s) for some SEH and
W=Wrl'" wrk with rtEn. Hence
Xr=nrl . . . nrkX8111 . . . nrl1
[
[...._--
, .
r--
L.
[
and so 84(Xr)= Xr.
Let r, SEn be fundamental roots which are joined in the Dynkin
diagram. Then r +S E <1>. Let
84. xr(t) =Xr(f(t)),
04.Xs(t) =xs(g(/)),
04. Xr+s(t) = Xr+s(h(1 )).
f
'-


-'_.
f
!
C.
GENEH.t\TIJlI.S, HELATIONS AND AUTOMOHPIIISMS IN CIIEVALLEY GROUPS 215
Since
[XsCu), Xr(t)]=Xr+s( -Nr, stU). . . ,
we have, on applying 04,
[XsCg(u)), Xr(J(t))] =xr+s(h( - Nr, sIll)) . . . .
But
[Xs(g(u)), xr(f(t))]=xr+s( - Nr, sJ(t) g(u)) . . .
and so
(
i,
i
I
i
I

t
i
t
i
I
f
i
I
Ii
f
f
I
I
1
f
i
f
j
!
!
!
h( -Nr, slU) = -Nr. sJ(t) g(u).
Now Nr, 8= :t 1, hence h(tll) = J(t) g(ll) for all t, UEK. Putting u= 1 we
havcJ(t)=h(t) and putting 1= 1 we have 'g(u)=h(rt). Hence
J(t) = get) =11(/)
for all t. We also haveJ(lu)=/(/)/(U) for all I, u. Since the map/: K-+K
is bijective and additive it must be an automorphism of K.
Now JL is simple and so its Dynkin diagram is connected. Thus there
exists an automorphism J of K such that
04. Xr(t) = x1-(J(/))
fnrall rErr.JCan be extended to a field automorphism ofG. Let 05=/-104.
Thcn
85.xr(t)=Xr(/)
for all rErr, tEK. Since these elemcnts Xr(t) generate G, 05 must be the
idcntity.
Now
05 = /-104 =/-1g-103 = /-1g-1d-182 = J-lg-1c/-1i210l
= J-] g-1d-1ii1 ill O.
Therefore 0 = hi2dgJ and the theorem is proved.
III
(We note that the proof that the functionJ on K satisfies
.I{tl) J(/2) = /(/1t2)
breaks down if 1£=Al since we cannot choose two distinct fundamental
roots. It is not difficult to find an alternative argument to cover this
C1.ISC.) . .
The result of 12.5.1 is also valid for perfect fields whIch are not finite.
Ilowc\'cr the proof is more dilTicult there as Sylow's theorem cannot
c used to show that the automorphism can be modified to fix U and V.

CHAPTER 13
The Twisted Simple Groups
'!'Ie lave shown that the Chevalley groups of type AI, BI, CI, DI can be
IdentIfied with certain classical groups. However only some of the classical
groups can be interpreted as Chevalley groups. Even over a finite field
there re classical groups which are not Chevalley groups, for example
the unltry group (cL .section 1.5) and the second class of orthogonal
groups In even dImensIOn (cf. section I .4). We shall describe in this
hcpter how to construct certain additional simple groups, the so-called
tWIsted tyes'; some of which can be identified with classical groUjJS.
Every classIcal grup over a finite field can be interpreted as a Chevalley
group or as a tWIsted group. In general, the classical groups which are
Chevalley groups or twisted groups are the linear and symplectic groups,
and the orthogonal and unitary groups corresponding to forms whose
Witt index is sufficiently large. The remaining classical groups can be
interpreted as 'non-split' groups of Lie type, but the discussion of such
groups is beyond the scope of the present volume. In addition to the
extra classical groups, we obtain as twisted groups several new families
of exceptional groups.
The twisted simple groups were discovered independently by Stein-
berg, Tits and Hertzig. The development we shall give follows Steinberg's
approach.
13.1 The Reflection Subgroup WI
The twisted groups will be obtained as certain subgroups of the Chevallcy
groups G = JE.(K). The twisted groups only exist in the cases when the
Dynkin diagram of JL has a non-trivial symmetry. It will be shown that
the twisted groups are also groups with a (ll, N)-pair and that the Wcyl
group WI of this (8, N)-pair is a rcllcction group which is a subgroup
of the Weyl group W of G.
Before discussing the twisted groups themselves we shall therefore
first consider the Wcyl group v in order to describe how the reflection
subgroup WI anses. W operates as usual as a Euclidean reOectiol1 group
on an [-dimensional space 1),
216
THE TWISTED SIMPLE GROUPS
217
I
t
;
Ii:
I
.

f


;
to


t
;
i

i
.

I
I
1

i

t
1
i
i
E
(

I

i
f_
i
f
j
}
j
!
j
Let p be a non-trivial symmetry of the Dynkin diagram of JL Then
there is a unique isometry T of 1J such that T(r) is a positive multiple
of p(r) = f for all fundamental roots r E 11. T satisfies the conditions:
T(r) = f if all the roots of 1. have the same length. A e J
t- ; if r is short, for 1L = Bz or F4,
T(r)= v/2
v/2 ; if r is long.
t- ; if r is short, for Jl = Gz,
7(r)= v/3
v/3 ; if r is long.
1l is clear that the order of T as an isometry of 19 is equal to the order of
p as a permutation of n.
-\
'i
Definition. We denote by 1JI the set of VE'}} such that r(v)=V.
For each v E1'J we denote by VI the projection of v on to the subspace
1)1. Then vI is the average of the vectors in the orbit of v under T. For
this average is certainly in VI, but since
(v, x) = (T(V) , x) = (TZ(V), x)= . . .
for XEVI the average has the same scalar product with x as v does.
We no consider the relation between 7 and W. Since -r{r) is a positive
multiple of ; for all r E n we have
"''''''''\
TWrT-I = W;"
TEn.
Since the fundamental reflections generate W, this s.?s_thaL1:.!!.<2!:.mal!s
Win !he group of all isometries of 'P.
Definition. We denote by WI the set of WE W such that T-l=
LEMMA 13.1 .1. ?Eat!-faIJ[l!!1._'2_'
PROOF. Let WE WI and VE1'JI. Then
T. w(v) = W7(V)=W(v),
Thus u'Cv) EVl, and so WI transforms 1)1 into itself.
-",
"'oa..

218
SIMPLE GROUPS OF LIE TYPE
""- '''-....,..-
Now suppose WE WI and w ¥' 1. Then there exists a root r E <1>+ such
tha 1'(r)E <1>-. However, the transforms of r by the powers of 7 are also
positive, .tus r1 >-0. Similarly w(r1) = w(r)l -<0. Thus w transforms
ome, posItIve element of 1 into a negative element, so cannot be the
identIty.
..
J
 '



&-A
We now show that WI is generated by elements of order 2.
f'
f
L-,;
PROPOSITION 13. 1 .2. Let J be an orbit of IT under p. Let W J be the
subgroup o! W generated by the elements Wr for r E J. LeI w be the element
of W J which transforms every positive root in <I> J into a negative root
Then .E wI, and WI is generated by the elements  for the differell;
p-orblts of IT.
r
r
I
l."
PROOF. Since 7r7-1=W#,EWJ for all rEJ, we have 7W7-1EWJ for
all WE W J. Thus 7W07-1 E W J. However, 7 preserves the sign of each root.
ili '
7%7-1(<1> j) = cI> J.
Hence 7%7-1=W'6, and it follows that %E WI.
Now let WE WI with W¥' 1. Then there exists a root rEIT such that
w(r) E ,<1>-. Let J be the p-orbit of n containing r. Then w(s) E (D- for all
S.E J SInce 7 presrves the sign of each root. However ,, changes the
sIgns of aJI roots In <D J but of none in <I> - cI> J. Hence
,,.......
!
c -.
1 (W%) = I (w) -I (ui).
We show that WI is generated by the elements  by induction on
I(w). If WE WI, w¥' 1, choose J as above. Then by indution we have
wW'6=W'61%2. . . %k
for certain orbits Jj of ll. Thus
W=%1%2... w-6k%
as required.
\-
,
We show next that the elements w-6 are reflections when restricted
to '}91.
LEMMA 13. 1 .3. »'({ coincides with Wr1 on 1 for each root r E J.
.
THE TWISTED SIMPLE GROUPS
2.19
PROOF. Let r, s E J. Since r, s are in the same p-orbit of n, r1 is a
posilive multiple of Sl. It follows that the projections r1 for rE (I)} are
all positive multiples of one another. Now
l'vi(r1) = wi(r)l
and H(r) E (1)7 for all rE <1>1. Thus H,itr1) is a negative multiple of r1
and, being an isometry, lv6 must transform r1 into -r1.
Now let VE'}J1 satisfy (r1, v)=O. Then (Sl, v)=O for all SEJ and so
(s, 1»=0 also. Hence w(v)=v. Thus 11"6 cQincides with the reflection
Irr1 on 1)1. ·
COROLLARY 13.1.4. The reflections WT1 of 1)1, for all rED, generate
,he group rVl of isometries of 111.
13.2 The System cI>1 in VI
We have seen that WI may be regarded as a reOection group operating
on 1)1. We denote by <I) 1 the set of vectors r1 ElF for all r E <1>; and by III
the set of vectors r1 ElF for all r Ell. We shall show that £1) I. behaves
rather like a root system for JY1 and that n 1 behaves like a funda-
mental system of <1)1. However, both in (1) 1 and 111 there can be positive
multiples or a vector distinct from the vector itself. In order to control this
!tiluation we introduce an equivalence relation on <I).
LEMMA 13.2.1. The sets w«I)f)form a partition of (1) as w runs through
,he elements of WI alld J runs through the p-orbits 0/ n, r alld s are ill
the same set if and only if r1 is a positive multiple of Sl.
i
t
t
\
1
t
i
1
!
i
[.
!
PROOF. We show first that each root IS In some set w«1)f). Let Wo
be the clement of W which transforms every positive root into a pegative
root. Since 7 preserves this sign of each root, 71+'07-1 transforms every
positive root into a negative root. Thus 7lV07-1 =wo and WoE V1.
Let r E (1)+. Then wo(r) E $- and by 13, 1 .2 we have Wo = \V1 . . . wtk
for certain p-01-bits J1, of II. Thus there exists an integer i such that
,,Hl . . . wk(r) E <1>+,
W6fW6H1 . . . lVik(r) E <1>-.

220
SIMPLE GROUPS OF LIE TYPE
However the only positive roots transformed by %i into negative roots
are those in <D, and therefore H't+l . . . Hl6k(r) E cJ:>.t.. Thus
r E 1I'k . . . H't+l«P ),
-rEH,k.. . Wt+lWf«PJ;).
Hence every root is in one of the given sets.
Now if r, s are two roots in <Pf, we have seen that r1 is a positive multiple
of SI. Transforming by WE WI we see that w(r1) = w(r)1 is a positive
multiple of II'(Sl) = w(s)1.
Suppose conversely that r, s E <1> and that r1 is a positive multiple of
Sl. Now we have shown that r E w«I>f) for a suitable element WE WI
and a suitable subset J of IT. Then 1V-1(r) E <1> j and so w-1(r)1 is a positive
element which is a linear combination of roots in J. However
w-1(r)1 = )V-1(rl) = w-1(As1) = Aw-1(s)l
for some A> 0, and so w-1(s)l is a positive vector which is a ]inear com-
bination of roots in J. The same must be true of w-1(s), as J is a p-orbit.
J-Icnce W-1(S)E <I>f and SE w«l>f). Thus s is contained in each of the
given sets in which r is contained. Therefore the given sets cover <I) and
any two of them either coincide or have no elements in common. .
We can now describe the extent to which <1)1 and TI1 act as a root system
and fundamental system for WI acting on 1P.
PROPOSITION 13.2.2. (i) <D1 spans 1)1.
(ii) Every element of (fJl is a linear combination of elements of TII with
coefficients allnon-Ilegative or all nOll-positive.
(iii) A basis of 1)1 may be obtained by picking one element of TI1 olll
of each set of positive /12l1/tiples.
(iv) If r1 E (1) 1 thell there is an element of WI 'which coincides with Wr1
on IP.
(v) If r1, Sl E (1)1 then WrI(Sl) E <1)1.
PROOF. Since <]) spans '-17 it follows immediately that <])1 spans l)1.
Since each element of <1> is a linear combination of elements _ of 11 with
coefficients all non-negative or all non-positive, the same is true for
(1)1 and n 1. Let it, . . . , Jk be the p-orbits of n. If we pick out one element
of 111 from each set of positive multiples we obtain a set r}. rJ,.. . . ,d,
where r1 E <1» by 13.2. 1. These elements are linearly independent, so
form a basis for tP.
THE 1 WISTED SIMPLE GROUPS
221
..-!"--'.
Let riE ()">1. We show there is an element of WI which coincides with
"'r1 on 'VI. Now r is contained in one of the sets w(<1)j) of 13 .2.1. We
may assllme that r=w(s), where SEJ, WE WI, since r will be changed
only by a positivt. multiple by taking SE J, and Wr1 wdl be unchanged.
Then
r1 = W(S)l = W(Sl).
Now H'81 coincides with an element of WI on -:pI by 13.1.3. Hence
Wr1 = WWs1W-1
coincides with an element of WI on 1)1 also.
Finally, let r1, Sl E <])1. There is an element WE WI such that Wr1 coincides
with IV on 1)1. Hence we have
WrI(sl) = w(Sl) = w(s)! E <1>1
and the proof is complete.
.
13.3 The Structure of Wi
.'\

i.
I
1

t



t
i
i
!
\
i
t
t
g

f
f
!
1

.
We now describe the structure of WI in the individual cases which arise.
Let rl. . . . , rk be a set of roots, one from each p-orbit of TI.. Then
1 r1 are linearly independent vectors in 1)1, and JVl, consIdered
'1"'" k . .
as a group of isometries of 'VI, is generated by Wl' . . . Wk' y conldef1g
the angles between the vectors rI, . . . , r1 it is usually possIble to IdentIfy
JV 1 with one of the Weyl groups of rank k.
13 .3 . 1 Type A r
Let the fundamental roots be Ph . . . , Pl, numbered as in the diagram.
1 2 [-1 [
0--------0-- - - -- ---0---0

t.
Suppose / is odd and write /=2k-1. Then the vectors r}, . . . , dare
'!(Pl + P2k-l), '!(P2+ P2k-Z), . . . , -!(Pk-l + Pk+1), Pk.
These form a fundamental system of type Ck. Thus WI is isomorphic
to W(Ck).
F""";t
\

222
SIMPLE GROUPS OF LIE TYPE
'--' J
Suppose I is even and write 1=2k. Then the vectors rl, . . . ,rl are
  1
'!-(PI + P2k), 'l(P2 + P2k--I), . . . , '!(pk + Pk+1),
which form a fundamental system of type Bk. Thus 1-VI is isomorphic
to W(Bk).
r
t:
[
r'
L.
,
l
''''''P'"'
lw'
--
;
L.
f!
j
I
le_
i
L.
,
'. [
-
r-

-
L
r--
'-,../-
13.3.2 Type D,
The symmetry in this case is as shown in the diagram.
1-1
---)
(
The vectors ri, . . . ,rl are:
PI, P2, Pa, . . . , PI-2, -!(Pl-I + PI).
These form a fundamental system of type B,-I. Thus WI is isomorphic
to W(B,-I).
13.3.3 Type E6
The symmetry is as shown in the diagram.
b8
4
The vectors ri, . . . , r are:
-!(PI + Pa), -l(P2 + P5), Pa, P4.
These form a fundamental system of type F4, thus WI is isomorphic
to W(F4).
13.3.4 Type D4
This time the symmetry has order 3.
2E<:)
4
THE TWISTED SIMPLE GROUPS
223
The vectors r1, . . . . rt are:
pI, -l(P2 + Pa + P4).
These form a fundamental system of type G2. Thus TVI is isomorphic
to W(G2).
13.3.5 Type B2
In this case k= 1 and so W is isomorphic to WeAl), a cyclic group
or order 2.
13.3.6 Type 02
k = 1 in this case also, and so W is isomorphic to J¥(A1).
13.3 .7 Type F4
The situation here is a littIe more complicated. Suppose PI, P2 are
the long fL1ndament1 roots and P3, P4 the short ones, as shown in the
diagram.

0-----<) 1----0
234
t .
Then the vectors r}, . . . , rl may be taken as:
t(PI + yl2p4), !(P2 + yl2pa).
The angle between these two vectors is given by
cos e _ _€_ + yl2€p + yl2p)__.
-I PI + yl2P4 1.1 P2 + yl2P3 I
Now it is readily verified that
(PI + yl2P4, P2 + yl2p3) = -I PI I 2,
I PI + yl2P4 I = yl2 I PI I ,
! P2+ yl2pal =21 PI I sin '/T/8.
-1
cos {}=--.
2y12 sin 7T/8
But 2 sin 7T/8 cos 7T/8=sin 7T/4= l/yl2, and so
cos {}= -cos 7T/8=cos 771'/8.
Hence
Thus 0 = 77T/8.

224
SCMPLE GROUPS OF LIE TYPE
THE TWISTED SIMPLE GROUPS
225
. It foUows that in this case WI is not one of the Weyl groups of the
Jmple root systems. t is generated by two reflections in a plane in axes
111c1med at 77T/8, and JS therefore a dihedral group of order 16.
13.3.8
13.4 Definition of the Twisted Groups
""'1
tct G be the ChevaUey group JL(K) and p be a non-trivial symmetry
or the Dynkin diagram of JL. We suppose that K is a perfect field of
characteristic 2 if 1L =Bz or F4 and that K is perfect of characteristic
3 if JL = Oz. Thus there is a graph automorphism g of 0 such that
g(X,,) = X;:
for all rEn, where f=p(r) (see 12.2.3, 12.3.3 and 12.4.1). Now g
commutes with each field automorphism f of -G. For if all the roots have
the same length we have
.
. The type ?f ':1 can conveniently be memorized by identifying nodes
III th Dynkm dlaran corresponding to fundamental roots in the same
p-orblt. The DynkIll dIagram of WI is given in the following table.
l1t
82
c;
f.;
gf(x,-(t ))=g. xr(f(t)) =x;:(f(t)) =f. x;:(t )= fg(xr(t))
A2k-1
Diagram of W
____>k-1 k
<>---0----0-----
2..f-1 ..f+ 1
------0--4
O:-O-o---...-
2k ..f+1
1-1
----
[
1 2 1-2.. /-1
0---0----0- -----a:::;co
Diagram of W 1
for rEn; whereas if JL=Bz, Oz or F4 we have
1 2 k-1 ..f
o---o---o-,----
gf(xr{t)) = g. xr(f(t)) =xi(f(t )A(i») =xr(f(tA(f»))
= f. Xi(tA(r») = fg . (xr(t))
for rEn.
Lct /l be the order of the symmetry p. Then 11 is either 2 or 3. If all the
roots of JL have the same length then
gn.Xr(t)=Xr(t), rEn,
and so gn = 1. Otherwise 1L = B2, 02 or F4 and 11 = 2. In this case
g2. Xr(t) = g .Xi( tAfT») =Xr(tA(T)A (i») =Xr(tp),
where p is the characteristic of K. Thus g2 is the field automorphism
\\hich raises every coefficient to the pth power.
Let a be the automorphism of G defined by a= gf, and suppose that a
non-trivial field automorphism f is chosen so that an = I-then a will
have the same order as p. Since an = gnfn the condition to be satisfied
by f may be stated as follows:
fit = I if all the roots of JL have the same length,
pf2 = I if JL = B2, Gz or F4, where p is interpreted as the pth-power
map on K.
A
2k
Li-o-____1) 
I .
)
! f! ..
f.-
r -

f
.
£6

;?;:
;
g
 '1'!L



I
I
f
r
l
(
J
j

o

o
PROPOSITION 13.4.1. Let G=1L(K) be a Chevalley group whose Dynkin
tliagram has a non-trivial symmetry p. Assume that K is perfect of charac-
laislic 2 ifJL=B2 or F4, and that K is perfect of characteristic 3 if 1!.=G2.
Lei g be the graph automorphism corresponding to p and f be a non-trivial
f
t
r
:J
w19I
16
..............,

226
SIMPLE GROUPS OF LIE TYPE
ft.
.... "'---"
field automorphism chosen so that a = gf satisfies an = 1, where n is the
order of p. Then we have
(l(U) = V,
a(V) = V,
a(H)=H,
a(N)=N
r"

and a operates on NIH';;. W according to the formula a(wr)=W,.. for all
rEn.
PROOF. It is clear from the definition of a that a(V) = Vand a(V) = V.
Thus a(VH) = VH and a(VH) = VH since VI-I is the normalizer of U
and VH is the normalizer of V. Also a(H)=H since FI= VH n VI/.
Now N is the subgroup of G generated by H and the elements nr fOf
rEn. Also
a(nr) = a(xr{l) x-rt -1) xr(I)) = xf{l) X-f( - 1) Xf(1) = 11,..
for rEll. Hence a(N)=N. It follows that a induces an automorphism
of NIH';;. Wsuch that a(wr)=w,.. for all rElI. .

['
-
t-'"
Po
L..
ThU.YlistIOu.p3!defi.? as cei!1_.o_g9.PS of t_valley
lY.hicll_ill_fj2'ed !emeiY.lse=-Y-__J!tQill2!.£!si1ii"-9f  type
22.!2gdered .i!L.1.A......Let G be a Chevalley group admitting an auto-
morphism a as in 13 .4.1. Then a fixes V, V, II and N. We define sub-
groups VI, VI, G1, ]{1, N1 as follows:
"tto 
- '---
t""
..- ;1:
f-
F.
L
Definition 13.4.2. (i) V 1 is the set of elements x E V such that
a(x)=x.
(ii) VI is the set of elements xEVsuch that a(x)=x.
(iii) G1 is the subgroup of G generated by VI and VI.
(iv) HI is the intersection of G1 and H.
(v) N1 is the intersection of 01 and N.
C-,_..
,.-"
.- L
:0-:1';
W
J
it
<t----'
t;
We shall show that Gl is (apart from a few exceptional cases) a simple
group and that the subgroups VI, VI, HI, Nl play an analogous role
in G1 to the subgroups V, V, H, N of G. Although every element of
Gl is fixed by a, Gl is not necessarily the subgroup of all a-invariant
elements of G.
Now the operation of a on W is the same as the operation of trans-
formation by the isometry T determined by the symmetry p. Thus the
subgroup WI consisting of all elements WE W such that a(w)=w is just
the subgroup considered in section 13.1. We shall show that WI plays
the role of the WeyI group of G1.
..,"'U..... :1
r-
'=...:-..v
r
.--..'
[
r,__
THE TWISTED SIMPLE GROUPS
227
. f (B N)-pair in the Twisted GroupS
13.5 EXIstence 0 a, 
. subsets weeD j) form a partition of (D
It was shown m 13.2.1 that the I 1 tl c -orbits of n. We shall
I h WI and J funs t HOUg 1 1 P
as w runs t lfOUg . 1- _ .1' sscs a little more closely. The
I . d' 'd al eqUIvU cncc C Ll
now look at t 1e m IVI u I tl '" POS'lt'lve rools in the system
. 1 1 r(tP--) are le
roots in the eqlllva ence c ass 1 J . I(}) The type of this
. t t the fundamental system H .
lr«I)J) with respec 0 _ I' tl  fllndamcntal system J. The
. h. 'lS the lypc u l L 0
root system IS t e SLlme '- ',' oJ be obtained by inspectIOn
various possibilities for the type 01 } may edSl y
of the Dynkin diagrams, and are as btHO}WS'f n has type Ai or Al x AI.
D or £6 each p-or I 0
IfJ[.=Azk-l, Z , A 1f1L=D JhastypeAl0r
Jf 1L=A2k, } has type Al x Al or 2. 4,
AIXAIxAl.
F. \1 .f 'JT - F4 J has
If 7( - G J has type Gz. ma y, 1 JL...- - ,
If 1!-=B2, J has type B2. jl..,- 2,
type Al x At Of .82: Is 'n (1) may be regarded as a positive
Thus every eqUivalence c ass I G
A A x A x A t A 2 B2 or 2.
system of type At, Al x 1, iI' ,
e one of the equivalence classes in (J) defined
LEMMA 13.5.1. Let S b b J G generated by the subgroups Xr
in 13.2. 1. Let X s be the Sll group OJ
for all rES. Then
i
\
\
i
\
i
t
t

Xs= 11 Xr,
rES
I root') in S ill any order. Also a(Xs)=Xs,
where the product is tak.en/over t e J X'-. fixed by a satisfies X1# 1.
and the subgroup X. of e enle!lts OJ S -
h t S is the positive system, with respect to
PROOF. We have seen t a . d. <D The subgroup X s
. f TIe root system contame III . .
some ordenng, 0 so . 1 Ch vaHey group correspondmg to
therefore plays the role of U n t.le e
this subsystem of <D. The factorIZatIOn
Xs= IT Xr
rES
5 F each rES we have
therefore follows from the results of chapter . or
a(Xr) = Xi'
for some rES, thus it follows t a7 ;"fnt elements of X s., If S is
Now consider the subgroup i  o..a t S such that rP. If S is of
of type AI, A2, B2 or G2, there IS a roo rE

228
SlMPLE GROUPS OF LIE TYPE
type B2 or G2 we have for such a root
a .xr(l) =x,{l)
nd so x,{J) is a non-unit element of X1. If S is of type Al or A2 we have
Instead
a.Xr(J)=x,(Yr), Yr=:!: 1,
by 12.2.3. If Yr= 1, xr(l) is the required non-unit element of Xl. If Yr= _I
then a has order 2 (since a cannot have order 3) and therefore [has order 2
also. Now there is an clement t=l-OEK such thatJ(t)= -to T'hus
a .Xt(t) =Xr( -J(t)) =xr(t)
and x,{t) is the required element.
Now Suppose S is of type Al x Al or Al x Al X AI. Then we have
a.xr(1)=x,(Yr), Yr=:!: 1,
for any root rES, by 12.2.3 and 12.3.3. We consider the elements
xr(1) X(Yr), x,.(J) Xf(Yr) X/(Yf) when S has type Al x AI, Al X Al X At
respectively. Now
a C1
xr(l) --)X;{Yr)- X,-(YrY;:),
X,.( I )Xf(Yr)xf(YrY;:) Xl-(YrY,Y/)
in the two above cases, and so we have YrYf= 1 and YrY;,Y,= 1 respectively.
Thus
a.xr(J) x,(Yr)=X;:(Yr) xr(YrYi)=Xr(I) Xf(Yl.),
a .Xr(I) Xf(Yr) Xi(YrYf) =Xf(Yr) X,(YrYf) Xr(YrYfYT)
= X,.( J ) X f(Yr) xf(Yrl'f)
and so we have a non-unit element of Xs fixed by a in each case.
II
The elements of X1 will be described in detail in 13.6.3.
PROPOSITION 13.5.2. (i) For, each WE WI there exists n wEN 1 such
tha. nw corresponds to w under the natural homomorphism from N bao JV.
(11) Nl/HI is isomorphic to VI.
PROOF. (i) Since by 13.1.2 WI is generated by the elements w' for
al.' p-orbits J of n, it is sufficient to prove the existence of elements nw  N 1
lth w=w. Now one of the equivalence classes of<D defined in 13.2.1
IS ai«Pf)=<PJ' Let S=(P;. Then by 13.5.1 there is a non-unit element
XEXl. We express this element in canonical form 8.4.4, and obtain
x=u'nwll, U'EU, liEU;;;.
THE TWISTED SIMPLE GROUPS
229
(We have chosen the representative 11w here, as we may do, so that no
clement of Ii appears in the expression for x.) Now w#-I, for w= 1 would
imply that xEUJ-I n V=1. Since a(x)=x we have
u'nwu= a(u') a(nw) o{u).
Comparing the double cosets BnB containing these elements we obtain
o(a') = 11', then WE W1. Now x is an element of the Chevalley grup
gcnc:rated by the subgroups Xr for all rE cDJ. Thus the element IV occurnng
in lhe above decomposition of x must be in W J. Since wi: 1, w transforms
some root in J into a negative root. But J is a p-orbit of nand WE W1.
Thus HI transforms all roots in J into negative roots. Hence w = %, the
only element of W J transforming an roots in J into negative roots.
We can now show that I1w is the representative we require. We have
a(U;;)=a(U nn;;IVnw)= U nn;;IVnw= U;;,
since WE V1. Thus in the decompositions
x =u'nwu= a(u') a(nw) a(u)
we have u', a(lI') E U and 1I, a(u) E U;;. By the uniqueness of such a de-
composition we have a(lI') = u' and a(u) = lI. Hence
11 w = ll' -1 xu-I E GIn N = N 1
by 13.4.2, and so 11 w is the required element.
(ii) Consider the natural homomorphism from N to W. Each element
of N 1 is fixed by a, so is transformed into an element of W fixed by a,
i.e. an element of W1. However for each WE W1 there is an element
II/LEN 1 which maps onto lV, as shown in (i). Thus the image of NI under
lhc above homomorphism is VI. There is therefore a homomorphism
from NI onto WI with kernel NI nH=Hl. Hence fII is normal in
N 1 and N 1/ If I is isomorphic to WI. .
'\
;
'-
PROPOSITION 13.5.3. Each element of G1 has a unique expression
g=rt'llI1wu, where U'EUI, hE HI, WE WI, nwENI and llE(U;;)l the set
J a-invariant elements of u;; .
PROOF. By 8.4.4 we may write
g=u'hnwu,
')
where U'EU, hEH, WEW,-UEU;. Since o{g)=g we have
u'hnwu= a(u') a(h) a(nw) a(u).
I

-
1M
L
[
',,", .
[
r=
"..".,...
L"
r---
\
-
;
i
t#
7
i

!
\
,
.
230
SIMPLE GROUPS OF LIE TYPE
Comparing the double cosets containing both elements we have a(w) = 1\',
thus }VEWI. Hence nw may be chosen to lie in NI. Also a(u')EU, a(h) Ell,
a(u) E U;;. By uniqueness of the canonical form we have
a(u') = u',
a{h)=/z,
a(u)=u.
Thus U'EVI and UE(V;;)l. Finally
h=(U')-Igu-In;;IEGl nH=HI
by 13.4.2. Thus g has an expression of the required form, and the unique-
ness is clear. .
We shall now establish the existence of a (B, N)-pair in the group
01. We define B1=01 nB. Then it follows from 13.5.3 that BI=[/llll
and that HI is in the normalizer of VI.
THEOREM 13.5.4. The subgroups BI, N1 form a (B, N)-pair in G1.
PROOF. We verify the axioms BN I-BN 5 given in section 8.2. By
13.5.3 we have G1=B1N1BI, thus BI and NI generate G1. Also
BI nNI=GI nB nN =GI nH =lfl.
Thus B1 nNl is normal in N1, and the factor group NI/Bl nNl is
isomorphic to WI, so is generated by a set of elements w of order 2.
The elements w E WI play the role of the fundamental reflections, as
J runs over the p-orbits of D. We show next that n(ui) B II1(w)-l of: B I.
Let S be the equivalence class <I> f of (1). Then by 13.5.1 there exists
xEXk with xi= 1. Now
n(w) xn(W')-I En(}) Xln(wt)-I = Xs'
Thus ll(}) xn(})-I is a non-unit dement of VI, so cannot be in Bl since
BI n VI=l.
Finally we must show that
BIn(H-'(D BI.B1nBl S BIn(H) nBl UB1nBl
for all nENl and all p-orbits J of D. The element } can be expressed as
a product of reflections corresponding to roots in J. Let
=WYIWY2 . . . WYk'
ri E J.
Then by 8.2.1 we have
Bn(wr,) B.BnB £ Bn(wr,) nB uBnB
THE TWISTED SIMPLE GROUPS
231
and by applying this repeatedly we see that B]/1(w)Bl.BII1Bl is in a
union of double cosets BI1(w) nB, whcre w is an element of the form
'II' II'. and (Sl S') .,. Sit) is a su bseq uence of (1'1, r2, . . . , r k).
"'1 6a'" /I ," . . 1 .
Aba, by 13.5.3, G1 intersects BllwB 111 BIl/wBI If WEW , and 111 the
empty set if w 1: WI. Thus B In(IV'b) B I, B InEI is in a union of double
(t)cts of form B In(w) nBI, where WE WI n W J. However, the onl;
ckmcnts in JVl n W J are 1 and 'rv. For every non-unit element of v
1Jl WJ transforms some root in J into a negative roo., so transforms
clch root in J into a negative root since J is a p-orbit of II. It must there-
fore be \lf. Thus
B l11(W) B 1. B InB 1 £ B In(w(j) nB 1 U B 1nB 1
and a1l the axioms for a (B, N)-pair are satisfied. .
The results on parabolic subgroups proved in section 8.3 are now
\'aliJ for the twisted group G1. The parabolic subgroups of 01 are the
uhgrollps containing B 1 together with the conjugates of thesel subgru?s.
It r,)l1ows from 8.3.4 that there are exactly 2k subgroups of G contallllllg
BI. where k is the number of p-orbits of n.
13.6 The Subgroup U1
We shall now describe in detail the elements of the subgroup VI of GI,
and of the related subgroups VI, (U)l and X1.
,
, '
PROPOSITION 13.6. 1. VI = IT x1, where the product is taken over
all equivalcl1ce classes S which are in (1)+. (The terms. ill the, pro{ct rnay
be taken in allY order.) Each element of VI has a wuque expressIOn as a
product of elemellts of X1 taken ill this order. Similar results hold for VI
ami for (U;;)l for each WE WI. JYe have. '.
VI = n x1 with uniqueness, taken over all equwalence classes S w/uch
arc ill (1)-, alld
(U;)l = II X. with uniquel1es., taken over all equivalence classes S
such that S is in (1)+ and w(S) is in (1:>-.
,
 }
. I
PROOF. The equivalence classes S in <1>+ form a partition of (D+. Thus
n Xs= D_ Xr= V.
Sc<l>+ rE+
Also each etement of V has a unique expression of the form
u= n Xs
Sc+

232
SIMPLE GROUPS OF LIE TYPE
with XSEXS. Now a(X )- X  '
if and only if a(xs)=x; fr ;:cr;a equl\alnce class S. Thus a(u)=11
and so . liE U \\e therefore have x 'E 1/1
oS A..,.
UI= n vI
1 $J
s::=. :.-
as required.
SimjJar proofs can be given for VI and for (U-)I
W . .
In or?er to give descriptions o[ the elements l' .
a possIble, it is useful to make' h' ofXs whIch are as sImple
wIth respect to which t1 a c OIce of the ChevalJey basis o[ JL
. 1e constants Y - + l' d .
be gIven explicitly. r- - mtro lIced In 12.2.3 can
LEMMA 13.6.2. Let 1L be a simple Lie al b
[he same length and let r - b 1 ge ra. all of whose roots have
. . ' r e trle map of (j)" . .
non-trIVIal symmetry 01" the D k. d" mto Itself ansmg from a
I 'J ')in 1fJ Iagrarn of 1L TI '.
c lOOse a Clrevalley basis 01" ';I(' h . len It IS possible to
d 'J :Jl... 111 SllC a wall that tl
etermined by J Ie automorphism of J[
. hrhi', erei', e-re-f (r E TI)
satisfies erYrei' where - 1 l
I 'Yr - un ess the equivalence l S
r 1aS type A2 and r = f, in which case Yr = -1. c ass con tmil ing
PROOF. We observe first that ive
of sign of any set o[ root t' g n a ChevaJIey basis, the changing
f . vec ors er rE (1)+ do t tIi
o bemg a ChevaHey bas' . i' '. cs no a  ect the property
. cIS, ProVIC ed the sIgns of th
negatIve root vectors erE (1)- . Ie corresponding
S r" al e clanged also
uPJ:ose the given symmetry has order 2 T .
morphIsm transforms Iz e I'nt t1 _ I . hen the square o[ the auto-
T r, r 0 lcmsc ves for aJI n "
llUs, for each r E <I> we 11a ve < r E ,so IS the identity.
erYrefYry,er = er
and so Y Y = J T1
r i' . lUS Yr=Yf. Suppose r=/:- - If
may change the sign of [' r. er -ef and ef -er we
None 0 er, ei' but not the tl T1
ow suppose instead that r _ - d _ 0 ler. len Yr = Y;: = 1.
-r an r=s+s [or Some sE<l\ Then
[eses] = Ns. ser.
Applying the automorphism we have
YsYs[eses] = Ns, BYrer
THE TWISTED SIMPLE GROUPS
233
-----...
and, since YSYi= 1, this implies that Yr= -1. Note here that the equi-
,-alcnce class S containing r is S= {s, S, r}, which has type A2.
Now suppose that rE <1>+, r=r, but r is not expressible in the form
r=s+5. Then by inspection of the root systems (described in section 3.6)
it can be seen that there is either a fundamental root ri = rt E II such that
r-rtE<I>, or a pair o[ fundamental roots rt, PiEIl such that r-ri, r-ft,
r-ri-r.iE<l>, or r is itself in n. If rEIl tht?n Yr=1. In the first case we
have
[er-re' ere] = Nr-re, reer.
Using induction on the height we may assume er-reer-re' Since ereere
we have erer, and so Yr= 1. In the second case we have
Her-re-pc> ere] e;:e] = Nr-re-fe. reNr-;:eo feer.
By induction on the height we may assume er-rcfeer-re-;se' Since erferf
and e;:ee;:e we have erer. Thus Yr= 1 in this case also. We have shown
that Yr= 1 for all poshive roots r with r=f, r=/:-s+s, and it follows immedi-
ately that Yr = 1 for all negative roots r with this property also.
Finally suppose the given symmetry has order 3. If r= r then erYrer,
where Y; = 1. Hence Yr = 1. If r =/:- r then
...--.
"
erYre;:Yry;:e;YrYfYrer = er.
Thus YrY;:Y/= 1. If two of Yr, YP, Y/ are -1 we can, by changing the sign
of one of the root vectors er, e;:, e;, arrange matters so that
Yr=Yi'=Yi= 1.
.
If al1 the roots of J[ have the same length, we shall assume that a Clle-
valley basis for JL is chosen as in 13.6.2, where this is relevant. We can
now describe the elements in the subgroups X.
PROPOSITION 13.6.3. Let S be an equivalence class in <1>. Then the
elements in the subgroup X1 are as shown below. (For the field automorphism
f we write I(t) = i if all the roots of JL have the sam.e length, alld j(t) = to
otherwise.)
(i) If S= {r} has type Al then X1 consists of the elements Xr(t) with
t=1.
(ii) If S={r, f} has type AIXAI, lvhere r, r have the same length, then
X1 consists of the elements Xr(t) x,(f) for all tEK.
(iii) If s= {r, P, f} has type Al x Al X Al then X1 consists of the elements
xr( t) x,( i) xr(l) for all t E K.


234
SIMPLE GROUPS OF LIE TYPE
I
'''' "-"
(iv) If S= {r, f, r+ f} has type A2 then X consists of the elemcl/lJ
xr(t) xiI) Xr+f(U), where U+u= -N.r -tt-
,r .
(v If S={r, f) has type Al x AI, where r is short and i is long, thell r1,
COllssts o.f the elements xr(t0) x;{t) for all tEK. 4 ,
(VI) If S= {a, b, a+ b, 2a+ b} has type B2 then X1 consists of the element!
xa(tO) Xb(t) Xa+b(tO+1+u) X2a+b(U20)
for all t, UEK.
(vi,i) If s= {a, b, a+b, 2a+ b, 3a+b, 3a+ 2b} has type G2 then :\,"1
consIsts of rhe elements J j
..1
(
f
t;.
[,'0--,
-,-
Xa(tO) Xb(t) Xa+b(tO+I + UO) X2a+b(t20+I + VO) X3a+b(U) X3a+21lv)
for all t, u, v E K. (The structure constants here have been chosen as in seelion
12.4.)
f''''
l=
PROOF. (i) Since a.xr(t)=xr(l) Xl is as stated
(ii) Since's .
',,-
l..
a.Xr(t) xrCu)=xj{l) xr(u)=Xr(ii) xf(f),
the a-invariant elements are those for which u= f
(iii) Since .
j
L__
a.xr(t) Xj{u) x;(v)=xr(l) Xi(ii) xr(v)=xr(ii) x;.(f) x;(ii),
the a-invariant elements are those for which u= I v=1.
(iv) Since ' .
a.xr(t) xr(V) XrH(U)=xi{l) xr(V) Xr+f( -it)
=x,(v) xr(l) Xr+i( -Nr, riv-ii),
the a-invariant eIements are those for which v = I and U + U = - Nr tf.
(v) Here K has characteristic 2 and 202= 1. Now ' f'
a .Xr(U) Xr(t) =x;{u20) xr(tO)=Xr(t0) Xj(u20).
Thu the a-invariant elements are those for which u= to.
(VI) In this case also K has characteristic 2 and 2{)2= 1. We have
a.Xa(ta) Xl{tb) Xa+b(ta+b) X2a+b(t2a+b)
=Xb(t8) xu(tg) X2a+b(tb) xa+b(t :a+b)
=xa(t2) xito) Xa+b(tOtg+tga+IJ X2a+b(tOto+t1)'
i
I
t
;
i
t._,.,
. i
i
.\i.
THE TWISTED SIMPLE GROUl'S
235
uing the commutator formula. Thus the a-invariant elements are those
(Of which
t to t t20 t ,20,0+,0
a = b' b = (, u + II = a b a .1. b'
20 20 20
/2u+b='a lb +ta+b.
Puuing t=tb, U=tgct+b' we obtain the required form.
(vii) In this case K has characteristic 3 and 3(}2=1. We choose the
structure constants as in section 12.4. Then
0, -"aU n) XIJ(tIi) X a+li(t a+b) X2a+b(t2a+li) X3a+b(t3atb) X3u+2b(t3a-f--2b)
=X/I;() xaU-Z) X3a+/J(tol_/J) X3a+2/J(t+/J) Xr1-t-/J(t;H-/J) XZtI.l-b(tgU.12b)
=xa(tZ) Xb(tO) X(+b(tOt) X2a'I_IJ(/D'iO) x:I(I_I.l-tOtO)
X X3a+2i _t{JtD) X3Ct+lJ(t°f-/J) X3a-I-2b(t( I-b) Xa+IJUga+b)
X X2a+b(tga+2b)
=Xa(t g) xuCtO) Xa+b{tOt 2 + t ga+b) X2a+b(tOt0 + / ga+2b)
( 3030 30) ( U030+t30 )
XX3a+b -ta tb +ta-t-b Xgu+21J -ta tb 2a+b'
TI1US the a-invariant elements are those for which
1(1=t2, tb=tO, ta_I'b=tOtZ+tga_I_IJ' t2a+b=t0/0+tga+2b'
t3a+b= _tOtO + tl-b' tga+2b= _/OtO + t+b'
Putting t =tb, u= t3a+b, v= t3a+2b we obtain the required elements.
II
The subgroups X1 of the twisted groups are of importance because
they are the analogue of the root subgroups of the Chevalley groups.
We describe next the way in which the elements in a subgroup Xl combine
together.
PROPOSITION 13.6.4. (i) {fS={r} has type Al and we write XS(t)=xr(t)
IJze.-'ll
XS(tI) XS(t2) =xs(1I + 12).
(ii) If S= {r, i} has type Al X AI, !'-vhere r, i have the same length, (me!
h'C 1I',ile x s(t) = xr(t) Xf(i) then
xs(lI) xs(t2)=xs(1I +t2).
(iii) If S= {r, f, P} has type A 1 X Al X Al and we write
x s(t) = xr(t) xf(i) x;(l)
lilt",
XS(tI) XS(t2) = x S(tl + t2).

236
SIMPLE GROUPS OF LIE TYPE
(iv) if S= {r f r+ -} /;
" r 7aS type A2 and we write
xs(t, u) =x,.(1) xf(i) Xr+f(ll),
l'vhere u+ii= -N ti tl.
r. f , len
XS(1I, Ih) :Xs(tz, uZ)=XS(tI +tz, Ul +uz-N, _[ t)
( ) I _ r, r I 2 .
v if S= {r, r} has type Al x Al wher . h
write X.s{/) = xr(t 0) xr(t) Ihen ' e r IS sort and f long and (flit
XS(tI) XS(t2) = X8(/1 + t2).
(vi) If S= {a b a+b 2 b} /;
" ,a + ws Iype Bz and we write
a{/)=Xa(t°) -XIJ(/) Xa+I,{IO+l),
(J(u) = X U+b(U) X2a-l-/;(1l20),
xs(t, u)=cx(t) (J(u)
then
XS(t1' u1) XS(t2' u2)=XS(tI +t2 li +u +tOt)
'I 2 12-
(vii) ( S {b b
= a, ,a+ ,2a+b 3a+b 3a+2b} h
write " as type G2 and 1ft'
then
cx(t)=Xa(tO) XIJ(t) Xa+b(tO+l) XZa+b(/20+1)
(J(u) =x a+b(UO) X3a+b(U),
y(v) =X2a-l-b(VO) X3a+2b(v),
X8(t, li, V) = cx(t) (J(u) y(v)
XS(t1, U1, VI) XS(t2, liz, vz)
=XsU +t U +ll t t30
I 2' 1 2- 1 2 , VI +V2-tZUI +tltO+l_tit.
PROOF. These are aJJ .
mutator formula 5.2.2. straIghtforward consequences of the Com-
.
We have seen earlier that the ChevaJIe .
root subgroups X X co 11 l-r y group G IS generated by the
r, -r {I rarE We now
result for the twisted gro . prove an analogue of (his
ups.
PROPOSITION 13.6.5.
all p-orbits J of n.
Gl .
IS generated by the subgroups Xl + Xl - fi
cDJ' (lJJ or
THE TWISTED SIMPLE GROUPS
237
PROOF. Gl is generated by U 1 and V 1 by 13.4.2. U I is generated
by [he subgroups x1 for aJl equivalence classes S in +, by 13.6.1, and
V 1 is generated by the X1 for all S in <D-. Thus GI is generated by the
lJbgroups X1 for all equivalence classes S of <D.
Now each equivalence class S has the form w(<Dj) for some WE WI
and some p-orbit J of n (I3. 2. 1). For each WE WI, let nw be an element
of N 1 chosen as in 13.5.2. Let Gu be the set of u-invari,mt elements of G.
"I hen \Iy'e have
nwX1n;1=nw(Xs nGa) n;l= X w(S) nGa= X:LJ(S)
jncc IIwEGO". Thus Gl is generated by the subgroups X.} and the elements
nlf for aIJ 1VEYI. However, Vl is generated by the elements w for aU
po-orbits J of II. Thus Gl is generated by the XJ and elements n(H-')
(or all J. Now consider the subgroup <XJ' XJ>' This is by 13.4.2 a
t\\istcd group whose Weyl group is, by 13.1.2, <w>. By 13.5.2 this
twisted group contains an element l1(W) EN mapping to w. Thus G1 is
generated by the subgroups Xj, XJ for all J.
..,eiI\
I
.. $,
"
!
13.7 The Subgroup 111
We now turn to a discussion of the subgroup Hl= H n Gl. It turns out
to be quite troublesome to decide which elements of H are contained in
this subgroup. We begin by describing the operation of u on H.
LEMMA 13.7. 1. fl all the roots of JL have the same length then
u. hr(t) = hf(l).
r there are roots of different lengths then
. u.hr(t)=hrCt(f)O).
PROOF. Suppose all the roots have the same length. Then
u . Xr(t)=Xf(Yri),
where Yr = 1: 1. Thus
u.l1lt)=a(Xr(t) x-r( -t-1) xr(t)
=x;{y,i) X-Fe _y_,.f-l) x;(yrl)
= I1f(Yri)

./

238
SIMPLE GROUPS OF LIE TYPE
'-
since YrY-r = 1. Therefore
Ie
a. hr(t) = a(nr(t) .nr( -1)) =n,(yrf) l1i{Yr)-l
=h,(yrf) n,(1) l1,(1)-lhj(rr)-l=h,(i).
Now suppose there are roots of different lengths. Then
a. xr(t) = X,(tA(f)O).
 ;
;
[

f.f'..JIt" f
Thus
""-j
'
.i
:J
"N!? ;
J
1
:=J
a .nrCt) = a(Xr(t) x-r( - t-l) Xr(t))
= X,(t A (1')0) X_,,( - t -1);\(f)0) X,(tA(l')O)
=n,(tA(f)O)
since ( -1 )A(r) = -1. (The latter statement is trivial -if K has characteristic
2, whereas if K has cbaracteristic 3 >"(i) is either 1 or 3.) Therefore
a. hr(t) = a. (nr(t) n.T( -1)) = n,-{t A(f)O) 11;( -1) = h,-{t A(t) 0). I

"'- .
!
Now the subgroup H of G consists of elements hex), where X is a A',
character of P, the additive group generated by the fundamental root5
PI, . . . ,Pl. The K-characters X which give rise to elements of Il arc
those which can be extended to K-characters of Q, the additive group
generated by the fundamental weights ql, . . . ,q" by 7.1.1. \Ve shalJ
prove an analogue of this result for the subgroup HI of Gl in the cac
when all the roots have the same length. In such a case the symmetry p
of the Dynkin diagram determines an isometry 7 of 13, the real vector
space generated by $, such that 7{Pi)rc:::: Pp(i). 7 transforms every roor
into a root and we have 7(r)=f. Now 7 transforms the fundamental
Co-roots by 7(lzp()=hpp(l)' Since ql, . . . ,q, is the basis of 1) dual to the
basis IzPI' . . . , hpl (see section 7.1), we have 7(qi) = qp(i). Thus 7 permutcs
the fundamental weights in the same way 111at it permutes the funda-
mental roots. In particular we have 7(Q) = Q. We write
7( a) = a
-,.....
; L..
 "
, [_...
- .
; t
; :t; }.'Fd
 I:; ro-
, i; 1
"OJ; -t 1t.
!
; t
L
.,-
'rn." -"t
for each OEQ.
[
Definition. Suppose aU the roots of JL have the same length. A K-
character X of P (or Q) is said to be self-conjugate if x(a) = x(a) for all
OEP (or Q).
r-
L
THEOREM 13.7.2. Suppose all the roots of 1L have the same length.
Theil hex) E H 1 if and only if X is a self-conjugate K-character of P which
can be extended to a self-conjugate K-character of Q.
t ---
l_
[
TI-IE TWISTED SIMPLE GROUPS
239
PROOF. Let hex) EH. By 7.1.1 X may be regarded as a K-character
of Q. Let x( qi) = Ai. Then we have
X = XPI. A1XP2, A2 . . . XPh AI
as in 7.1.1, where Xr, A is the K-character of Q defined by
Xr, A(a)=,.\2(r, a)/(r. r), aEQ.
We consider which K-characters X of Q are self-conjugate. .The neces-
ry and sufIicient condition for this is that X(t/i) = X(qi) for 1 = 1, . . . , I.
0W
XU/i) = X( q pet») = Ap(i),
. \  ror all i Thus every self-conjugate K-character
thus we reqUIre "p(i) = "i ('. 0 .
of Q is a product of characters correspondmg to the p-orblts J of 11,
and these characters have the form:
Xr. A' ,.\ = X,
Xr, AXt. X,
Xr, AX;, AXi, A,
if J={r},
if J={r, i},
if J={r, f, n.
Let H2 be the group of automorphisms of JL]{ of the for hex), where
X is a self-conjugate character of Q. We recall from sectlO 7.1 that
Ir,(t)=h(xr. t). Thus }[Z is generated by elements correspondmg to the
p-orbits J of 11 as follows:
hr(t), t=l,
h,-(t) h,(f),
hr(t) hf(l) hi(r),
if J={r},
if J={r, f},
if J={r, f, P}.
We shaH show that }[Z is contained in HI by showing that each of
these genera tors of 11 Z lies in }J 1.
I 'l S=<I>+ where J is a p-orbit of n. If S has type Al and t=I, then
x,(;) and x;( - t-1) are in Gl. Hence n,-(t) =Xr(t) X-r( - t-1) x,-{t) EGI
'and h,-{t)=f1,.(t)nr(-l)EGl. If S has _type AIXAI r A_IXAl>:Al' :
1imilar argument shows that h,-{t) hf(t) or hr(t) hf(t) hi(f) are 111 G ,
I f X X COnliTIlltes with each of Xr-, X-r- and each of Xi, X-p.
Incc cac lOr, -r .
Suppose S has type A2. Here things re not so simple, for tle subgroup
< \' V" X X-) of G is isomorphIc to the Chevalley glOUp A2(K).
, r, .... t, -r, j' ) . <v X X X)
By 11 .3.2 there is a homomorphism from SL3(K II1tO .... r, f, -r, -t

.L."fU
such that
SIMPLE GROUPS OF LIE TYPE
1+ te12-+- xr(t),
1 + te23xf(t),
r + le13Xr+f(t),
1 + te21 X-r(t),
I+le3_2x-f(t),
1+ te31 x-r-f(t).
Matrix multiplication shows that
(_-1 
U _1
G -1
U :
(t 0
t-If
o
 ) -+X,(I) x-.-( _1-1) "'.{I)=n,(I),
 ) -+X,(I) L,( _1-1) X,(I) =n,(I),
 ) -+n,(t) n,( -I )=h,(I),
 ) ni(t) I1r( -1)=h,(t),
t-I
 ) hr(t) h;{i).
i-I
Now :Xr(t) xf(i) Xr+f(U) is in Gl whenever U+ - - - b
(\Ve have cl h u- -tt, y 13.6.3 (iv).
, . lOsn t e structure constants so that M _
obtdIn the matnx representation in the above form.) j:o-l in order to
x-r(t) x-rei) X-r-f(U)
is in GI whenever u+u=ti, since N_r, -p=-1.
THE TWISTED SIMPLE GROUPS
241
Now we have
----..,
G
C
o -: ) -+x,.{l) x,(l) xm(u),
o
 ) -+x-.-(I) x-r<i) x-.--;(u).
i
Suppose ", t are elements of K satisfying ,,-I+X-I=ti. Then the matrices
( At :i )- ( 1 0 )- ( Xt i )
-I
0 X-I -t 0
arc mapped into elements of GI. Thus their product (from left to right)

,
(L 0 D
- ,,-IX
0
is mapped into an element of GI. This holds for a given " whenever
there exists tEK such that tl=,,-l+X-I. We show in the following lemma
that each clement "1=0 of K is expressible in the form "=\Xi\ where
).1, "2 arc such that the above equation can be solved for t. Thus
( AIX  )-( 
o 0 X-I XII
:1 ) C;1
o
o
2 )
- "IIXI
o
- "i1X2
o
is mapped into an element of GI. Hence hr(") hf(X) EGI, as required.
LEMMA 13.7.3. Let K be a field admitting an automorphism of order 2.
l.et L be the subset of K consisting of elements" such [hat there exists
tEl\. with ti="-I+X-I. Then each element "EK is expressible in theform
,\ = \ Ai 1. where "1' "2 E L.
i .-.<{
-"'1
-'""
I
/
:/

242
SIMPLE GROUPS OF LIE TYPE
.' !!."-'
f
\ t:
-j' 
i
h
i

.-1-:
I
t- f{
. 
E
,
Ij
PROOF. Suppose A =1= X. Define Al = A - X, A2 = (X - A)/X. Then
All + XII = 0 and A;l + X;l = 1.
Thus AI' A2EL and A=A1X;1.
Now suppose A=X. There exists f-LEK such that f-L=I=O amI p,= -p..
Define Al = Af-L, A2 = p,. Then All + XII = 0 and A;l + X;l = O. Thus
AI, A2 EL and A = A1X;1. 31
' C''''
j, ','-
r
"""""'L
ii
We now return to the proof of 13.7.2. We have now shown that
H2 is a subgroup of HI in all cases, and shall now prove that HI is
contained in 1[2.
Let N2 be the subgroup of N1 generated by H2 and the elements nWI
chosen as in 13.5.2, for all WE WI. We have
nwh(x) n;;l=h(x'),
where x'(r)=x(w-1(r», by 7.2.2. Suppose h(X)EH2. Then X is a self-
conjugate character of Q. Thus, for all a E Q, we have
x'(a) = X(w-1(ii»= x(w-1(a»= x(w-1(a)= x'(a)
and so X' is also a self-conjugate character of Q. Thus flw normalizes
H2, and so H2 is normal in N2. Also N1=111N2 and HI nN2=H2.
Thus
1\ [,'.,'
i!.
!
............;.
r
I
1 \
lL
--,
(
l r'o
".- \: t 
. r
N2/H2=N2/HI nN2H1N2/HI=N1/HI WI.
We next define B2= UIH2. Since H2 normalizes VI, B2 is a subgroup
of G1. We shall show that for each p-orbit J of 11
XJ S B2 uB2n(w) B2.
Since all the roots have the same Jength, J has type AI, Al x AI,
Al X Al X Al or A2. If J has type Al the elements of XJ have form
x-r(t), where t = f. Now
X-r(t)=xr(t-I) hr( -t-I) nrXrU-I), t =1= 0,
and we may choose n(w)=nr. Since hr( -t-1)E1l2 we have
x_r(t)EB2 uB2n(}) B2
\1
'--'
) l
whenever t = f.
If J has type Al x Al we have
x-r(t) x-,(f)=Xr(t-I) x,(i-I) hr( -t-I) h,( -i-I) nrntx,{t-I) Xf(f-I)
and we may choose n(})=nrnf' Also hr( - (-I) hr( - i-I) EH2 and so
X-r{t) x-t(i) E B 2 U B 21l(W) B 2.
- .
I.
'-
t:=
.  ,-,.,
TI-lE TWISTED SIMPLE GROUPS
243
. . t works if J has type Al x Al X AI.
A SlIndar argumen .. k of the homomorphism from SL3(K)
If J has type A2 we agam nM e use .
T X X X) Under this homomorplnsm
to (); r. ;-. -r, -j'.
c : ) x-,(I) x-,(f) X4-'(U).
Now we have
(:  ) = ( r:-1 :1) (:-' U-1 ) ( _  ) ( fl' 1)
-1 001 00 u 1000
u t _ _ Th we have seen that the images of all the matrices
Suppose u+u=tt. en
. . . Gl except for
in this equatIOn are In ,
( - ).
100
. ... G1 also However, this image is in
h. of this matnx IS 111' 1 t1.s
Thus t e Image ,.1 f W Thus we may c 100se 1\
N and corresponds to th? element H'o 0 .
image as 1I{IVo). Since the l\1(lfu:,  )
" J (--1) h-(U-I)Efl2 we have
IS II' Ll r J 2
X-r(t) x-r(f) X-r-r(U) EB 2 U B 2n(lIfo) B
whenever u+u=ti. Thus
X1- c B2 UB2/l(W) B2
<DJ -
in all cases. . 8 1 4 that B 2 U B 2n(HIo) B 2 is a subgroup
It noW follows exactly as In R . f the argument of 8. 1 .5 we then
of G1 for each p-orbit J of n. epea mg
have J) B2 B2 B2
B 2nB 2. B 211(W) B 2  B 2/1Il(Wo U 11

L,<,> SIMPLE GROuPS OF LIE TYpE ,
I
for each nEN'. It then follows as in 8.2.2 that B'N2B2 is a SUbgroup i
of Gl. However X and XbJ are contained in B2N2B2 for all p-orbils ,
J of n and these subgroups generate G' by 13.6.5. Tl,uS Gl=B'N'B'. j
It is now easy to show that H' is Contained in H'. Let hEHl. Then !
hEB2N2B2= U1H2N2H2UI= U1N2UI.
Thus We may write I, =u'nu, where u, u' E V' and nEN'. Hence
11 = (u')-lhu-1 EB' nN'=Bl nN' nN'=Hl nN'=H'.
Thus (u')-I.hu-1h-1h EH'. It follows that (U')-I.hu-1h-1= I and hE H'.
This completes the proof. .
We now discuss the subgroup H' in the cases where there are roots
of different lengths. Then 1L has type B2, G, or F..
THEOREM 13. 7.4. Suppose 1L eOJIloins roots of rwo different lengths.
Then the elements hex) E H which are fixed by a are the ones for which
X(f) = x(r) AUW, rEI1.
If 1L has type B, or F., [hen an element hex) lies in H' if and only if i[ is
fixed by a. The Same holds if1L has type G2 provided [he field K is finite.
(It is not known Whether H' coincides with the set of a-invariant elements
of if if J[ has type G2 and K is infinite.)
PROOF. We first describe the elements hex) which are fixed by a. As
before, X is a K-character of Q and we have
where X( qt) = If. Tl1 us
X = XPl, llXP2. t2 . . . XPI. t"
Hence
l
heX) = [I Izp/lt).
i=I
a. hex) = IJ hii.(tA Ui,) 0) = h (17 Xii,. ,;<i'tJO ).
Thus hex) is a-invariant if and only if
X = 11 XPL, t;(Pt)o.
i
Comparing the values of these two characters at qp('J We obtain
/ p(i) = /: CPt) 0.
TWISTED SIMPLE GROUPS
THE
245
-----\
Th ) ( )A(PdO.
us x(qp(l) =x q, 'gIlts
ld 0 f the fundamental wel
h w that an analogous result ho S e recall from section 7. I
We s 0 the fundamental roots Pi.
q, are replaced by
that I
Pi =  Ajiqj.
;=1
Thus
i
!
J
!
f
,
I
I
!
I
) = IT X( q pUJ)Ap(J), p(f)
(pp(i») = X( ApU)p(i)qpU) j
X; ( )AJi A(pp(f»O
= IT X(qj)Ap(j), p(1) A(pp(j)° = 9 X qj
j
= X(PtY(PP(i»O
since (pp(j)
Afi =Ap(i), pU) =Ap(j), p(i) >(Pp(i»)"
I and short roots.) Thus
II that p interchanges ong
(We reca ) ( )'\(Pp(i)0.
X(p p(i) = X Pi
. Of and only if
I ( ) is a-invanant 1
Hence 1 X _ )A(;')O r E n.
X(r) = X(r ,
d we consider whether
. HI'S fixed by a, an h( ) fixed
I ment hex) m I of elements X
Now every." e It is evident that the group _ Aif,.) for each
the converse IS trueb the elements of form h(x. 're p-orbit of n
b a is generated y B or G2 there IS JUS F. the
y . { -} of n. If JL has type 2 bOt If JL has type B2 or 4
p-orblt f, r F there are two p-or I s"
d if 3L has type 4
an A XA, or B2. Th n we have
p-orbits have type  bOt of type Al x AI. e
Suppose J = {r, f} IS a p-or I 1) (t) X (tA(;')O) =nr(t) n;,(tA(f)ff)
-1 x (_(tA(i')O)- Xr I' 0
. (t) X-(tA(f)O) X-r( - t ) -f h (-1) nf( _ I) IS
iXr. , . t I we ave nr
o . nt of Gl. Puttmg =-
arrd thIS IS an eleme h characteristic 2.) Thus
in G'. (Recall that K as _ )hr(t) hf(tA(f)IfJh(xr. 'X,. '>('J")
nr(t) l1j{tA(f)O) nr( - I) n,( 1
f Gl nH =Hl.
. and this is an element 0
t
t
F
'1
1- ':I'!t

1
j
f
I
I
I
t
I

,
..-."
/-

....
f,
246
'-
SIMPLE GROUPS OF LIE TYPE
Now suppose that J={a b} is a b'
root and b is long. We sho that Iz p-or It o type ]Bz, where a is a sho
representat ion Th b (X a, lXv, t ) E H by means of a maIm.
. " e su group of G generated by X x: X X '.
:omo:phlc to the ChevaHey group B2(K) = C2(K). n;en y lIa,) 2-i';
e.re IS a homomorphism from SpiK) into (X X X v," II
whIch a, b, -a, A -vJ unucr
l'
6 
I+t(elz-e_z, -I)-+Xa(t),
1+ tez, -2-+Xv(t),
I+t(el, -2+e2, -l)-+Xa+b(t),
I + tel, -1-+X2a+b(t),
1- tee_I, -2 -e21)-+X-a(t),
1+ te-2, 2-+X-b(t),
1+ t(e-I, 2 + e-2, I)-+X-(a+b)(t),
1+ te-I, I-+X-(2a+b)ft).
Since K .has characteristic 2 the signs are irrelevant and the ' . .
fact an Isomorphism. It ma be h . nMp IS 10
multiplication that und th.Y' c eckd by a straIghtforward matrix
, er IS IsomorphIsm, we have
)
.,0)
L
',--
l_
n 0 0  ) -+h.(t) h.(t2li). j
t 20-1 0
0 t-1 
t:

0 0 tl-2O :1
'Ie
Define J
UI(t) =Xa(tO) xv(t) Xa+b(tO+1)
and
V2(t) =X-(a+b)(t0)X-(2a+b)(t).
Then UI(t), V2(t) Jie in GI for aJl tEK by 13 6 4 ( ") W .
that ' .. VI. e may now venfy
A=(i t1-O t o ) 
1 tf} t 2f}-1 i
0  -+ U1(t 20-1). 1
0 t1-()
C 0 0 i ) -+02(1-1).
B- 0 1 0
t-1 t-f} 1
t-O 0 0
p.
'F,
L.,
1.\
"""""'
i
THE TWISTED SIMPLE GROUPS
247
Now ABA-I is the matrix
n-1
o
o 0
o 0
t 1-20 0
t201 )
and so the image of this matrix is in GI. Putting t = 1, the image of
'}
n ! i D
is in GI. Multiplying these two matrices together we see that
ha(t) hb(tZO)EGI,
and so is in HI, as required. Thus HI coincides with the set of a-invariant
elements of H when JL has type B2 or F4.
Finally suppose JL has type G2. As we lack a matrix representation of
conveniently small degree (the smallest has degree 7) we argue in a diCTerent
manner. We assume in this case that the field K is finite. We define two
clements u(t), UoEU1 as follows:
li(t) = X2a+b(tO) X3a+2b(t),
Uo = x (t(l) Xb(l) x a+v( I) X2a+b{l).
The fact that these elements are in U 1 follows from 13,6.4 (vii), Consider
the effect of these elements on the element e-3a-2v of a Chevalley basis.
We have
u(t).e-3a+2b= -t2e3a+2b+al,
lio.e-3a-2ll=Nb, 3a+bM2a+b, -3a-2b, 3e3a+2b+a2,
where aI, a2 are linear combinations of elements of the Chevalley basis
other than e3a-l--2b. Using the structure constants given at the end of
M:clion 12.4 we have Nv, 3a+bfYha+b, --3a-2b, 3 = 1, hence
Uo . e -3 a-2b = e3 a+2b + a2.
Now Gl=BINIBI= U1NIUl and we have similarly GI= VINIVI. Let
110 and u(t)=v'nv, with v, V'EVI and /ZEN 1. Then
v'nv. e-3a-2b = - t2e3a+2b + al

248
SIMPLE GROUPS OF LIE TYPE
and it follows that.
n.e-Sa-2b= -/2e3a+2b+tern{s involving different er's.
If we argue in a similar manner for Uo we have Uo = vnovo' where
vo, VE VI, no EN I
and
no. e-3a-2b = e3a+2b + terms involving different er's.
Now n and no are elements of NI corresponding to the element Wo of
WI. For Wo is the only non-unit elements of WI. Thus n, no operate
monomially on the root vectors and we have
n.e-3a-2b= -t2eSa+2b,
no. e -3a-2b = e3a+2b.
Also no-l.n must be an element of III satisfying
/lOIn. e-3a-2b = - t2e-3a-2b.
However, the a-invariant elements of II are those of the form
ha(uO) hb(U) for all U EK. Such an element operates on e-3a-2b as follows:
ha(uO) hb(u).e-3a-2b=(uO)Aa. -3a-2buAb. -3"-2be-3a-2b
= u-Ie-3a-2b.
We now compare coefficients and see that
l1oln=ha(( _1-2)0) hb( -t-2).
Thus
haeC _t-2)0) hb( -1-2)EHl.
Putting 1 = 1 we have
ha(( _1)°) hb( -1) EHI
and by multiplying these two elements we obtain
ha((t-2)O) hb(t-2) EHI.
Thus the group of a-invariant elements of II will be generated by elements
of III provided the multiplicative group of K is generated by the squares
in K and the element - 1.
Now K is assumed finite and so the non-zero squares in K form a
subgroup of the multiplicative group of K of index 2. The multiplicative
group is therefore generated by the squares together with any singlc
THE TWISTED SIMPLE GROUPS
249
-'.
non-square. We show that -1 is a non-square. Suppse t2 = -1. Then
12°=( -1)0= -1. Thus 12°=/2 and so 10=:!: I. Hence 1° =1 and
1302=/3= -I.
But the automorphism 8 of K satisfies 382 = 1. Hence t3011 = t and we
have a contradiction. This completes the proof. ·
I t is not known whether the result of 13.7.4 holds if Jl = G2 and K is
infinite. For further information on the situation in this case see Ree [3]
and Steinberg [15].
.--.
!
i
I

1- -
,,-  \.......
f-

t..
CHAPTER 14
Further Properties of the Twisted Groups
,--
t
[
In th.e ast chapter we defined the twisted groups and gave a detailed
descnptlOn of the subgroups VI HI NI WI In the t I
. ' , ,. presen C 1aph:r
,;e sl:al denve the orders of the finite groups in the families, prove the
sImplIcl.ty o the t:visted groups, and show that certain of these groups
can be IdentIfied wIth classical groups.
[.--.
...
14.1 The Finite Twisted Groups
Suppose GI is a twisted group constructed as a subgroup of the Chevallcy
group G = JL(K), and supose that the field K is finite. if JL has type AI,
where I  2; then K admits an automorphism of order 2, so must be the
field GF(q ) for some prime-power q. The same applies if JL has type
Dl or E6..If JL has type D4 and the graph automorphism of G used in the
construction has order 3 then K must be a field ad .tt'
.' ml II1g an auto.
morphIsm of order 3. Tus K = GF(q3) for some prime-power q. If 1. has
type B.2 or F4, then K IS a field of characteristic 2 admitting an auto-
mrhlsm e s.uc.h that 2e2= l. If JL has type G2, K is a field of charac-
ten.stlc 3 .admlttIng an automorphism e such that 3e2 = 1. We 1etermine
whIch finite fields have these properties.
\.
.-
L
f"-
'11
l)
[N 

["
LMA 14. 1 . 1. Let K = GF(pn) be a finite field oj characteristic p
adnlltrmg an automorphism e satisfying p()2 = 1. Then n is odd and 0 hOJ
the form
"" -.
j
(-.-',
}"o= APTn,
where n=2111+ 1.
r-
PROOF. We have AO=}"pr for some r. Thus }"p02=AP2r+1 d
}"p2T+1_ A '. . ' an so
- for all A E K. Thus K IS contaIned In GF(p2T+1), hence n divides
2r+ 1. It follows that n is odd, and we write n=2m+ 1. Let
i
r-
L
2r+ 1 =(2m+ 1).(2s+ 1).
250
f
{
FURTHER PROPERTIES OF THE TWISTED GROUPS
251
Then r=s(2111 + 1)+/11 and we have
ApT = AP(2111+1)6t-m = AP(2"'+1)B pm = (AP(2m+l)B)pTn.
l3ut AP2m+1 = A, thus AP(21/1+1)8 = A and APT = Apm. Hence AO = Apm as
required. ..
Lemma 14.1.1 shows that if Jf. has type B2 or F4 then K=GF(22m+1)
for some m, and if JL has type G2 then K = GF(3 2m+1) for some 111.
The notation GI = tJf.{K) will be used for the twisted groups, where
the superscript i denotes the order of the symmetry of the Dynkill diagram
used in the construction. The finite twisted groups will be denoted by
2Al(q 2), I 2,
2Dl(q2), 14,
2E6(q 2),
3D4(q3),
2B2(22m+1),
2G2(3 2m+!),
2F4(22m+1).
It is convenient to write the three latter groups also as 2B2(q 2), 2G2(q 2),
2F(q 2) and so we define q in these cases to be pm+l/2, where p =2 or 3.
Note however that q is an irrational number in these cases rather than a
rational prime-power.
The groups 2B2(K) were discovered by Suzuki [1, 2] and are called
Suzuki groups, and the groups 2G2(K), 2F4(K) were discovered by Ree
[2, 3} and are called Rec groups.
In order to determine the orders of the finite twisted groups we first
calculate the orders of their subgroups VI, (V;:)l and HI.
LEMMA 14.1.2. (i) I VI I = q N.
(ii) I (U;:)ll =ql(W), WE WI.
(iii) I HI I =1 (q-r}1)(q-1]2) . . . (q-rll),
,rhere 7')1, . . . , 1]l are the eigenvalues of the isometry T of 19, and d is the
ordn of the group of self-conjugate K-characters of Q/ P in the case when
"II roots have the same length, alld d = I otherwise.

252
SIMPLE GROUPS OF LID TYPE
PROOF. Let S be one f tl .
Then 13.6.3 shows that l 1valence cIasss in (j) defined in 13.2.1.
and I (U)ll=ql(w). I sl-g . By 13.6.11t foJlows that I Ulj=q'"
We now consider I HI I Su ose Jl
Then a. h,(t) = h-(i) by 13 7. 1 PLP I a the roots have the same length.
. r . .. et I(X) be an ele t f H
a A-character of Q. Then men 0 ,where X is
a . hex) = a hex )
. Plo Al . . . :x Ph AI
as in 7.1.1 where At = x(qt). It follows that
a. heX) = h(X1h. Xl . . . XPIo xJ = hex),
where (ijt) = x(q£). Thus heX) is invariant unde' ..
self-coo!ugate K-character of Q. r a If and only If X IS a
Let HI be the subgroup of H fl'
a. I fII I is the number of self, o. e emets whIch are invariant under
d ' -conjugate K-cl1'lracters f Q I
eten1llne the number of If' c: o. n order to
. se -conjugate K-charact .
p-Ol'blts of n and the corre' .1' . ers we consIder the
f d sponull1g p-Ol'blts of th t
un amental weights. Let J b' 1. . e se ql,..., ql of
If . C SUC 1 el p-orblt If I J I 1 I
se -conjugate K-character . t th " = t 1e value of a
X a e correspondl11g b
zero element of G F(cl) so . b h . qt can e any noo-
I l' celn e c osen 111 q 1 If I
va ue of X at qt can be a - ways. J I =2 the
(j{ is then determined Tc: hnuysntoln-zcrlo element of GF(q 2) and the value at
. . Ie va ues of 0 th' b' -
111 q2 _ 1 ways. Similarly if I J 1_ 3 th Ix n IS or It can be chosen
b' - e va nes of :x on tl ' .
or It can be chosen in q3 _ 1 . 1e COI respondlllg
K-characters of Q is ways. Thus the number of self-conjugate
TI q( I J 1- 1)
J
over all p-Ol'bits J of n H
, . owever, the roots of xlJl 1 .
of the Isometry T of 1) on the sub' . - are the eIgenvalues
roots in J. Thus space spanned by the fundamental
T} (qIJI-l)=(q-7)1)(q-7)2)... (q-r;l),
where 7)1. . . . , 7)i are the eigenvalues of T on 11
Now HI i th .
s e set of elements h( ) \ 1 .
character of P which Call b t d X 1-./ lere X IS a self-conjugatc K.
e ex en cd to a If' r
of Q (13.7.2). Thus there is a I . . se -c?I1Jugate A-charactcr
b .. 101110morphism of HI 111 .
Y restnctl11g X from Q to P. The ker . onto I. obtalllcd
set of hex) where X is a S If . nel of thIs homomorphism is the
, " ( e - -conjugate K-ch' t f ..
IdentIty on P. Thus I }]l . HI 1_ I I. . c:\rac cr 0 Q wlw.::h IS the
. - ( , w 1ere d IS the order of the group of
FURTHER PROPERTIES OF THE TWISTED GROUPS
253
-
\
s.elfconjugate K-characters of Q/P. Therefore
1 1
I HI I =d l!I (q-7)t),
as required.
Now suppose that there are roots of different lengths, i.e. that GI is
a Suzuki or Ree group. Then HI consists of all a-invariant elements of
/l by 13.7.4, since we have a finite field. Also hex) E HI if and only if
X(r)=X(r)A(f)O, ren.
As before the values of X on a p-orbit J of IT can be chosen in qlJI-l
ways. (Note that all p-orbits now have two elements.) Thus
l
l HI I = II (qIJI-l) = II (q-r;t)
J i=l
for the Suzuki and Ree groups. ·
""'""".
Note. The number d in 14.1.2 takes the value (/+1, q+l) for
2At(q2); (4, ql+ 1) for 2DL(q2); (3, q+ 1) for 2E6(q2); and 1 for 3D4(q3).
For if JL= Ai, DZk-I-I or E6, the group Q/P is cyclic with a generator a
satisfying a= -0. Thus X is a self-conjugate K-character of Q/P if and
only if x( - a) = x(a)q, i.e. X(a)Q+1 = 1.
Thus d=(tl, q+ 1), where tl= \ Q/P I. For the groups 2D2k(q2), Q/P
is elementary abelian of order 4 and the map aii interchanges two
non-zero elements and fixes the third. Thus d=(2, q+ 1) in this case,
and the case 2 Dl(q 2) can be summarized by writing d=(4, ql + 1). Finally
for 3 D4(q 3), Q/P is elementary abelian of order 4 and the map aii
permutes cyclically the three non-zero elements. Thus d= 1 in this case.
These facts may easily be verified using the information about the group
Q/P given in section 8.6.
..........
:";
_ r1..
PROPOSITION 14.1.3.
1 l
I Gl 1=- qN II (q-r;t) L ql{W).
d i=l WE IVI
PROOF. This follows from the double coset decomposition of GI
with respect to BI. The number of elements in the double coset BlnwB 1,
wE WI, is I BI \.\ (U;)I\ by 13.5.3, and this is equal to
1
_ qN IT (q-r;f).ql(w)
d i=I
by 14.1.2. The result follows.
---.
.

-
["
.
f'-<-
L,,,
r-
[
[
L.
r-
)
...
....<:!::Ii'!!.....
'f._
254
SIMPLE GROVPS OF LIE TYPE
14.2 Factorization of the Polynomial PW1(t)
As with the Chevalley groups it is ossibl '.
obtained for the order of th 't . t dp e to sImplIfy the expression
e WIS e groups by find' r. '.
of the polynomial mg a Jactonzat:on
PWl(t)=  tZ(w>.
W E JVI
To do this we use a modification of the ar
used in section 9.4. gument for the Chevalley groups
Let 3J=JR[Ir,..., lz] be the ring of Pl' I . .
isometry of lJ 0 ynomla mvanants of W. The
r operates naturally on 3J by
-r{P)(x) =P( r-1(x))
for PE], XEtJ. Since P is an invariant I . I
r(P) must also be invariant N Pro ynomIa and r normalizes fV,
. ow r translorms the sub JJ
geneous polynomials of degree n into itself for ead i:ce b n of hon.lO.
the base field from IR to C h  11. us yextendIng
. we can c oose a basIs of 3J . .
elgenvectors of r Hence th . n consIstmg of
. . e generators Ir I of 3J I'
nng can be chosen to be eigenvect f V./.., Z as a po ynol1ual
h, . . . , I, are chosen in this way orsho r'd e suppose the basic invariants
, were eg It =dt and
-r{It) = €tlt, €t EC.
PROPOSITION 14.2. 1.
 I tZ(W) = IT (1- €ttrl,).
WE w i=l I-T}tt
PROOF. (a) Let
PWl(t)=  tZ(W) and FWl(t)= IT (l-€ltdf)
we WI i=l I-T}-tt .
 P: a;sPaW}(t) enot the corresponding polynomials for the groups
J> ny p-mvanant subset of IT F h. .
J let OJ be the number of b't' J . or eac p-lI1vanant subsct
p-or I s In We sh n sJ h
PwJ(t) satisfy the following identities:' a)Ow t at PW1(t) and
 (-I)oJ Wl(t) =tN
J PWj(t)'
 (-I)oJ !:.w1(t) =tN
J Pwj(t)'
FURTHER PROPERTIES OF THE TWISTED GROUPS
255
where the sums are taken over all p-invariant subsets of IT. (Compare
these identities with the ones in section 9.4.)
(b) The identity involving Pw1(t) is proved as before. We have
PW1(t) .,
2: (-1) OJ ---- . = 2: (-1) OJ PDI(t)
J PWJ(/ ) J J
= 2: (-1)0J ( 2: tl(W»
J WE JY
w(J) S <1>+
= 2: ( 2: (-1) OJ) tl(w).
UlEW' J.
w(J) s 11>.
Let Jw be the set of roots rETI such that w(r) E <1>+. Then the coefficient
of (l(w) in the above sum is
2: (-1)°J=(1-1)n, where n=OJw'
JsJw
This is 0 unless Jw is the empty set. If Jw = cp then IV = WOo Thus the above
sum is t'(wo) = tN.
(c) We now consider the polynomial PW1(t). In order to show that
this polynomial satisfies the above identity we must consider the operation
of the Weyl group on the Coxeter complex. Note that the isometry T
of 1) also operates on the Coxeter complex. Now each orbit of the Coxeter
complex under the operation of W contains just one element CJ by
2.6.3, where
CJ={v; (v, r)=O for rEJ, (v, r»O for rETI- J}.
For each p-invariant subset J of n, let f1.r(WT) be the number of elements
in the orbit containing CJ which are fixed by Wr. We show that
2: (-l)oJ n.r(wr)=det w,
J
WE WI.
Let '(!I be the subspace of 1) of elements fixed by WT. Then the elements
of the Coxeter complex which intersect 'C1Ll are prccisely those which are
, fixed by WT. For if an element 11;.. of the complex intersects 'ill it contains
a point in common with wr(1[\), and so wr(1k) = 11;... Conversely, if
w-r(1\.)=1L\., WT must fix some point of 1k since Wr is an isometry. Thus
'k intersects W.
We now consider the complex in W whose elements have form 1k n W,
where 1lt is an element of the Coxeter complex with wr(1K) = j[{. We

256
SIMPLE GROUPS OF LIE TYPE
wish to find the dimension of it n W, and consider first the case in which
1f.{=CJ for some J. Since W7(CJ)=CJ we have w(Cp(J})=CJ and so
p(J)=J by 2.6.3. Thus 7(CJ)=CJ and w(CJ)=CJ. Therefore )II fixes
every element of C.J by 2.6.2 and so IVT acts on C.J in the same way
as 7. It follows that the elements of C.J fixed by WT are the elements of
C.J fixed by 7. However, dim CJ = I II - J I and each p-orbit of II - J
contributes 1 to the dimension of the 7-invariant elements of C.J. Thus
we have
dim (C.J n W) = On-.J.
We now consider an arbitrary element 1k of the Coxeter complex with
W7(1k)=1k. We have 1k=w'(CJ) for some w' E v and some J, by 2.6.3.
Then
W7w'(C.J)=w'(CJ)
and so
1V'-lW. 7W' 7-1. T( C.J) = C.J.
However, TW' T-1 E W, and so we have
dim (C.J n Thl') = On-J,
where W' is the subspace of 1) of elements fixed by W'-lW71V'. Now
W = w'(ll') and so
On-J=dim (CJ n w'-l(OO))=dim (w'(CJ) nW).
Thus dim (ltt nW)= OU-J.
We now apply 9.4.4 to the complex induced in ij;1. We have seen that
the number of elements of this complex of dimension i is given by
fit = 1: /lAwT).
°n-J=i
Thus we have
l.: (-l)on-JnJ(wT)=( _l)dlm W.
J
Hence
L; (-l)oJ nAwT)=( -1)Ofl.( _1)111m i!{.
J
Now w, T and WT, being orthogonal transformations, have eigenvalues
1, -lor pairs of complex conjugates. Thus
det (IV7)=( -l)l- dim W,
det T=( -l)l- dim ))1=( -l)l- On.
FURTHER PROPERTIES OF THE TWISTED GROUPS
257
L_\
Hence
det w=( _1)011.( _1)dlm i!{
and we have
1: (-I)oJ nAwT)=det w.
J
(d) We prove next an analogue of 9.4.7 by showing that
1: ( -1)0J tr{JJ}n T= tr In T,
J
where (31",,)n is the space of homogeneous polynomials on t) of degree
n invariant under WJ, 3ln is the space of homogeneous alternating poly-
nomials of degree 11, and the sum extends over a}l p-invariant subsets J
of n. Note that the terms dim (J):n and dim :n in 9.4.7 have been
replaced here by the trace of 7 acting on these two subspaces. .
Consider the group < W, T) generated by Wand T. The remarks 111
(c) show that the stabilizer of CJ in < W, T) is < WJ, 7). Therefore the
unit character of < WJ, T) induced up to < W, T) satisfies
1 )(WT) = nAwT),
just as in 9.4.2. Now let X be the trace function of < W, T) acting on
s&n, the space of homogeneous polynomials of degree fl. Then we have
x:)(WT) = 1 g:)(WT). X(WT)
=nAw7) X(W7),
as in the proof of 9.4.7. By (c) it follows that
1: ( -1 )oJXH)(WT) = det IV. X(WT).
J
-----.,
We now average over Wand obtain
1:(-l)OJ  X:)(WT)=-II 11: detw.x(w7).
.f I WIWEJV W WETV
But
1 I
-\ I L; xg)(WT)= -\ W----I 1: X(WT),
W WEIV J welVJ
as in the proof of 9.4.7. Thus
L (-l)OJX(---- 1: WT) =-!-I 1: det w. X(WT).
J I WJ IWE WJ I W WE JV


.;
258
SIMPLE GROUPS OF LIE TYPE
'---
Let M be any (W, T)-module and let
1
T=, Wlwww,
Then M has a direct decomposition M=MoffiMI, as in 9.3.2, where
Mo is the set of elements annihilated by T and MI is the set of elements
fixed by T. Since T commutes with T, TT leaves Mo and !'vII invariant
In fact TT=O on Mo and TT=T on MI. Thus the trace of TT on Mis'
equal to the trace of T on Ah.
Applying this with M =n we have
X(j WI I 2: WT)=tr()J)nT
J WE JVJ
f"--
L.
b.-
[
0-
[J
......
: ,,"'0'
--
and also
It foHows that
;.:(-, 1 ,2: det w.wT)=trinT.
W we 1V
 ( -l)0J tr() ) T= trj T
J J n n
as required. We observe that trin T= trJn_N T since by 9.4.6 we have
.3In=O for n<N,
1Jn= ( n Hr) .3Jn-N for nN
r E (1)+
f"
1 .
.,;1 L"",
and
 1,
T( n Hr)= n Hr.
r e (1)+ r E (1)+
:11' r-
Lw_ i!! t.:,
.' j'
; {
: ,i!
'IL
(e) The coefficient of tn in
1 1 1
1 - IT 1 td,
IT (1-'f}tt).PWl(t) i=I -Et
iI
;i !
 !
i l..
'I
r!
I (
f !
 
is the trace of T on 3Jn, as in 9.3.3. Thus the coefficient of tn in
1 tN
}.!I (1 -'f}tt) .Pw1(t)
iL
'j
Ii
 .'
is tr )n_NT = tr in T.
Now consider the coefficient of tn in
1 1
1 (1- EJltdJ1). . . (I - EJktdJk)(1-'f}k+1). . . (I-TJl)'
IT (1-'f}tt).Pw(t)
i=l
FURTHER PROPERTIES OF THE TWISTED GROUPS
259
where
k (1 - EJttdJI)
PWl(t)= IT .
J i=1 1-1]t
S'nce "'., = l' ffi}).L where WJ operates trivially on 17j-, the basic in-
I v J J, .. . f W
variants of WJ on V may be taken as the basIc ll1vanants 0 J on
VJ together with / -k additional elements _ satisfying T(Xt) =1]tX, where
are the eigenvalues of l' on Vj-. Thus the coefficients of
7].t+l, . . . ,1]1
IF! in
1
(1- EJltdJ1) . . . (1- EJktd./k)(1-1]k+1to) . . . (1 77ltO)
is the trace of Ton (3JJ)n, again as in 9.3.3. By (d) it follows that
1 tN
 (-I)oJ Pw}(t) =PwIU)'
(f) We have now proved the identities
 (-I)oJ PW1(t) =tN,
7 Pw(t)
PW1(t) N
L (-1)0J _-j-=t ,
J Pw.lt)
and so may deduce that PW1(t)=PJi/l(t) by induction on I J I.
completes the proof.
This
.
14.3 The Orders of the Finite Twisted Groups
THEOREM 14.3.1. The orders of the finite twisted groups are given
by
I GI I =!qN (qd1_ q)(qd2_ E2) . . . (qdl_ El),
d
where d is as in section 14.1 and dt, q as in section 14.2.
PROOF. We have shown in 14.2.1 that
l (1 - Ettdl) l (t-dl- Et)
2: tl(w) = n ---- = tN ,n t--1- -; ,
WEJVl i=l 1-1]tt t=1 1]

260
SIMPLE GROUPS OF LIE TYI)E
sinc dI + . . . +dl=N+/ by 9.3.4. Replacing I by 1-1 we have
2: IN-l(w) = fl (d1_ £i).
WE W' 1=1 t-YJt
However, WoE WI, and for each IV we have
I(wow) = N -l(w).
It follows that
 IN-l(w) =  tl(w).
WE W' WE W'
Thus
1 1
.n (t - YJt).  tl(w) = n (tdl- £i).
t=l WE W' i=l
The result now follows from 14.1.3 on replacing I by q.
.
e shall now determine the numbers £1,..., £1 for the individual
tWIsted groups. If we put t = 1 in the identity
 Il(w) = iI (.=!),
WE W' i=l 1 -YJ[t
we see that the number of £t equal to 1 is the same as the number of
eq ul t? 1. If the isometr T of 1) has order 2, then £i = :i: J and YJI = :t 1J/
for dB l. Thus t01, . . . , q IS a permutation of 1]1, . . . ,YJ1. If T has order 3
then JL=D4 and "71,. . . ,YJ4 are J, 1, w, w2, where w=e2"i/3. Thus two
of £1, . . . , 4 are I and the other two must be w, w2 since T is a real
transformatIOn of order 3. Thus in aJl cases q,. . . , €[ is a permutation
OfYJ1,...,YJI. _
.The numbers ch, . . . , ell and EJ,. . . , t01 are now known; however we
stIll have to determine which tOt is associated with which c1,. If 1L has
type A I, Dl (I odd!, or EI3, then T = - 11'0. (This follows easily from the
fact that 11'0# - 1 In these cases.) Thlls T operates on each invariant 111
the same way as - J. Hence T(Jt) = ( - I )ti'lt and q = ( _ 1 )d,.
If JL has type Dl we may take for the fundamental roots
P1=C1-e2,p2=e2-ea,... ,PI-l=e1-1-el,PI=el-1+el,
as in section 3.6. Then T operates on the orthonormal basis (el) of 1)
by
T(et) = et,
i=I,...,I-I,
T(C,) = -el.
FURTHER PROPERTIES OF THE TWISTED GROUPS
261
I-
.-...
\
Each XElJ may be expressed in the form
X=X1e1 + . . . +X1el
and the basic invariants may be taken as the first 1-1 elementary sym-
metric functions in xi, X, . . . , X[, together with X1X2. . . xl' Thus the
el, and q have values
d1=2, d2=4,. . ., d1-1=2(1-1), d1=1,
€1=1, £2=1,..., £1-1=1, £1=-1.
JfJ[.=B2 we have d1=2, d2=4 and the £'s are I, -1. Now the invariant
x7
i
of degree 2 has £=1, since T is an isometry of 'P. Thus €1=1, £2=-1
for B2. If J[.=G2 we have d1=2, d2=6 and the €'S are 1, -1. Thus we
have £1 = 1, £2 = -1 as before.
Finally suppose Jl= F4. Then
d1 = 2,
d3=8,
d4= 12
\
i
d2 = 6,
and the £'s are 1, 1, - 1, - 1, where £1 = 1. We show that the other basic
invariant with € = 1 is the one of degree 8. The roots of F4 may be written
with respect to an orthonormal basis e1, ez, ea, e4 in the form
:t et :t ej,
i =f j,
:t et,
.H :te1:t ez:t e3:t e4).
Let x E'P be written as
X=X1e1 +X2e2+X3e3+X4e4
and define the polynomial 1 by the formula
(v'2 )8
1= .(:tXt:tXJ)8+  (v'2(:tXt))8+ 2: - --- (:tX1:tX2:f::X3:i:X4) .
l-FJ 't 2
There is one term in I for each root, and the terms corresponding to the
short roots have an additional factor yl2. Since the elements of W permute
the long roots and the short roots, I is a polynomial invariant of W.
Since T maps long roots into multiples of short roots by a factor of yl2
\\c have T(I)=I. Also, I is not a multiple of the quadratic invariant
x+x-I-x+x. For substituting (Xl' X2' X3' X-i)=(I, i, 0, 0) in the formula
j

f'
i
.........
nf ..
 
L.
\ ,
'''''-
\.h>
\_-
\.-
262
..J
SIMPLE GROUPS OF LIE TYPE
for I gives a non-zero value Thus I
of degree 8. Hence £ = I '_ 1 may be taken as the basic invariant
WI, £2 - - , £3 = 1, £ = - I
e bave now obtained 'ft  4 .
twisted groups. They are as s:s. ormuIae for the orders of the finite
THEOREM 14.3.2.
/ 2AI(q2) / = (I + 1, Iq + 1) ql(l+1)/2(q2 - I )(q3 + 1)
2 2 1 x (q4-1) . . . (ql+l +( -I)'),
/ Dl(q) / = (4, ql+ 1) ql(l-I)(q2_1)(q4_1)
1 x(q6-1)...(q21-2_1)(ql+I)
I 2E6(q2) / = 36( 2 1)( 5
(3, q+ 1) q q - q + I)(q6-1)(q8_1)(q9+ 1)(qI2_1).
13D4(q3) /=qI2(q2-1)(q6_1)(q8+q4+1),
/2B2(q2) /=q4(q2-I){q4+1),
I 2G2(q2) 1= q6(q2 -1)(q6+ 1)
I 2P4(q2) / = q24(q2_1)(q6+ 1)(q8_1)(qI2+ I)
q2 =22m+I,
q2=32m+I,
q2=22m+l.
14.4 The Simplicity of the Twisted Groups
We saIl now show that the twisted groups are aU sim Ie
exceptIOns over very small fields. p , with a few
THEOREM 14.4.1. The twisted
2A2(22), 2B2(2), 2G2(3), 2P4(2). groups are all simple, except for
B PROOF. . (a) By 13.5.4 each twisted group GI contains BI NI
 I' INtplhr. We shaH therefore use the criterion for simplicit giveS i
. .. ree out of the four cond"f . .
checked B I . '. . I IOns given III II. I . I are easily
Let GI b IS soluble Since It IS a subgroup of the soluble group B= Ul/
. 1 e a normal subgroup of GI contained in B I Th GI' .'
In VI HI and its con' VI II' . en I IS contamed
proof of I I 1 2 Jugate H, so GI IS contained in Ill. As in the
. . we see that every element of Gl commutes with eve
element of VI and every element of VI. Let h(X)EGf. Since ry
hex) Xr(/) h(X)-I =Xr(x(r) t),
FURTHER PROPERTIES OF THE TWISTED CROUPS
263
it follows from 13.6.3 that X(r) = 1 for each root r E <1>. This is because
each root r occurs in some equivalence class S, and so there is an element
in X  involving r. Hence O = I and
n gBIg-I=I.
(J E Q'
The Weyl group of the (B, N)-pair Bl, NI is VI, and the set I of distin-
guished generators of V 1 is the set of elements III as J runs over the p-orbits
of n. We have seen in section 13.3 that these elements either form the
set of fundamental reflections in some indecomposable Weyl group, or
III =2 and the two elements of I generate a dihedral group of order 16.
In either case it is impossible to decompose I into non-empty compli-
mentary subsets which commute with one another. It is therefore suffi-
cient by 1 1.1.1 to prove that 01 =(01)'. We shall show in fact that
(Gl)' contains X. for each equivalence class S of <1:>.
(b) Suppose all the roots have the same length. Then the equivalence
classes S of <I) are of type AI, Al X AI, Al X Al X Al, or A2. Let hex) be
an element of I/l. We use the following relations:
hex) xr(t) h(x)-I =:\Ax(r) I)
if S={r} has type AI.
hex) Xr(t) x;:(i) h(X)-1 =Xr(X(r) t) x;:(X(f) i)
jf S= {r, f} has type Al x AI.
hex) xr(t) x;:(i) x;(l) h(X)-1 =xr(x(r) t) x;:(X(f) i) Xf(X(f) l)
if S={r, f, P} has type Al x Al X AI.
hex) Xr(t) x;:(i) Xr+i(U) h(x)-I =Xr(x(r) t) x;:(x(f) i) XrH(X(r+ f) u)
if S={r, f, r+f} has type A2 and u+u= -Nrfti.
Now we can find, given any rE<1> and Ii=OEK, a K-character x of Q
such that x(r)=t2 (see 11. 1.3). In fact, since the exceptions Al and Cl
ac not among the cases being considered here, we can find a K-character
X of Q such that X(r)=t.lf r=f we can find, given any ti=O in the fixed
field Ko of the automorphism t------. i, a self-conjugate K-character of Q
such that x(r)=/. If ri=f we can find for each ti=O in K a self-conjugate
K-character of Q such that x(r) = t.
If S={r} has type Al and Koi=GF(2) we can choose a self-conjugate
. character X of Q with x(r)i=l. Then x,{(x(r)-I)t)E(OI)' for aU tEKo.

264
SIMPLE GROUPS OF LIE TYPE
Let u be any non-zero element of Ko and define
t=--
x(r) - 1"
Then Xr(U)E(Gl)', and so X1  (Gl)'.
If S has type Al x A or A A A .
even if Ko= GF(2). 1 1 x 1 X 1, we see similarly that X £ (GI)',
1 f S has type A2 the above relations show that
. r«x(r)-I) t) xf((x(r)-l) i) x'"+f«x(r-I-f)-l) u+Nrf(X(f)-I) tl)
lIes In (Gl)'. If Ko"l= GF(2) we can choose a self-con'
of Q such that x(r + F) --.L 1 F .f () \ Jugate K-charactcr
--r-. or 1 X r = 1\ then ( -) - X d ( -
Suppose iX(r +f) = I for all'\ Then ,\"-1 ft X r -. an X r+r)=,Xt
for all '\"1=0 in Ko. Thus Ko is a finite field O:l ,\ "1=_0 m K and so ,\2 = I
and A<+1= 1 for all A"'O. K T! . Ko-GF(q). Then A=A'
Ko=GF(2), the excluded ce.' 1US fL= 1 for al1 fL"I=O in Ko, so thai
Let t1, Ul be elements of K satisfying t + - - -
self-conjugate character of Q 'I I' 11 1- - Nrrtltl. Choose a
by X sue 1 ! 1at X(r+ r)"I= 1 and define t, uEK
t =  u =  _ N"fCr/l.2._- 1 ttJ
x(r)-l' x(r+f)-l x(r+p)-l.
Thenu+il=-N-ti-andx(t) (I) ( ).
as . d rr "r 1 Xf 1 XrH Ul IS in (Gl)' Thus Xl c (01)'
c require. . s - ,
(c) Sllpose all the roots have the same Jen th and -
each eqUIvalence class S of <1> we d fi I g Ko - GF(2). For
. 13 e lJ1e e ements x .(1) or x (I ) f vi
as m .6.4. We recall that it 1 'b . S S , U 0 A.
ifShastypeAlxAlorAlxAl;l.een shown m (b) thatx.s(t)E(Gl)'
Now the equivalence classes in <I> are in 1
the elements of a root system with W I -1 correspondence with
section 13 3 ley group WI. We have shown in
. . _ t 13 t this root system has type:
Ck if Gl=2A2k-l,
Bk if Gl = 2A2k,
Bl-l if Gl = 2 Dl,
F 4 if G 1 = 2£6,
Gz if Gl = 3 D4.
In each case, except Gl =2A2, the root system with We I
of two different lengths and t . 1 yl group v has roots
, wo eqUlva ence classes correspond to rools
FURTHER PROPERTIES OF THE TWISTED GROUPS
265
of the same length if and only if they have the same type. Now Wl operates
transitively on roots of a given length, and since
-,"",-"
Xl -1 Xl
I1w snw = w(S) ,
WE WI,
we see that Gl operates transitively by conjugation on the subgroups
XJ of a given type. It is therefore sufficient to show that one such sub-
group of each type lies in (Gl)'.
Suppose the Dynkin diagram of WI has a double bond. Let Sl, Sz
be two equivalence classes corresponding to fundamental roots joined
by this double bond. If Gl is not of type zA2k these classes are of type
Al x Al and AI. Let Sl = {r, f} and Sz = {s}. Then the commutator relations
show that
(xsiu), xs1(t)] =xs3(tu) xsiliu),
where S3={r+s, f+S} and S4={r+f+s}. (We have used the fact that
K has characteristic 2.) Thus xsaCtu) xsi1fu)E(Gl)' for all tEK, uEKo.
However, XS3(tu)E(Gl)' since S3 has type Al x AI. Thus xsil)E(Gl)'
and so x14 s; (Gl)'. Hence (Gl)" contains a subgroup XJ of type Al as
well as one of type Al x AI' thus (Gl)' contains x1 for all S.
If Gl is of type 2A2k, the equivalence classes SI, S2 are of type Al x Al
and A2. Let Sl={r, f} and. Sz={s,- S, s+s}. These fundamental roots
r, ;, s, s of 1(. may be chosen so they are related in the manner shown in
the Dynkin diagram of 1L.
-----
r 5 5 .,
0------0---- _____-o-----<:J
The commutator relations show that
{xs2(u, v), xs1(t)) =xs3(tfJ) xsitu, tiv),
where S3={r+s+s, f+S+s} and S4={r+s, F+s, r+f+s+s}. Thus
xs3(tv) xsilU, IlV) E(Gl)'. However, XS3(tV)E(Gl)' since S3 has type
Al x A!, thus xsltu, tlV)E(Gl)' also. Putting 1= 1 we have xsiu, V)E(Gl)'
and so X4 s; (GI)'. It now follows as before that xl  (G1)' for all S.
If Gl has type 3 D4 the equivalence classes correspond to a root system
of type G2. Let SI, Sz be classes corresponding to short roots of Gz whose
sum is short. Sl, Sz both have type AI. Let SI = {r}, Sz = {s}. Then the
commutator relations show that
(xslu), xs1(t)] = xs3(tu),
where S3={r+s}. Thus XS3(1)E(Gl)' and so X3E(Gl)'. Hence XJ S (Gl)'
for all S as before.
'.,

266
SIMPLE GROUPS OF LIE TYPE
.,"" 
(d) Now suppose that <I> contains roots of different lengths. Then the
equivalence classes S of <I> are of type Al x A!, Bz or Gz. Suppose thal
Gl=zB2(K) or ZF4(K). Then the equivalence classes have type Al XAI
or Bz. Let hex) be an element of HI. Then x(i) = x(r )"-(f) 0 by 13.7.4.
Let S= {r, i} be an equivalence class of type Al x AI, where r is shon
and f long. Then
.
--"I.
£i1
hex) xs(t) h(X)-I=xS(X(f) t)
and so
[-..
-'"'"
hCx) xs(t) h(X)-l xs(t)-I=xs«x(f)-I) t)
by 13.6.4 (v). Now by 13.7.4 we can find an element h( x) E HI for which
x(i) takes any prescribed non-zero value in K. Thus if K /; GF(2) we may
choose X so that x(f)/; 1. It follows that Xl s (G1)'.
Now let S={a, b, a+b, 2a+b} be an equivalence class of type B2.
Suppose that aCt), f3(1I) are defined as in 13.6.4 (vi). Then
heX) f3(1I) h(X)-I=fKx(a+b) 1I).
r
t,,,,,
r-'
t
t"k
Thus we have
\

L..
hex) (J(u) h(X)-l (J(u)-l={J«x(a+b)-I) 1I).
By 13.7.4 we can choose h(X)EHl so that x(a+b) takes any non-zero
value in K. Thus (J(u) E(Gl)' for aU u, provided K /; GF(2). We also have
hex) a(t) h(x)-1 =a(x(b) t)
L_
tr--
t
 .
and
L_
hex) a(t) h(x)-1 a(t)-l=a(x(b) t) a( -t) (J(t0+l)
=a«x(b)-I) t) (J«X(b)O-I) tfH1)
by 13.6.4 (vi). Since (J(U)E(Gl)' for all U we have a«x(b)-l) t)E(GI)'
for all t. As before we may choose X with x(b)/;I provided K/;GF(2).
Thus aCt) E(GI)' for all t. Hence
xs(t, u)=a(t) (J(U)E(GI)'
and X1  (G1)', as required.
(e) We suppose finally that Gl=2Gz(K). In the proof of 13.7.4 we
showed that HI contains the element
,
!
t.-
\....-
f
(-,.-..
h(x)= T1a« - >,-Z)O) hb( - >,-2)
r
L
FURTHER PROPERTIES OF THE TWISTED GROUPS
267
for all .\ E K. Since
X=Xa, (_",-2)OX b, _",-2,
we have
x(a) = ( - .\-Z)O. Aaa. ( - .\-Z)Aba = - .\-40+Z,
x(b) = ( - .\-Z)O. Aab. (- .\-2).'1bb = - .\60-4.
Let a(t), f3(u), y(v) be defined as in 13.6.4 (vii). Then
hex) y(v) h(X)-l =y(x(3a + 2b) v)
and
h(x) y(v) h(X)-l y(v)-1=y«x(3a+2b)-1) v)
by 13.6.4 (vii). Now x(3a+2b)=-.\-z. Suppose xt3a + 2b) = 1 for all.\.
Then _.\-z= 1 for all .\/;O in K, hence .\'1= 1 for all .\:;1:0 in K. Now K
is a field of characteristic 3, so if K i= GF(3) we can choose X so that
x(3a+ 2b) =1= 1. It follows that y(v) E(Gl)/ for all v E K.
Now consider the elements f3(u). We have
hex) f3(u) h(x)-1=f3(x(3a+b) u)
and so
hCx) f3(u) h(X)-1 f3(u)-1=f3{(x(3a+b)-1) u)
by 13.6.4 (vii). Now x(3a+b)=.\-60+Z. Suppose x(3a+b)=1 for all .\.
Then .\-6,7+Z = 1 and so .\-2+20 = 1 since 6(J2 =2. It follows that .\-6+60 = 1
and so .\-4 = 1. Thus .\4 = 1 for all .\ =F 0 in K. ] f K /; G F(3) we can therefore
choose X so that x(3a+b):l1. Then f3(U)E(GI)' for all UEK.
Now consider the elements a(t). We have
Iz(x) aCt) h(X)-1 =a(x(h) t)
and so, by 13.6.4 (vii), we have
h(x) a(t) h(X)-1 a(t)-I=a{(x(b)-1) t) (J(u) y(v)
for certain elements u, vEK. However, f3(u) and y(v) belong to (G1)' and
therefore a({x(b)- J) t) is in (G1)' also. Now X(b) = _.\60-4. Suppose
;.:(b) = 1 for all .\. Then _.\60-4= 1 and so _.\Z-40= 1 since 6()z=2. It
follows that .\120-8=1 and _.\6-120=1, whence _.\-z=1. Therefore
.\-1=1 for all .\/;O in K. If K/;GF(3) we can choose X so that X(b)/;l.
. Thus a(t) E(G1)' for all t E K.

268
SIMPLE GROUPS OF LIE TYPE
We have now shown that
xs(t, ll, v) = a(t) f3(U) y(v) E(Gl)'
for all t, U, vEK, hence Xl S; (Gl)' as required.
(f) We have now shown that x1 S; (01)' for all equivalence classes S
of (1), except when 01= 2A2(22), 2fl.,{2), 2G2(3), 2F4(2). Since the sub-
groups Xh generate 01 we have 01=(01)' except in these cases. Thus
01 is simple by 11.1.1. .
Note. The four twisted groups which have not been proved to be
simple are aU in fact not simple. 2A2(22) is a soluble group of order 72,
2B2(2) is a soluble group of order 20, 202(3) is a group of order 1512
which has as its commutator subgroup the simple group Al(S) of order
504, and 2F4(2) has as its commutator subgroup a subgroup of index 2
which is simple of order 211.33.52. 13. (2.f"4(2)' can be proved to be
simple by an argument along the lines of the preceding discussion (cr.
Tits [12]). This group is not isomorphic to any of the other simple groups
of Lie type which we have discussed.
14.5 Identification with some Classical Groups
We show now that the twisted groups 2Al(K) and 2D1(K) are isomorphic
to certain classical groups.
THEOREM 14.5.1. 2A1 is isomorphic to the unitary group PSUl+1(K, f)
leaving invariant the Hermitian form
f= <:(XQXl-XIX1-1 +X2XI-2- ...)
with matrix
-1
1
A=<:
-1
Here € is defined to be 1 if I is even and an elernent of K satisfying E + i = 0
if I is odd.
FURTHER PROPERTIES OF THE TWISTED GROUPS
269
-\
PROOF. By 3.5.2 there is an automorphism of the Lie algebra JL such
that
er--+e"
hr-hr
for r E Il or - r E n. If we transfer to the field K we obtain a corresponding
automorphism of the Lie algebra JLK. If we then combine this auto-
morphism with the field automorphism t--+i of K we obtain a 'semi-
automorphism' tj; of JLK which satisfies the conditions
tj;(hr)=hf, tj;(er)=e,.,
tj;(AX + fLY) = Atj;(X) + p.1j;( y),
rE:!:n,
A, fL E K; x, Y E'1LK.
We now consider the map
0_ tj;Otj;-l,
where 0 is an element of the Chevalley group G= J[(K). The transforma-
tion tj;Otj;-l is easily seen to be an automorphism @f JLK and we have
tj;Xr(t) tj;-l = tj; exp (ad ter) tj;-I
=exp ad (tj;.ter)=exp (ief)=Xi{i)
for rE:!: n. Thus tj;Xr(t) tj;-lEG, and since the elements xr(t) for rE:!: n,
tEK, generate G it foHows that the map O--+tj;Otj;-l transforms G into
itself. It is in fact an automorphism of G, for it is invertible and satisfies
\
) ) -".
!
\
J
r po_
i
i

i
t
tj;Ol OZtj;-l = tj; 01 tj;-I . tj;02tj;-1.
However, this map coincides with the automorphism (j of G on the
generators xr(t) for r E :!: 11, so is equal to u. Thus the elements of G
invariant under u are those which commute with the semi-automorphism
';1.
In the present case '1LK may be taken as the Lie algebra of (1+ 1) x (1+ 1)
matrices of trace O. The fundamental root vectors may be taken as e1, HI
for i = 0, 1, . . . , 1- 1 and the fundamental co-roots are then
{.:: of"'!'lI-'?:!
g-
¥
I'. .....ur,.
f
f
:

eu - eHI, HI,
i = 0, 1, . . . , I-I.
The automorphism of JLK given by the symmetry of the Dynkin diagram
acts as follows:
1'
e1, HI-el--1-I, [--1,
eu- eH1, H1-el-t-1, 1-1-1 -el-1, [-1.
K
....

"'-
.;
tw};!!_
ft---"

i,»-;.
r'
L;
'="-' ,;
[
f"
L

t...._
r
I
t..
i
L..
[--
f
I
L_
....-
'-...
270
SIMPLE GROUPS OF LIE TYPE
FURTHER PROPERTIES OF THE TWISTED GROUPS

Now it is easiJy verified that the maps l"f--+ -M' (where M' is the trans-
pose of M) and A1--+A'-1!vJA are both automorphisms of '1[.J{. Com.
bining them we obtain an automorphism Jvf--+ -A-1Al 'A. Under this
automorphism we have
THEOREM 14.5.2. 2 DI{K) is isomorphic to the orthogonal group
PD.u{Ko, f), where Ko is Ihe fixed field of K under lite field automorphism
IIsed to define 2 DI(K) and.f is the quadratic .form
eij--+ ( - 1 )H1+ 1ez_j. Z-i.
Thus this automorphism behaves in the same way as the automorphism
induced by the symmetry of the Dynkin diagram when operating on
Ci, HI and eU-eHl, HI. Thus these two automorphisms coincide. Thl:
semi-automorphism .p defined above is therefore given by
M -A-lAf'A.
Now we have seen that G consists of the automorphisms of '1[.K given
by
XlX-1+X2X-2+... +XI-1X-(1-1}+{XI-CXX-I)(XI-exX-I),
where ex is a generator of K ovcr Ko.
(Note that the quadratic form f is defined over Ko. It has index [-I
regarded as a form over Ko and I regarded as a form over K.)
PROOF. It was shown in I I .3.2 tha t the Chevallcy group G = D I{K)
is isomorphic to the orthogonal group Pfhl{K, fn), where fn is the quad-
ratic form
M--+TMT-l,
)'1)'-1 + )'2)'-2 + . . . + )'1)'-1.
TESLI+l{K).
We consider which of these automorphisms commute with .p. In order
for this to be so, T must satisfy the condition
-A-l{T-l)'M'T'A= -TA-lM'AT-l,
Let
A=G, ,)
which impJies
be the matrix of fn. The group D..'2/(K,fD) is the commutator subgroup
of 021(K,fn). It is generated by matrices exp (te,.), where rEIl or - n
and tEK, and all its matrices T satisfy T'AT=A. If we use the matrix
representation given in 11.2.3, the matrices exp (ter) for r E n are
M'T'ATA-l=T'ATA-lM'.
As this hoJds for aU M EJLK we must have
T'ATA-1=)./
for some 'AEK. Thus T'AT='AA.
Suppose T is an upper unitrianguJar matrix. Then by comparing the
(0, I)-coefficients on each side we have 'A= 1. Similarly, if T is lower uni.
triangular a comparison of the (I, O)-coefficients shows that 'A = 1. Thus
the matrices T giving rise to the elements of VI and VI are precisely the
upper and Jower unitrianguJar matrices of the group SUz+1(K,f). However,
SVz+1(K, f) is generated by its upper and lower unitriangular matrices.
Thus G1, the group generated by VI and VI, consists of all transforma-
tions Joyf--+TMT-l, where TESUI+1(K, f). Therefore G1 is isomorphic to
PSU1+1(K,j). .
l+t(e12-e-2, -1), I+t(C23-e-3, -2),..., I-H(el-1, z-e-l, -(I-I}),
1+/{el-l, -l-CI. -(I-I,).
These matrices correspond to the nodes 1, 2, . . ., I respectively in the
Dynkin diagram
1-1
L--LL----
,
We note that the index v(f) of the Hermitian form f is (l+ 1)/2 jf I is
odd and //2 if / is even. Thus v(j) is as large as it can be in the Jight of
Witt's theorem.
Now consider the map
exp (ter)--+exp (tef)
rEi IT.
271
/0
(f[ .,.. . i.
i'

272
SIMPLE GROUPS OF LIE TYPE
FURTHER PROPERTIES OF THE TWISTED GROUPS
(a is a generator of Kover Ko). Then the matrix of the formln with respect
to Xl, . . . , Xl, X-I, . . . , X-I is S'AS, where
A....."l.t. p::;: 1- t!-e ,_( - (-{,e -.f'll -l_r./
fn;>".MJ.. : It n-:f'l'. jd_' -....
ce-t(.e '\..\I>'> -,  '0
[A a.".L.to k.L",,_ _ L;I..t<k).
The operation of transformation by the matrix B below induces this
map.
1
2
S= ( [2-2
O'
B= 0 1 I
1 -1
. 1
0 -I
A=
So= G
We note that a matrix M satisfies
Transformation by B leaves the first 1-2 matrices exp (te..) invariant
and interchanges the last two. I t is convenient to write the rows and
columns in the order 1,2,... ,I-I, -1, -2,..., -(/-1),1, -l. Then
o ),
o
[l-I
o
/z-I 0
o
o
l
1
o
-).
-a
/ll'(S'AS) M=S'AS
if and only if T=SMS-I satisfies
T'AT=A.
( /21-2 .)
B=
0
where
BoG ).
;} ,
We consider the conjugate subgroup S-ID.2l(K,jn)S and investigate
which matrices 111 in this subgroup correspond to matrices TED.21(K,fn)
such that BT= TB. Let
M_(Mll
- /1121
Since the map exp (ter)-4exp (fer) is induced by the transformation
T-4B-ITB,
MI2)
M22
2/-2
2
Then, using the fact that BoSo = So, we see by matrix multiplication that
BT= Tn if and only if
/ Thus the subgroup of D.z1(K, In) of elements T invariant under a IS
/ SD.21(Ko,j) S-I. Since G is isomorphic to PD.2I(K, flJ) it foHows that
\ Gl is isomorphic to PD.21{Ko.J).
we shall need to consider the matrices fixed under this transformation.
These are the matrices TED.2l(K, In) which satisfy BT= TB.
We now change the basis of the underlying 21-dimensional space over
K so that the point which originaUy had coordinates
Mll = Mll,
MI2=MI2,
YI, . . . ,YI, Y-I, . . . ,Y-I
now has coordinates Xl, . . . , Xl, X-I, . . . , X-I, where
Yt =Xt,
i = 1, . . . , 1- 1, - 1, . . . , - (/ -1),
YI = XI- aX-I,
./
i/
.;;.
Y-I =Xl- a;X-l
2/- 2 2
M21 = M2I.
M22=M22.
\ '-
t "',
273
------.
J
i
;
i-
(
'1 
/
i
(.'. \

kp
--."
j
: ..",...,.
r

w---,
"
- 1 '''=-
I
J
! ["
I .""'"
".."., i
j
I
I
i
CHAPTER 15
Associated Geometrical Structures
---
i
L"
Certain geometrical structures, called buildings, on which groups with
a (B, N)-pair operate as groups of automorphisms, have recently been
introduced by J. Tits. We shall describe these structures here and
demonstrate the connection with groups with a (B, N)-pair. Closely
connected to these geometries are somewhat simpler structures on which
Weyl groups (or more gl1erally Coxeter groups) operate as groups of
automorphisms. These structures are the Coxeter complexes, which we
have described in earlier chapters. However, in ordedescribe the
buildings we first need an axiomatic system which picks out the essential
features of the Coxeter complex. A geometry satisfying these axioms
wiII be called an abstract CC!.?:l!.!er complex. We shaH introduce these
ideas by means ofa-serle-s--ofdefiiilo-nsot Increasing complexity.
 ..-
f .
t""
't--
.  .
; I I
1.-_...

i
i ._
. .
 ,t ;
 '; i
 i t
, I
 i l
\ t ;
j 'I' 
, :: t_
 \} 
(
Lr
; 1 ,
! --I 
",",- I -I
, i
: I
. !
i __.
15.1 Chamber Complexes
I
J
i:
i"-
1- '
I. 
;,
0jl
;\\,
We consider a set 8 endowed with a relation S of partial ordering.
o is called a simplex if 0 is isomorphic to the set of all subsets of some
set, partial1y -ofdred by inclusion. 8 is calIed aS2!.!}J!.lex if:
(a) For each A E0 the set of elements BEe such that B S A forms a
simplex.
(b) Each pair of elements A, BE0 have a greatest lower bound A nB.
It follows from (a), (b) that a complex 0 contains a unique minimal
element, which will be caIled O.
We define the rank of each element of a complex 8. Rank A is the
number of elements B such that B is minimal with respect to the pro-
perties B S A, B-I=O. Thus the set of elements B with B s A is isomorphic
to the set of subsets of a set with cardinality rank A.
A subset 8' of 0 with the induced partial ordering is called a sub.
complex if, for all A E0', BE0 with B s A, we have BE0'. There exist
subsets of a complex 0 which are complexes, although not subcomplexcs
of 8. For example, let A E8 and define St A (tbe star of A) by:
8t A={BE0; B  A}.
274
l (:"< /
\ 
ASSOCIATED GEOMETRICAL STRUCTURES
275
Then St A is a complex contained in e but not a subcomplex of 8 (unless
A=O).
For any two elements A) B E8 with A s B we define the codimension
of A in B by
codimn A = rankst A B
In particular codimn A = 1 if and only if A -1= B and there is no element
X with AcXcB. Also codimnA=O if and only if A=B.
A complex (0 is called a chamber complex if:
(a) Every clement or 8 is containeu in a maximal element.
(b) Given any two maximal elements C, C' of 8 there exists a finite
seq lIcnce
C=Co, Cl, Cz,..., Cm=C'
of elements of 0 such that
codimcl_l (C1-1 n CI)=codimcf (Ci-l n C1)  1
fori=I,...,m.
The maximal elements of a chamber complex will be called chambers.
The above condition on the codimension means that Ci-1 is either equal
to Ct or that C1-1-l= Ci and the intersection has codimcnsion 1 in each.
LEMMA 15. 1 . 1. An element of a chamber complex has the same co-
dimension in all the chambers containing it.
PROOF. Let A E0 be contained in two chambers C, C'. Then there
exists a sequence
C=Co, CI,..., Cm=C'
as above. Let B=Co nCl n ... nCm nA. Then
codimcf_l B=codimcf B for ;=1,..., In
and so
codimc B=codimc' B.
Now it is readily verified that codimc (Co n Cl n . .. n Cm) IS finite
and it follows that codimA B is finite also. Hence
codimc A =codimc B-codimA B
=codimc' B-codimA B=codimc' A.
.

276
SIMPLE GROUPS OF LIE TYPE
ASSOCIATED GEOMETRICAL STRUCTURES
277
'""-)
It follows from 15.1. I that the terms of a sequence Co, Cl, . . . , Cm of
the type described above are aJJ chambers. A sequcnce of chambers of
this type wiJ1 be called a gallery. Two chambers C, C' are said to be
adjacent if co dim (CnC')=l. Thus a gallery is a sequence of chambers
in which every pair of consecutive chambers are either identical or
adjacent.
We now consider maps from one chamber complex to another. Let
8, 8' be chamber complexes. A map 0: : 8-48' is called a morphism of
chamber complexes if:
(a) o:(C) is a chamber of 8' for each chamber CE0.
(b) For each chamber CE8, 0: induces an isomorphism between the
simplexes determined by C, o:(C).
It is clear that a morphism of chamber complexes preserves the partial
ordering (8  A implies o:(E)  o:(A)) and leaves invariant the rank of
each element. A morphism of 8 into itself is calkd an endomorphism, and
an endomorphism which is invertible is called an automorphism.
A chamber complex is said to be thill if every clement of codimension 1
is contained in exactly two chambers and thick if every element of co-
dimension 1 is contained in at least three chambers.
0:, (3 coincide on Ct and on all faces of the element Cl-l n Cl of codimension
1 in Ci. Since the faces of Ct form a simplex and 0:, f3 are isomorphisms
between the simplexes determined by Ct and o:(Cl) it follows that 0:, {3
coincide on all face of Ct, and we have a contradiction. .
. " 15.2 FoIdings
Let 8 be a thin chamber complex. A folding of 8 is an endomorphism 0:
of 8 satisfying:
(a) 0:2=0: (Co is idempotent).
(b) Each chamber in 0:(8) is the image under 0: of exactly two chambers
of 8.
The idea of a folding is the key to the definition of an abstract Coxeter
complex. We shall now elucidate some properties of foldings.
LEMMA 15.2.1. Let 0: be a foldillg of 8. Then there exist adjacent
chambers C, C' of 8 such that CEo:(8), C'if=0:(8). rr C, C' are any two
slich chambers we have o:(C')=C.
............,
LEMMA 15. 1 .2. Let 8, 0' be two chamber complexes in 'which each
element of codimensioll 1 is contained in at most two chambers, and let
0:, f3 be two morphisms oJ8 into 8' injective on the set oj cham.bers. Suppose
there exists a chamber CE8 such that o:(A) = f3(A) for all A  C. Theil rx=f3:
PROOF. Since 0: is idempotent there exist chambers Cl, C2 with
CIEo:(8), Cz rto:(0). Since Cl, C2 can be joined by a gallery there exists
a pair of adjacent chambers C, C' in such a gallery with CEo:(8),
C'f{0:(8).
Now all the faces of C are in 0:(8) and so 0: acts as the identity on
them. In particular o:(C nC')=C nC'. Thus o:(C')::> C nC'. Since 0
is thin this implies that o:(C')=C or o:(C')=C'. But C'if=0:(8), thus
C1= .
PROOF. The elements A satisfying A  C will be called the faces of C-
Suppose there is a chamber C' E8 such that 0:, f3 do not coincide on the
faces or C'. Let r be a gallery of minimal length joining C to C'. Then
r={c=co, CI,..., Crn=C'}.
Now there exists an integer i such that 0:, f3 coincide on the faces of Cl-l
but not on all faces of Ct. We have
{J(Ct)::> (3(Ct-l nCd=o:(Ct-l nCt).
But o:(Ct-l n Ci) is contained in only two chambers, which are o:(Ct-I)
and o:(Ci) since 0: is injective on chambers. Thus (J(Ct)=o:(Cf.-I) or
(Ci)=o:(Ct).
Suppose (3tCi)= o:(Ct-l). Then (3(Ct) = {3(Ct-l) , contradicting the fact
that f3 is injective on chambers. (Ct-l, Ct are distinct since r is a
gallery of minimal length joining C to C'.) Thus {3(Ct)=o:(Ct). I-Icnce
A set of chambers in a chamber complex is called convex if every gallery
of minimal length joining two chambers in the set has all its terms in
t he set.
LEMMA 15.2.2. Let 0: be a folding of 8. Then the chambers in 0:(0)
form a convex set in 0.
PROOF. Let C, C' be two chambers in 0:(8) and
I'={C=Co, Cl,. .., Cm=C'}
be a gallery of minimal length joining C, C'. Suppose r contains some


'--
t
r--

L,
Ii.--
r.

t."",
!--.
W
L
r
t..
'-
r
L..

i
f
. -
I
I
i
L.
r--
I
I
L..
f-
!
'-
L.
278
SIMPLE GROUPS OF LIE TYPE
ASSOCIATED GEOMETRICAL STRUCTURES
279
term C{ £/:ex(8). Then we can find two consecutive terms Cj-l. Cj in r
such that Cj-IEex(8), Cj£/:ex(8). Now ex(r)={ex(Co),....ex(Cm)} is also a
gallery joining C to C'. By 15.2.1 we have ex(Cj)=Cj-1 and so
ex(Cj-l) = ex(Cj).
Then ex(Ct-I)=Ct by 15_2.1 and so cx(Ct)=Ct-l. Thus a:(Ct)=cx(Ct-1)
and the gallery cx(f') has two consecutive terms which are equal. By
omitting one of these we would obtain a gallery joining C, C' which is
shorter than r, a contradiction. .
Thus two consecutive terms of ex(r) are the same, and by omitting one
we could obtain a gallery joining C to C' of shorter length than r, a
contradiction. II
Let C, C' be chambers in a chamber complex 8. The distance bctween
C, C' is defined as the shortest lcngth of a gallery joining C, C'. Thus
if r={c=c(J, CI,..., Cm.=C/} is a gallery for which /II is minimal we
define
LEMMA 15.2.3. Let ex be a folding of 8. Then the chambers not ill
a:(0) also form a convex set in 8.
elist CC'=m.
In particular, dist CC' =0 if and only if C = C' and dist CC' = 1 if and
only if C, C' are adjacent.
PROOF. To establish this result we introduce a map a on the set of
chambers of 8. If C is a chamber not in ex(0) we define cx(C)=C; while
if C Ea:(8), C is the image under ex of exactly two chambers. One of these
is C and the other is defined as cx( C).
Let C, C' be two chambers not in ex(8) and
r={c=co, CI,.. . , Cm=C'}
LEMMA 15.2.4. Let a: be a folding of the thin chamber complex 8 and
C, C' be adjacent chambers of G such that CEa:(0), C'cta:(0). Then for
any chamber D of e we have:
dist C'D=dist CD+l if DEa:(G),
dist CI D =dist CD -1 if D cta:(0).
be a gallery of minimal length joining C, C'. We show that iY{r) is also
a gallery. To do this we must prove that the images under a of two adjacent
chambers D, D' are either adjacent or identical. This is clear if neither
D, D' is in ex(0). So suppose D Eex(0). Then a:(cx(D)) = D. Let A be the
face of lieD) such that a:(A) = D n D'. Let D" be the chamber containing
A other than lieD).
PROOF. Since C, C' are adjacent it is clear that
dist CD-ldist C'Ddist CD+1.
Suppose DEa:(0) and r is a gallery of minimal length joining D to C'.
Then r contains two consecutive terms, one in ex(0) and the other not.
By 15.2.1 a:(r) is a gallery joining D to C which contains two consecutive
terrns which are equal. Hence dist CDdist C'D-l, and we must have
equality.
If D cta:(8) we take a gallery r of minimal length joining D to C. By
the proof of 15.2.3 li(r) is a gallery joining D to C' which contains two
consecutive tcrms which are equal. Hence dist C' D  dist CD - 1, and
we again have equality. .
Ci(D) D"
V
A
o 0'
V
DnO'
a
If D"Eex(0), we have A Ea:(8) and so ex(A)=A. Then li(D)=D' since
o is thin and a(D)cta:(0). Thus cx(D') = D' and li(D)=li(D').
If D"£/:a:(8), we have ex(D")=l-D (since D"-j;a(D). Thus ex(D")=D' and
D" =li(D'). Hence lieD), lieD') are adjacent.
Thus lieD), cx(D') are either adjacent or identical and so
li(r) = {cx(Co). li(CI),.. ., li(Cm)}
is also a gallery joining C, C'. Suppose some term of r is in ex(8). Then
we can find two consecutive terms Ct-l, Ct of r with Cf-l fj:a:(8), C, Ea:(G).
: PROPOSITION 15.2.5. Let ex be a folding of G and C, C' be adjacent
chambers of 8 such that CEex(G), C' fj:a:(G). Then ex is the only folding
.such that a:( C') = c.
PROOF. We showed in 15.2.1 that a:(C')=C. Let {3 be any folding
of 8 such that {3(C')=C. Then 15.2.4 implies that, for any chamber

280
SIMPLE GROUPS OF LIE TYPE
ASSOCIATED GEOMETRICAL STRUCTURES
281

"
D EG, D belongs to a{G) jf and only jf D belongs to [3(8). Since 0:(8), [3(0)
are chamber complexes we have a{8) = [3(8). 0: and [3, being idempotent,
both operate as the identity on 0:(8).
Let 0:(8) be the subcomplex of 8 consisting of all faces of all chambers
of the form (i(D) for D EG. The chambers of form (i(D) for D E8 are
those which are not in 0:(8), and so form a convex set by 15.2.3. l--lcnce
(i(8) is a chamber complex. Similarly (8) is a chamber complex. Dy
15.2.4 a chamber belongs to (i(8) if and only if it belongs to (8), hence
(i(8) = peG).
Now consider the two morphisms:
p2 is injective on chambers also. Let C, C' be a pair of adjacent chambers
such that ex(C')=C and [3(C)=C'. Then p2 fixes C and also fixes each
face of C n C', hence p2 fixes each face of C. By 15.1.2 p2 is the identity.
In particu lar p is invertible, so is an automorphism. .
15.3 Abstract Coxeter Complexes
(i(8)8,
fJ
0:(8)8.
An abstract Coxeter complex is a thin chamber complex L such that,
given any pair C, C' of adjacent chambers, there exists a folding 0: of L
with o:(C') = C.
Since 0:, {3 are foldings, they are injective on chambers when restricted
to (i(8). Moreover, ex, (3 coincide on the chamber C' E 0:(8) and on all the
faces of the element C n C'. Thus 0:, {3 coincide on all faces of C'. Dy
15. 1 .2 it follows that 0:, (3 coincide on a(8). But 0:(8) U 0:(8) = 8, thus
0:, (3 coincide on 8. _
LEMMA 15.3. 1. Let L be an abstract Coxeter complex and A be an
element of L. Then Stk A i.s al.so an ab.stract Coxeter complex.
PROOF. Stk A is certainly a complex. To show it is a chamber complex,
let C, C' be chambers containing A. There exists a gallery
PROPOSITION 15.2.6. Let 0: be a folding of G and C, C I be adjacent
chambers of G such that ex( C ') = C. Suppose there is a folding {3 slich that
(3( C) = C'. Then (3 has the same property for any other pair oj adjacent
chambers oj this type, viz., if D, D' are adjacent and ex(D') = D, then
(3(D) = D'.
r={c=co, Cl,..., Cm=C'}
-\
in L joining C, C', where m=dist CC'. Let ex be the folding of L such
that exeC) = Cl. Then
PROOF. By 15.2.4 a chamber is in ex(8) if and only if it is not in [3(8).
Thus D'Ef3(8), Drj{3(8), and so (3(D)=D' by 15.2.1. 1/1
dist C1C'=dist CC'-I,
hence C' E ex(L) by 15.2.4. Thus all faces of C n C' are in ex(L). Consider
the minimal non-zero faces of C. These are all faces of C n Cl except
one, which we call V. Now V s C and so ex(V) s Cl, hence ex(V) #- V.
Thus V i'ex(L), and so V cannot be a face of C n C I. It follows that
C n C' s C n Cl. Hence A is contained in Cl. Using induction we see
that A is contained in each Ct. Thus St A is a chamber complex. Stk A
is certainly thin, so it remains only to check the existence of all possible
foldings.
Let C, C' be adjacent chambers both containing A. There is a folding
0: of}: with ex(C') = C. ex fixes A, so induces an idempotent endomorphism
of Stk A. The same applies to the opposite folding (3 of ex. Let D be any
chamber in ex(1:) containing A. Then (3(D) also contains A, and
The folding {3 of 15.2.6 which, when it exists, is uniquely determined
by ex, is called the opposite folding of ex.
PROPOSITION 15.2.7. Let ex, [3 be opposite foldings oj G. Then there
exists an automorphism p of 8 )-I,,'hich coil/cides wilh (3 0/1 ex(8) and with ex
011 (3(G). Also, p2 is the identity.
PROOF. ex, f3 both operate as the identity on ex(G) n f3(G), so p is weIl-
defined. It is clearly an endomorphism of e. Now p is injective on
chambers. For if p(C1)=p(CZ) Eex(G) we have C1, C2E{3(8) and
ex( Cl) = ex( C2),
wl1ence Cl=C2. The same applies if P(Cl)=p(C2)E{3(G). It follows that
exf3(D) = p2(D) = D
by 15.2.7. Thus the two chambers D and f3(D) which ex transforms into
D both contain A. Hence ex induces a folding of Stk A. .
\

filf;N
.
282
SIMPLE GROUPS OF LIE TYPE
ASSOCIATED GEOMETRICAL STRUCTURES
283
'-
We shall now prove some general properties of abstract Coxeter com-
plexes, showing first that such a complex can be 'folded down' into a
single chamber.
leaves invariant all faces of C. It is equally cIcar that pc is the only endo-
morphism with this property. II
:j11!5O" 
i
PROPOSITION 15.3.2. Let  be an abstract Coxetcr complex, C a
chamber in  and S(C) the simplex of all faces of C. Then there exists a
unique idempotent morphism of L onto S(C).
The map pc is called the retraction of :L; on to the simplex of faces
of C.
We now introduce an equivalence relation on . Given A, BE we
write A",B if pc(A)=pc(B).
:.' r
'\.t'jt
i 
:[
" t,
Note. An idempotent morphism will be called a retraction.
LEMMA 15.3.3. The equivalence relatioll defined on L is independent
of the chamber C.
I
1'r
;, \""',
PROOF. Let r={c=co, C1,...,Cm} be any gallery beginning with
C. We show there is an endomorphism y of  leaving invariant all faces
of C such that y(r)={C, C,...,C}. We use induction on the length
m of r, the result being clear if 111 =0. If C1 = C the result is clear by
induction, so we assume C1#C. There is a folding a of  with a(Cl)=C.
Thus a:(r)={C, C, . . . ,a:(Cm)}. By induction there exists an endomorphism
o leaving invariant all faces of C such that oa:(r) = {C, C, . . . , C}. Then
y = oa has the required properties.
It follows that, given any finite set C, of chambers of , there exists
an endomorphism y of 1: leaving invariant aJl faces of C snch that
y(Ci)=C for each i. For there exists a gallery beginning with C and
containing all the Ct.
Now for each A E there exists an endomorphism y leaving invariant
all faces of C such that y(A) ES(C). For if we choose a chamber C' con-
taining A there is an endomorphism y fixing all faces of C with y( C ') = C.
We now show that the element y(A) ES(C) is uniquely determined, i.e.
that if y, 0 are endomorphisms of  leaving invariant all faces of C such
that y(A) ES(C) and O(A)ES(C), then y(A)=o(A).
Let C' be a chamber containing A and r={c=co, C1,.. ., Cm=C'}
be a gallery joining C to C'. Consider the galleries y(r) and o(r). There
is an endomorphism E" of  which leaves invariant all faces of C and
maps each chamber in y(r) and in ocr) to C. Consider the endomorphisms
«:y, «:0 of 1:. They agree on all faces of C=Co. Suppose by induction that
they agree on all faces of Ct-1. Then they agree on C'l and on all faces
of Ct-1 n Ct, thus (by the usual argument) they agree on all faces of Ci.
In particular «:y, «:0 agree on all faces of Cm = C '. Thus «:y(A) = «:o(A).
Since y(A), o(A) are in S( C) this gives yeA) = o(A).
We now define pc(A) to be the common value y(A) for all endornorph-
isms y of 1: leaving invariant al1 faces of C and such that y(A) ES(C).
It is dear that pc is an endomorphism from L into S(C) and that pc "
PROOF. Let C' be another chamber of L. We show that the equivalence
relations for C, C' arc the sa me. Since any two chambers can be joined
by a gallery we may assume C, C' are adjacent. By 15.2.7 there exists
an automorphism 0 of  such that o( C) = C '. Also 0 maps each face
of C n C' into itself. Thus 0 coincides with pc' on each face of C n C '.
Since o(C) = pc'(C), 0 coincides with pc' on each face of C. Similarly
8-1 coincides with pc on each face of C '. Thus 0pc is an endomorphism
of  into S(C '):
 ':
LS(C)S(C')
and 0pc leaves invariant each face of C'. Hence opc= pc'. It follows that
pc(A) = pc(B) if and only if pc'(A) = pc'(B). .
\... -
=_' Ii L
:w [--
It'!
-iti -
it!.1fl
:  -
''' I
rh
"M r-
:r l
.- f,i "
fi!
"'I r-
II L
II rN
 ,
 I ,
-J,,-
! l
'Ii
_ ui 
\ ,--
; :
:;, L__
We shall say that equivalent elements of L have the same type. It is
clear that each chamber has exactly one face of each type.
LEMMA 15.3.4. Let L be an abstract Coxeter complex and C be a
chamber of L. Let y be an endomorphism of L leaving invariant the type
of each face of C. Then y preserves the type of each element of L.
PROOF. Let C' be another chamber of L. We show that y leaves
invariant the type of each face of C '. Since any two chambers may be
joined by a gallery we may assume that C, C' are adjacent. Now y leaves
invariant the type of each face of C n C' and it also ]eaves invariant
the type of C'. (All chambers have the same type.) 'Thus, since y(C')
has just one face of each type, y leaves invariant the type of each face
of C'. .
Endomorphisrns and automorphisms of the kind discussed in 15.3.4
will be called type-preserving.

284
SIMPLE GROUPS OF LIE TYPE
ASSOCIATED GEOMETRICAL STRUCTURES
285
:\
LEMMA 15.3.5. The automorphisms p defined in 15.2.7 are type-
preserving.
PROPOSITION 15.3.6. Let  be an abstract Coxeter complex. Then
JV(L) is the group of all type-preserving automorphisms of L.
the Wi. We show that for each chamber D of L there exists WEH such
that D= w(C). We do this by induction on dist CD, the result being
clear if D is identical with c;: or adjacent to it. Otherwise there exists a
chamber E adjacent to D such that
dist CE=dist CD- I.
By induction E= w'(C) for some w' EH. Now W'-1 (D) is adjacent to C, so
has form H'f(C) for some iEI. Thus w'-l(D)=w.t(C) and so D=W'Wi(C),
where w'wiElf as required.
Let w be any element of W(L). Then there exists w' EH such that
w'(C)=w(C). Thus w-1w' is a type-preserving automorphism of 
fixing C, so must be the identity by 15. 1 .2. Thus tv E Hand H = rV(L).
We now show that the reflections Wi generate W(L) as a Coxeter group.
Each clement WE rV(L) can be expressed as a product
PROOF. Let <x, f3 be a pair of opposite foldings of  with respect to
which p is defined. Let C, C' be a pair of adjacent chambers such that
CE<X(L), C/Ef3(L). Then p leaves invariant each face of C nC/ and
interchanges C, C /. Thus p preserves the type of C and of each face
of C n C /, hence p preserves the type of each face of C. By 15.3.4 p is
type-preserving. ..
Let rV(L) be the group generated by a]] the automorphisms of  of the
kind described in 15.2.7.
PROOF. By 15.3.5 each e1ement of W(L) is a type-preserving auto-
morphism of L. Conversely, let 0 be any type-preserving automorphism
of L. Let C be a chamber of L. Since any two chambers can be joined
by a gallery it follows from 15.2.7 and the fact that L is a Coxeter complex
that there exists yE J-V(L) such that y(C)=o(C). Thus y-ID is a type-
preserving automorphism of L which fixes C. But a chamber C has just
one face of each type. Thus y-1o fixes each face of C, so must be the
identity, by 15.1. 2. Hence 0 E W(L). II
W=WiIWiz' .. Wik' iaEf.
Let l(w) be the minimal length of any expression of W in this form. We
show that
l(w)=dist (C, w(C)).
...."""" -
f "...,='
Let C be a chamber of the abstract Coxeter complex L. Each chamber
adjacent to C intersects C in a face of codimension 1. Now the number
of faces of codimension 1 in C is rank C (which may be finite or infini,c),
and each such tlce is contained in just one chamber other than C. Thus
there are rank C chambers of L adjacent to C. Each of these chambers
gives rise to an involutary type-preserving automorphism of L: as in
15.2.7. These automorphisms will be called the reflections in the faces
of codinlcnsion 1 in C. -- -_.--.--__h_.
Suppose !(w)=k and W=WiIWiz' . . Wik with iaEI. Then
{C, Wil(C), Wi1Wt2{C)" . ., Wi1Wi2' . . Wi,lC)}
is a gallery joining C to w( C) of length k. Thus dist (C, 11'( C))  k and so
dist (C, w(C))I(w). Suppose conversely that dist (C, w(C))=k. Then
1 here is a gallery
THEOREM 15.3.7. Let  be an abstract Coxeter complex and C be
a chamber of L. Then the reflections ill the faces of codimensioJ1 1 in C
generate the group JV(L). .ftloreover V(r.) is a Coxeter group with respect
to this set of generators. (cf. 2.4.2.)
r={c=co, Cl, . . . , Ck=W(C)}
joining C to w(C). Since C, Cl are adjacent we have Cl = Wil(C) for some
hEf. Since Cl, C2 are adjacent we have C2=Wi1(C/), where C, C' are
adjacent. Hence C2=WiIWiZ(C) for some i2EI. Arguing in a similar way
we see that Ck=W.tIWiz' . . Wik(C) with each iaEf. Hence
W(C)=Wi1Wi2. . . Wik(C),
and since wand Wil Wtz . . . WtA; are both type-preserving we have
W=WiIWi2' . . H'tA;'
PROOF. Let the rellections in the faces of codimension 1 in C be
denoted by Wi, ic1; and let 1-l be the subgroup of rV(L:) generated by
Thus l(w)  dist (C, w(C)), and so we have l(w) = dist (C, w(C)).
We show next that if l(WilWiz' . . Wtk) <k and l(Wi2... wiA;)=k-l,
then there is some j 2 such that
.'
I""
Wiz. . . Wij= Wi! . . . Wij_l'

d
! 1
_: '--
o I
286
SIMPLE GROUPS OF LIE TYPE
This wilI be sufficient, using a theorem of Matsumoto, to show that
W() is a Coxeter group. Let W=WilWi2... Wik and write
Cr=WilWi2. . . WirCC)
, I
J
. I:.t
,-..;: ! 'I
.  1[.-'1
i .
I , 
,
i .i
; ! ir-
- ; .;Il::
-,  
I <i
 ....
- ! ;: [----
 i 
i,;[ ",...
..,- : ;: J
;;ITLI" ·
:. '.
....w_ ";'-1
; i  !:
-F/:I to-
:1  jl
'Iir-.-.
...- -Ii!:
J! r __
:1' L
JI
I t_
1>/
'i!
-I f-
H !
! L
II
tf f
h '
'I 1
If ..
"
F
:r--
 t
 \
 ,.-

j
;
for r=l, 2,..., k. Then
dist (Cl, w(C))=dist (WilCC), Wil. . . WiA;(C))
=dist (C, lI'i2 . . . wiiC))
= I(Wi2 . . . H'iA;)
=k-l.
Also
dist Cc, I1'(C))=I(w)::::;k- 1.
dist (C, Iv(C))::::;dist (Cl, w(C)).
Let  be the folding with (Cl)=C. By 15.2.4 we have W(C)E(1:).
Since Cl (1:), there exists j 2 such that Cj-l a:(L) but Cj EO:(L). Then
(Cj-l)=Cj. By definition of  we have o:(Cj-l)= wilCj-1), hence
Wi1(Cj-l) = Cj. Thus we have
Thus we have
Wi l' Wi 1 Wi2 . . . WtJ_l( C) = Wi 1 Wi2 . . . Wij( C).
Since all the automorphisms involved are type-preserving it follows that
whence
Wi2 . . . Wij_l = Wil Wi2 . . . Wi}'
Wi 1 . . . Wij_l = Wi2 . . . Wir
The proof of 15.3.7 is completed by establishing the foIIowing result,
due to Matsumoto.
THEOREM 15.3.8. Let W be a group generated by a set of involutions
Wi. For each WE W let l(w) be the shortest length of any expression of w
as a product of the involutary generators. A product Wil . . . Wik is called
reduced if l(wi; . . . Wik)=k. Suppose W satisfies the condition that, when-
ever Wi2 . . . Wik is reduced but Wil H1i2 . . . Wik is /lot reduced, there exists
an integer j with 2  j  k such that lVi2. . . Wij = H'il . . . Wij_l' Then JV is
generated by the Wi as a Coxeter group.
(The group W(1:) considered above satisfies the hypotheses of this
theorem.)
PROOF. Any relation between the generators Wi can be expressed In
the form
Wil . . . Wi,. = 1.
"
ASSOCIATED GEOMETRICAL STRUCTURES
287
\Ve show first that any such relation is a consequence of relations of form
w¥= 1 and
H.'it... Wj6=JIlk1,.. IVka,
where both expressions are reduced. \Ve prove this by induction on r
Since lI'il'.' Wt,. is not reduced there exists ex such that II't"'+1'.' Wi,.
is reduced but lI't",. . . Wi,. is not. Thus there exists un integer j r such
that Wt",... WiJ_l = Wi",+! . . . Wlr Both expressions in this equation are
reuuced. Using this relation and JV.l= 1 we deduce
Wi 1 . . . Wir = Wi 1 . . - lVi"'_llVi"+1 . . . lVij_1 Wii+1 . . . Wir.
Since the expression on the right has fewer than r terms the result follows
by induction.
We show next that each relation
Wit. . . H'ja= Wkl . . , Wks'
where both expressions are reduced, is a consequence of the Coxeter
relations (WiWj)m/i= 1. We again use induction on s. Since wit... Wjs
is reduced but WklWh . . . Wjs is not reduced, there exists   2 such that
IV leI Wit . . . I\Ij"_l = Wit . . . Wj",.
It follows that
wklWit... Wj"'_lWj",+!,., IVjs=IVkl... H'ks'
Both expressions in this equation are reduced, and by cancelling Wlel
we see by induction that this relation can be deduced from the Coxeter
relations.
We now distinguish two cases. Suppose  < s. Then
WklWit. . . Wja_l = wit. . . Wj"
can be deduced from the Coxeter relations, by induction. Thus
Wit. . . }\'js=WklWit. . . Wj"'__lIVja+1' . . Wj,
can be deduced from the Coxeter relations, and therefore so can
Wit. . . Wjs=Wkl. . . Wk,.
Suppose o:=s. Then we have
Wit . . . Wjs= WklWh . . . Wjs_l = Wlel . . . Wk,.
By induction, the relation
Wk1Ii'it. . . Wj,_l = Wlel . , . Wk,

288
SIMPLE GROUPS OF LIE TYPE
can be deduced from the Coxeter relations, so the required relation
Wit . . . Wis = Wkl . . . Wks
can be deduced from the Coxeter relations, provided that
wh . . . Wia = H'k1H.'it . . . Wi8_l
can be deduced from these relations.
We now repeat the argument. The relation
Wit. . . Wia= Wk1l-Vit. . . Wi8_l
can be deduced from the Coxeter relations provided that
Wk1H'h'" Wi8_l=whwk1Wh'" WiS_2
can be deduced from these relations. Repeating the argument a number
of times, we eventually have to show that
WhWklWh' . . =Wk1WhWkt. . .
is a consequence of the Coxeter relations, which is obvious.
.
15.4 The COllll}lcx ::E( W, IT)
We shall now show, as is to be expected, that the Coxeter complex as
defined in section 2.6 is an abstract Coxeter complex. .
Let <1) be a root system and W be the Weyl group of (D. Let IT be a
fundamental system of roots in (I>.
THEOREM 15.4. 1. The following two partially ordered sets are abstract
Coxeter complexes which are isolJlorphic to one another:
(i) The Coxetcr complex h ofJV (defined as ill section 2.6), with partial
order  defined by KI < K2 if and ollly if KI is contained in the closure
£2 of Kz.
(ii) The set "LUV, 11) of af! cosets wWJ for all WE Wand all subsets
J of 11, with partial ordering  defined by KI  Kz if and only if K2 is a
subset of KI.
PROOF. We show that Land 2:(11/, 11) are isomorphic as partiaJJy
ordered sets, and then show that 2:;UV, I1) is an abstract Coxeter complex.
We recall some facts about L. Let C be the chamber corresponding to
the fundamental system n. Then the elements of L contained in the
ASSOCIATED GEOMETRICAL STRUCTURES
289
closure C are the elements CJ defined in section 2.6. The stablizer of CJ
under the action of W on L is WJ, by 2.6.1. Moreover, each element
of L can be expressed in the form w(CJ) for exactly one subset J of IT,
by 2.6.3. Let w!, Wz be two elements of W. Then WI(CJ)= wz(CJ) if and
only if WIWJ=WZWJ. Thus the map
w(CJ)-+w WJ
is a bijection between 1: and L(l1/, IT). We show that this bijection pre-
serves the partial order relations.
Suppose wWJ > w' WK. Then wWJ is a subset of W'WK. It follows that
w'-lw,1/J is a subset of WK, and so W'-IWE WK and WJ is contained
in WK. This implies that J is a subset of K, since the fundamental roots
are linearly independent. Hence W V J > w' W K if and only if w'-Iw E W](
and J is a subset of K.
Now suppose w(CJ) > W'(CK). This means that W'(CK) is in the cIosue
of w(CJ), and so w-IW/(CK) is in CJ. In particular w-Iw'(C!d is in C.
However, the only elements of the complex contained in C are those
of the form CL, for some subset L of n, and if w-IW'(CK) = CL then
we have L=K by our earlier remarks. Thus w-Iw' E WK and CK is con-
tained in CJ. Jt is clear from the definitions of CJ, CK that this implies
that J is a subset of K. Thus w(CJ) >W/(CK) if and only if w-Iw' E WK
and J is a subset of K.
Thus we have shown that the bijection between 1: and L(W, TI) pre-
serves the partial orderings . We must now verify that '1:( W, TI) and
hence 1: also) is an abstract Coxeter complex. We observe that the maxImal
clements of (W, TI) under the ordering  are the elements of JY. Two
elements WI, wz of Ware adjacent if and only if W2 = WIH'r for some
r E n. Since the fundamental reflections Wr generate W, it follows that
l:( W, IT) is a chamber complex. 'L( W, IT) i-s thin, because the elements
of codimension 1 are sets of the form {w, WWr} for some WEW, rEn.
Thus each element of codimension 1 is contained in two chambers. To
show that L( Y, I1) is an abstract Coxeter complex it remains to prove
the existence of all possible foldings.
. Let IV' and W'Wr (rEn) be two adjacent chambers of 1:(W, TI). We
show there exists a folding a such that o:(W'Wr)= w'. Let s =- w'(r) and
defIne
---..
)
''1
WI = {w" E W; W"-I(S) E <t>+},
W-z = {w" E W; W"-I(S) E (])-}.
Thus W= WI U Wz and WI n Wz is empty. Let ex be the map of '1:(W, IT)
1

',-,
f
'b..
[
f--'

'_jJ
I!0'
i
l,.,
.Oi;=-","
{ C
i. 

[
\. -'
'- 
\
,......
290
SIMPLE GROUPS OF LIE TYPE
into itself given by
(WIYJ if wWJ $ W2,
o:(wWJ)=
wswVJ if wVJ s V2.
0: dearly maps chambers into chambers. If WE WI we have o:(wWJ)=JIIVJ
for all subsets J of IT. Thus 0: operates as the identity on all the faces
of the chamber w. Now suppose WE 'V2. Then o:(w)=wllw, and
0:(11' WJ)= WsW WJ
whenever wVJ S W2. We show that o:(wWJ)=wswWJ even if wWJ $ Wz
and this will prove that 0: induces an isomorphism between the faces
ofw and the faces of wsW. Now WE W2 and so w-](s)E<I)-. AJso wWJ $ W2.
so there is some element WJE WJ such that WWJE VI. Then wJIW-](S)E <1>1,
whence W-I(S)E VJ«I}+). Thus we have
W-I(S)E cD- n JYJ(<D+)=<DJ
by 9.4. I. It follows that
WW-1(S)=W-IWsWE WJ.
Therefore
0:( W W J) = W W J = wsw W J.
We have now shown that 0: is an endomorphism of L.(W, IT).
If WE WI, 0: operates as the identity on all faces of w. If W E V2 then
0:(11')= Wsw is in WI. Thus 0: operates as the identity on aH faces of o:(w).
Hence 0:2 = 0: and 0: is a folding. Finally W'Wr E f,-V2 and so
O:(W'H'r) = wsw'wr= w'.
Thus 0: is a folding which maps w'wr to w'.
.
We determine next the group of type-preserving automorphisms of the
abstract Coxeter complex L.( W, IT).
PROPOSITION 15.4.2. The group of type-preserving automorphisms of
1:( W, D) is isomorphic to W.
PROOF. Let 0: be the folding of L.(W, D) which maps w'wr to w'. The
effect of 0: on the chambers is given by:
(11' if WE WI,
a(w) =
WsW if WE W2.
ASSOCIATED GEOMETRICAL STRUCTURES
291
(The notation is as in 15.4.1.) By replacing w' by w'Wr we may determine
the opposite folding f3 of 0:. f3 operates on the chambers by:
(WsW if IV E J1I1,
f3( 11') =
w if \II E W2.
Let P be the involutary automorphism of 2:( f,-V, D) determined by 0: and
f3. Then
pew) = wsw
for all WE W. Thus P operates on the elements of IY by left-multiplication
by the reflection Ws. Now the automorphisms c:f this kind gl'.ncrate the
whole group of type-preserving automorphisms of 2:.(11/, 11), by 15.3.6.
Since the Weyl group vV is generated by its rcllections it fo]]ows that the
group of type-preserving automorphisms is isomorphic to vV. .
We now show that L.( W, D) is, to within isomorphism, the only abstract
Coxeter complex whose group of type-preserving automorphisms is
isomorphic to W.
THEOREM 15.4.3. Let W be a lYeyl group and L: be an abstract Coxeter
complex lV/lOse group of typc-preserving aut011lorpldsl11s W(2:), whcn
regarded as a Coxeter group as in 15.3.7, has the same relations as W
}-t-'hen generated by a system offundamental reflectiolls. Then L. is isomorphic
to L.( W, ll).
PROOF. Let C be a chamber of L.. Then by 15.3.7 W(L.) is generated
as a Coxeter group by the reflections in the faces of codimension 1 in C.
In the present instance the Ilunlber of such reflections is l=rank W.
Let these reflections be pl. pz, . . . , pl. Then
W W(Z) = (PI. pz, . . ., Pl).
Now rank C=I and C has faces AI. A2,..., Al of codimension I in
natural 1-1 correspondence with PI, p2,. . ., pl. Let J be any subset
of 0, 2, . . . , /} and
CJ= n At.
.ieJ
Then the elements CJ are all faces of C, and every element of L is expres-
sible in the form w( CJ) for some WE W(L.) and some J.
Consider the complex St CJ. This is also an abstract Coxeter complex
by 15.3.1. The group of type-preserving automorphisms of St CJ is

292
SIMPLE GROUPS OF LIE TYPE
isomorphic to the subgroup W.1 of V(L;) generated by the elements Pi
[or i E J, also by 15.3. 7. In particular, WJ operates transitively on the
chambers of L containing CJ, and I VJ I is the number of such chambers.
Now VJ stabilizes CJ since each of its generators has this properly.
Suppose the stabilizer H of C.1 in V were greater than W.1. Then II
would operate on the chambers containing C.1, and so the chambers
lr( C) for IV E II could not all be distinct. However, this contradicts 15. I .2.
Thus W.1 is the stabilizer of C.1 in V.
Let WI, H'2EJ;V. Then Wl(C.1)=1V2(CJ) if and only if WlWJ=WZWJ.
Thus the map
I;L;UV, 11),
W(C.1)wW.1'
is a bijection. This bijection transforms the chamber H-{ C) in L into the
chambcr w in L( J;V, rI). Moreover, it induces an isomorphism between
thc faces 11'( CJ) of w( C) and the [aces IV JFJ of w. Thus it is an isomorphism
bctween Land L( W, D). .
Note. \Ve ha ve shown that there is just one abstract Coxeter complex
L; (to within isomorphism) whose group [V(L.) of type-preserving auto-
morphisms is isomorphic to W, i[ J;V is a Weyl group. In general W()
is a Coxeter group. Now there are Coxcter groups which are not Wcyl
groups. (A classification of the finite Coxcter groups is given in Bourbaki
[I ].) Although such groups will not be discussed in the present volume
it is possible to extend the result mentioned above to all Coxeter groups.
Given any Coxeter group V there is, to within isomorphism, a unique
abstract Coxeter complex whose group of type-preserving automorphisms
is isomorphic to JY.
15.5 Buildings
A building is a pair (.0, s1') where .0 is a chamber complex and.sd is a
set of sllbcomplexes, called apartments, satisfying the folJowing axioms:
Bl. .Q is a thick chamber complex.
B2. The apartments of .0 are thin chamber complexes.
B3. Given any two chambers C, C' in .0 there exists an apartment
LEd such that CEL and C'E2::.
B4. If A, A' are elements of n which are both contained in each of the
ASSOCIATED GEOMETRICAL STRUCTURES
293
apartments L, L' Ed, there exists an isomorphism between L, I;' leaving
invariant A, A I and all their faces.
---..,\
It follows from these axioms that any two apartments of a building
Q are isomorphic. For let L, I;' be apartments and C, C' be chambers
with CEI; and C/EL'. Let f={C=Co, C1,..., Cm=C/} be a gallery
joining C, C I. There exist apartments Ll containing Co, Cl; LZ containing
Cl, C2; etc. By B4 we have
LLlL2 ... LmL'.
Example 15.5. 1. The building .o(G; B, N).
Let G be a group with a (B, N)-pair. Then there exist two subgroups
B, N o[ G such that:
EN ). G is generated by Band N.
BN 2. B n N is a normal subgroup of N.
BN 3. The group W = NIB n N is generated by a set Wt of involutions
(i E l).
BN 4. If ni EN maps to H'i under the natural homomorphism, and if n
is an element of N, then
BniE . BnB s; BninB u BnB.
BN 5. nlBni=l-B, iEI.
The subgroups of G containing B are in 1-1 correspondence with
the subsets J of I, by 8.3.2, and have the form P.1=<B, nt; iEJ) by
8.3.1. Let .0 be the set of left co sets gP.1 for all gEG and all subsets J
of 1. We introduce a partial ordering on n which is the reverse of set-
theoretical inclusion. Let 2::0 be the subset of .0 which consists of the
elements nP.1 for aU n EN and all J. Then gLO is the set of co sets gnP .1
for all n EN and all J, and we define sl to be the family of subsets g LO
of .0 for all g EG... We shall show that .0 is a building and that sf is a
set of apartments in .0. The building constructed in this way will be called
n(G; B, N).
We show first that .0 is a chamber complex. The maximal elements
(chambers) of .Q are the elements gB and the elements of codimension 1
are those of form gP .1, where J is a I-element subset of J. Thus Band
iE are adjacent chambers if i EI. It fo)]ows that the chambers .B, nB
can be joined by a gallery for all 11 EN. Let g be any element of G. Since
G=BNB we have g=bnb' with b, b' EB and 11 EN. Thus gB=bnB. Now
B, nB can be joined by a gallery and so bB, bnB can be joined by a gaJlery
also. Thus B, gB can be joined by a gallery. Let gl, gz be arbitrary elements
......"")


ft
294
SIMPLE GROUPS OF LIE TYPE
\
"'-""
of G and put g=gzlgl. Since B, gB can be joined by a gallery it follows
that glB, g2B can be joined also. Thus ,Q is a chamber complex.
We now show that the chamber complex ,Q is thick. Consider the
dement PJ=(B, ni) of codimcnsion 1 in 0, where J={i}. It is clear that
Band niB are chambers containing P J. However they are not the 0nly
chambers containing P J. For if they were, B would be a subgroup of
index 2 in P J and hence normal in P J, and we would have lltBIl! = B,
which contradicts BN 5. It follows that gB and glltB are not the only
chambers containing gP J. Thus ,Q is thick.
Now P J=B u BIl-IB and the only elements of N contained in PJ arc
1, 11£ by 8.3.1 and 8.3.4. Thus Band IltB are the only chambers con-
taining P J of the form nB for n EN. It follows that IlB and ImiB are the
only chambers in Lo containing liP J. Thus Lo is thin, and hence all the
other apartments of ,Q are thin also.
We verify next that any two chambers of n are contained in some
apartment. LetglB, g2B be two chambers and let g=gllgZ. Since G=BNB
we have g=bnb' for b, b' EB and 11 EN. Thus gB=bnB. Now Band IlB
are in LO, and so Band gB are in bLo. It follows that glB and g2B are
in the apartment glbLo.
Finally, suppose we have two elements of n and two apartments which
each contain both these elements. By multiplying on the left by a suitable
element of G we may assume one of the apartments is LO and that the
chambers are P J and nPK, where n EN. Let the other apartment be go.
We show there is an isomorphism between LO and gLO which leaves
invariant PJ, IlPK and all their faces. Since PJEgLo we have g-lPJEL,o
and so g-lPJ=n'PJ for some n'EN. Thus gn'=PJEPJ, and gLO=PJL.O.
Now nPKEPJLO and so p",inPK=n'PK for some n' EN. It foHows tha+
P JnPK=P In'PK.
We now require the foHowing lemma.
(-,.,
-::;:51
r '
L,
[
",>I
"..,--
c.,
r
{
to,"
r"
L..
r
L_
f
L
LEMMA 15.5.2. Let G be a group with a (B, N)-pair. Then for each
IlEN we have
PJnPK nN=NJnNK.
(The notation is as in 8.2.2.)
r
I ,
PROOF. Axiom BN4 applied repeatedly shows that, for any subset
No of N, we have
\.
r-.
}
t.__
NJBNo  BNJNoB.
r
L
ASSOCIATED GEOMETRICAL STRUCTURES
295
By inverting both sides we also obtain
NoBNJ  BNoNJB
for any subset No of N. By applying these formulae we have
P,JnPK nN=BNJBIlBNJ(B nN
 BNJI1BN](B nN
£ BNJnN]{B nN
=NJI1NK
by 8.2.3. The reverse inclusion is obvious.
II
We can now complete the argument to show that O(C; B, N) is a
building. We have PJI1PK=PJI7'PK and so) by 15.5.2, we obtain
NJnNK=NJn' NI<. Thus
N JI1N Kn-1 = NJ17' N]{11-1
and, in particular, we have
NJ nn'N]\n-1 =f-cp.
Let FIENJ nn'NKn-l. We show that left multiplication by the element
pJn gives an isomorphism from 2:0 to g1:o wiLh the required properties.
We have
PJI1Lo= PJo=g'£O
and so we have an isomorphism from Lo to gLo.
Also
PJllp J = PJP J=P J
and
PJIl.nP]{ = PJI1'p]( = nP [(.
Thus this isomorphism fixes P J and nP]{ and clearly fixes also all faces
of these two eJcments, since the faces arc larger subsets of C. Thus axiom
B4 has been established, and so D(G; B, N) is a building.
15.6 Retractions onto an Apartment
We have shown above that there is a building associated to each grollp
with a (B, N)-pair, and have therefore established the existence of a large
number of buildings. We shall now prove some further general properties

296
SIMPLE GROUPS OF LIE TYPE
of buildings, concentrating on the relationship between a building and
its apartments.
Let 12 be a building, L an apartment of Q and C a chamber in . For
each element A En there exists an apartment ' containing A and C,
by axiom B3. By B4 there is an isomorphism ' --+  which leaves invariant
all faces of C. By 15. 1 .2 there is only one such isomorphism. The image
of A under this isomorphism is an element of  which is independent
of the choice of ', by B4. This image will be called retr1:, dA).
LEMMA 15.6. 1. The map
D--+ 1:,
A --+ retr 1:, c(A)
is a retraction frnm D onto 1:.
PROOF. The map is clearly a morphism, and since it acts as the identity
on  it is idempotent. .
We use retractions of this kind to prove the following important
result.
THEOREM 15.6.2. The apartments of a building are abstract Coxeter
complexes.
PROOF. Let L: be an apartment of a buiJding D. We know that L is a
thin chamber complex and so must prove the existence of all possible
foldings. Let C, C' be adjacent chambers of 1: and let A=C nC'. Since
.Q is thick there is a third chamber C" containing A. Let L:' be an apart-
ment containing C, C". Let ex be the map of L: into itself given by
ex = retr 1:, c'. retr 1:', c.
The retraction\) arc restricted to the apartments being considered, which
are as shown beJow.
reLrE' 0 rclr'\' 0'
L ') 1:' , , ) 1:
Then ex is an endomorphism of L, and we have
ex(C) = retr1:, c'(C) = C,
0:( C') = retr 1:, c'( C ") = c.
ASSOCIATED GEOMETRICAL STRUCTURES
297
--.....,
We may define similarly another endomorphism {3 of :E by interchanging
the roles of C, C'. Thus (3(C')=C' and fJ(C)=C'. Furthermore 0: and fJ
both leave invariant all faces of A. We shall show that 0:, f3 are a pair of
opposite foldings of L, thus proving that L is an abstract Coxeter
complex.
We define a set r:E(A) of galleries of L. A gallery I' of 1: lies in f:E (A)
if:
(a) The first term of I' contains A.
(b) There is no gallery of L shorter than r whose first term contains A
and last term coincides with the last term of f.
Thus r L(A) is the set of galleries of 1: which start from a chamber
containing A and reach their destination as quickly as possible. We
shall show that if I' Er:E (A) then 0:(1') Er1: (A) also.
Now (I'
0:(1') = retr 1:, C' retr 1:'. C ).
The map
retrE' 0
1: ') L'
is the unique isomorphism from 1; to 1;' which leaves invariant all the
faces of C. Since I'E1'1:(A) it follows that retr1:', c(r) Er1:' (A). Similarly
the map rclrE C'
L' .} L
1
is the unique isomorphism from L' to 1: which leaves invariant all faces
ofC'. Since retr1:', c(I')E1'1:'(A) it follows that
retr:E, C' retrL', c(1') E1'L (A).
Thus 0: maps 1'1: (A) into itself. Similarly 1'L (A) is mapped into itself
by fJ.
Let 1'={c=co, CI,..., Cm} be a gallery in rL(A). We show by
induction on m that all faces of Cm are invariant under 0: and under
0:8 but that Cm is not fixed by fJ. These facts are clear if m=O, since
o:(C)=C, fJ(C)=C' and o:(3(C)=o:(C')=C. Suppose therefore that m>O
and write y=o: or o:fJ. By induction y fixes all faces of Cm-I. Now y(C7II)
contains Cm-I n Cm, and so must be either Cm-I or Cm. However,
y(Cm)=Cm-I would imply that
y(1')={c=y(Co), y(CI), . . . , Cm-I, Cm-I}
is not in f L (A), contrary to the fact that 0:, {3 transform r:E (A) into itse]f.
Hence y fixes Cm. It also fixes all faces of Cm-I n Cm, and so it fixes all
faces of Cm. -

f"
_ 1"-- >I
c
1!!mI!Ji :-
;
-;
r
bt
.
r'
L
f' '>
,
l"",
f-
t-,..
'--
t...-

f
1-
L...
r-'
,
i:I
t,,,
c--
i
L,
L
C'
t _
L
r -
\. -
l_
298
SIMPLE GROUPS OF LIE TYPE
Now suppose by way of contradiction that {J( Cm) = Cm. Then we
have
{J(r) = {C'={J(Co), ..., Cm}Er1;(A)
and so {J fixes all the faces of Cm. In particular {J fixes Cm-l n Cm. Hence
{J(Cm-l) is either Cm-l or Cm. However, {J(Cm-I)=Cm-1 is false by
induction, and {J(Cm-I)=Cm contradicts the fact that {J(r)Erk(A).
Thus we have a contradiction, and hence (3(Cm)-=/:- Cm.
We are now able to show that a, f3 are a pair of opposite foldings of
L. We show first that a is idempotent. Let D be a chamber in 1: and let
r E r k (A) be a ga]]ery whose last term is D. Then ex(r) is also a gallery
in rk (A) and has first term C and Jast term ex(D). Thus ex fixes all the
faces of ex(D), as shown above. Hence a2=ex.
We now show that each chamber in a(L) is the image of just two
chambers in },:. Let D be a chamber in a(L) and let D = ex(E), where
EEL. Now we have shown above that for each chamber EE},:, either
a(E)=E and {J(E)-=/:-E, or a(E)-=/:- E and {J(E)=E. If ex(E)=E then E= D.
If a(E)-=/:-E then {J(E)=E and {Ja(E)=E, as shown above. Thus
E= {J(ex(E»= {J(D).
Hence there are just two chambers in L such that ex(E) = D, viz., D and
{J(D). Therefore a is a folding. It follows by symmetry that fJ is also a
folding, and by definition that a, {J are opposite foIdings. Thus L is an
abstract Coxeter complex. .
Definition 15.6.3. Let A, A' be two elements of a building O. Then
A, A' are said to have the same type in 0 if they have the same type in
any apartment containing A, A'.
We observe that this condition is independent of the apartment chosen.
For let LI, L2 be two apartments containing A, A'. Let C, C' be chambers
of Ll containing A, A' respectively. Let y be the retraction of },:l onto
the simplex S (C) defined in 15.3.2. Then A, A I have the same type in
},:l if and only if y(A')=A. Now by B4 there exists an isomorphism
S : },:I-+},:2 which leaves A and A I invariant. Thus the retraction of L2
onto the simplex S(S(C) is given by
S(X)-+Sy(X),
X E },:l.
A and A' have the same type in L2 if and only if A' is mapped to A under
this retraction, which holds if and only if y(A') = A.
AssocrATED GEOMETRICAL STRUCTURES
299
15.7 Groups of Type-Preserving Automorphisms
We shall be concerned with groups of type-preserving automorphisms
of a building .0, and first give an example in the building .o(G; B, N)
constructed from a group G with a (B, N)-pair.
PROPOSITION 15.7.1. III the building .o(G; B, N), the group G operates
by leji multiplication as a group of Iype-prescrving autol1lorphisms which
is transitive 0/1 the pairs (C, L), where C is a chamber ami}: is an apartment
containing C.
PROOF. We use the notation of 15.5.]. The elements of .0 are co sets
of the form gP J. Consider the apartment 1:0 of .0. The chambers of
LO have form nP J, where 11 EN, and the map
nP J----->'P J
is an idempotent morphism from LO to the simplex of faces of the chamber
B. It is therefore the retraction described in 15.3.2. Thus IIIP J and Jl2P K
have the same type if and only if J = K. Similarly in the apartment Z = g1:o
of .0, the elements glliP J and gn2PK have the same type if and only if
J=K. We now consider any two elements glPJ and g2PK of .0. They
have the same type in .0 if and only if they have the same type in some
apartment containing both. Thus glP J and g2PK have the same type if
and only if J = K. It is now clear that the map of .0 into itself given by
gPJ----->'xgPJ,
xEG,
is a type-preserving automorphism of .0.
Let C be a chamber and 1: an apartment of Q containing C. Then
L=g},:o for some gEG, and C=gnB for some I1EN. Let x=gn. Then
C=xB and L=XLO. Thus the element x of G transforms the pair (B, Lo)
into the pair (C, L). It follows that G operates transitively on the pairs
(C, },:) with CE},:. .
We shall now prove a converse of this result, namely that a group
of type-preserving automorphisms of a building which is transitive on
the pairs (C, L) with CEL is a group with a (B, N)-pair.
THEOREM 15.7.2. Let (.0, .91) be a building and G be a group oj type-
. preserving Gutomorphisms of Q which is transitive on the pairs (C, L.;) with

300
SIMPLE GROUPS OF LIE TYPE
C E L:, where C is a chamber and L: an apartment of n. LeI Co, Lo be a
fixed chamber and apartment with Co E LO, let B be the stabilizer oj Co
in G and N be tile stabilizer of L:o. Theil the subgroups B, N form a (8, N)-
pair in G. .A10reover, V=N/B nN is isomorphic to the group V(2:) of
type-preserving aulomorphisms of each apartment L of.o.
PROOF. We verify the last assertion first. N operates on LO as a group
of type-preserving automorphisms. The transitivity hypothesis shows
that N operates transitively on the chambers of La. Thus N induces on
2:0 the full group V(2:o) of type-preserving automorphisms, by 15.1.2.
Thus we have an epimorphism N-+ 11/(2:0) with kernel B nN. For a
type-preserving automorphism of 2:0 which fixes Co must be the identity,
again by 15.1.2. Thus HI B n N is isomorphic to V(2:o). This is isomorphic
to HI"(2:) for any apartment L:, since any two apartments are isomorphic.
We now show that the subgroups B, N satisfy the axioms for a (B, N)-
pair. It has already been verified that B n N is normal in N and that
NIB nN is generated by a set of involutions. We choose for the set I of
generating involutions the reOections of 2:0 in the faces of codimension 1
in Co (see 15.3.7).
Let gEG and let L be an apartment containing Co and g(Co). Then
Co is contained in 2: and g-I(2:), so the transitivity condition shows there
exists bl EB such that g-I(2:)=bl(2:). Also, since Co is in 2:0 and L, there
exists b2EB such that b2(2:0)=L. Thus g-lb2(2:0)=brb2(LO) and it follows
that b21gb1b2EN. Hence gEBNB and we have G=BNB.
Let 11! be an element of N which induces on o a generating involution
lI'i of W (iEI). We show niBrwl-B. Suppose this is false, so that ntB=Fni.
Then we have
B. ni( Co) = l1i. B( Co) = ni( Co).
Thus B stabilizes both Co and l1i(CO)' Now Co and ni(Co) are adjacent
chambers and, since n is thick, there is a third chamber C / containing
Co nni(Co). Let 2: be an apartment containing Co and C/o By transitivity
there exists bE B such thal b(2::o) = 2:. Since B stabilizes both Co and
ni(CO) we see that Co, nl(Co), C/ are all in L:. This contradicts the fact
that 2: is thin.
Finally we check the axiom
BntB. BnB  B/li/1B u BnB.
Given elements nl., n EN and b EB we consider the adjacent chambers
Co and l'Ii(CO). Let A=Co nt/t(Co). The element b fixes Co and is type-
'\
ASSOCIATED GEOMETRICAL STRUCTURES
301
.-Ic,
,
preserving, so fixes all faces of Co. In particular bA=A and bl1tCo contains
A. Thus the chambers nCo, nn/,Co and nblltCo all contain nA.
Consider the set of galleries whose first term is Co and last term is a
chamber containing nA. Let r be such a gaJIery with as few terms as
possible and let
1'={Co, CI, . .., Cm}.
Let L be an apartment containing Co and nbntCo. Then there exists
blEB such that blL:=O.
Now L contains Co and nA, and we show that  contains each term
of the gallery r. Suppose this is false, and let Ct be the first term in r
not in 2:. Let D be the chamber of L other than Ct-l containing Ct-l n Ct.
Then
retr. D (Ci-l) = Ci-l,
retr. D(Ci)=Ct-1
and retr. D (1') is a gallery whose first term is Co, whose last term contains
nA, and which has two consecutive terms identical. This contradicts
the fact that the length of r is minimal. Thus  contains each term
of r. Similarly L:o contains Co and nA, so contains each term of r.
We shall show by induction on i that blCi = Ci. This is clear if i =0.
Assume inductively that blCi-1 = Ci-l. Then bl(Ci-1 n Ci) = Ci-l n Ct
since bi is type-preserving. Thus blCi contains Ci-l n Ci. Now Ci E L
and b1CiELo. The two chambers of o containing Ci-l nCi are Ci-l
and Ci. Now b1Ci#-Ci-l since bi is bijective, hence bICi=Ct. In particular
b1Cm=Cm. Since bl is type-preserving we have also blJ1A=nA.
Now nbniCO is a chamber in L which contains nA and so b1nbniCO
is a chamber in Lo which contains /lA. Thus b1llbl1iCO is either flCO or
/llliCO. If b1nbniCo=nCo we have nblliEBnB, and if blnbniCo=/lflf,CO we
ha ve nbl1t E Bm1tB. Thus
-
,
-,'
i
nBIl-/,  BnfliB u BnB.
By taking inverses it follows that
l1iB/1 s; BlltllE u BnB.
Thus G has a (B, N)-pair.
.
Note. The above proof shows clearly the geometrical meaning of the
axiom BN 4. It was shown that some element of B transforms the chamber
nblllCo into a chamber of Lo containing nCo nn/ltCO, which must therefore
. be either nCo or nntCo.
-
L

\-.,. ..
iL.--:;;I_
[
L
.....-
L.
r
L;
1'-
i
L_,-
f -
;i
L".
r'
L_
"'"--,_.
302
SIMPLE GROUPS OF LIE TYPE
The relation between the algebraic and geometric structures discussed
in this chapter may be summarized in the following scheme.
CHAPTER 16
.A It'b/'(Ji(' Slrlli IliI'!'
Geometric structure
Sporadic SiInple Groups
Group with (B, N)-pair .(
1
) Building
1
) Abstract Coxeter complex
The simple groups of Lie type which we have described in the earlier
chapters include almost all the finite simple groups at present known.
In adition to the, cyclic groups, alternaling groups and simple groups
of Lie type over finItc fields, there are nineteen further finite simple groups
klOwn at the time of writing (February 1971). I n order to complete the
plctue .regading. the finite simple groups as it now stands, we give a
d.escnptlOn Il1 this final chapter of these nineteen so-called 'sporadic'
s]ple groups. This description will be exceedingly brief in comparison
wIth the amount of information available about these groups. Further
information .can be found in the survey articles of Tits [21] and Feit [2],
whose notatlO11 we foHow here, and in papers dealing with each group
individually.
Coxeter group .(
J. Tits has recently carried out a detailed investigation of the buildings
of the various different types. In particular he has shown that a finite
building whose associated Coxeter group is an indecomposable Weyl
group of rank at least 3 must be a building O(G; B, N), where G is a
finite Chevalley group or twisted_jP"oup. Tits is able to deduce from this
that the only finite simple groups with a (B, N)-pair of rank at least 3
are the finite Chevalley groups and twisted groups which have this property
(see Tits [22]).
16.1 The Mathieu Groups
Five sporadic simple groups, which we denote by Mll, M12, M22, M23, M24
were described by Mathieu in papers in 1861 and 1873 [I, 2]. They can
be described most easily as groups of automorphisms of Steiner systems.
A Steiner system of type (r, s, t) is a set A together with a family of
subsets Bi of A, called blocks, such that I A I =t, I Bt I =s, and each
subset of A with r elements is contained in exactly one block Bt. It is
known that there is, to within isomorphism, exactly one Steiner system
of each of the types (5, 6, 12) and (5, 8, 24) (see Witt [1, 2]). These Steiner
systems will be denoted by S(5, 6, 12) and S(5, 8, 24).
An automorphism of a Steiner system A is a permutation of A which
maps each block into a block. It can be shown that the groups of auto-
'morphisms of S(5, 6, 12) and S(5, 8,24) are simple and we write
Nh2 = Aut S(5, 6, 12),
A1z4 =Aut S(5, 8, 24).
The groups M12 and M24 have the property of being quintuply transitive
permutation groups. They are in fact the only known quintupJy transitive
303

304
SIMPLE GROUPS OF LIE TYPE
SPORADIC SIMPLE GROUPS
305
...-'\.
permutation groups other than the symmetric and alternating groups.
In M12, the only element which fixes five symbols is the identity, but in
A124 the elements stabilizing each of five given symbols form a subgroup
of order 48. Thus the orders of these groups are given by
Thus the Mathieu groups are uniquely determined in terms of the classical
groups MlO and M21.
I M121=12.11.10.9.8,
I M241 =24.23.22.21.20.48.
Given a Steiner system of type (r, s, t) we can obtain a Steiner system
of type (r-l, s-I, t-l) by taking all the blocks Bi of A which cntain
a fixed element aEA, and then taking the sets Bl-{a} as blocks m. the
set A-{a}. Steiner systems S(4, 7, 23) and S(3, 6,22) can be obtall1ed
from 5(5, 8, 24) by this process, and a Steiner system 5(4, 5, 11) from
S(5, 6, 12).
We defme M23 to be the subgroup of M24 stabilizing one symbl
aEA. Thus 11-123 is a permutation group on twenty-three symbols, and IS
a group of autOl11orphisms of S(4, 7, 23). Define M22 to be tbe stbilizer
of one symbol in M23 and !V/u to be the stabilizer of one symb?lm 1I-h2.
Then Jl,122 is a group of automorphisl11s of S(3, 6, 22) and Mu IS a group
of automorphisms of S(4, 5, 11). It is known that 11-123, k!u are the full
automorphism groups of S(4, 7, 23), S(4, 5, 11) respectively, and that
MZ2 is a subgroup of index 2 in Aut S(3, 6, 22).
Let MIO, 11-121 be the stabilizers of one symbol in Mll, 11-122 respectively.
Then !vIto has a subgroup of index 2 isomorphic to PSL2(9) and operates
as a permutation group on the ten points of the. projective line ?ver
GF(9). By 12.5.1 Aut PSL2(9) is generated by the muer automorpls'11S
and by a diagonal automorphism d of order 2 and a field automorphism!
of order 2. Thus
16.2 Sporadic Simple Groups Characterized by the Centralizer
of an Involution
I Aut PSL'2(9) : PSLz(9) 1=4.
MlO is the subgroup of Aut PSL2(9) generated by PSL2(9) and the auto-
morphism £If. .
11-121 turns out to be isomorphic to PSL3(4) and operates as a permutatIOn
group on the twenty-one points or the projective plan .over GF().
The Mathieu groups may also be dcfineJ as transItive xtcnslOn.. A
transitive extension of a permutation group G on a set A IS a ta.nsltlvc
permutation group G on a set A =A u {ii} such that the stabilIzer of
fi in G, when restricted to A, is G. It can be shown that 11-111, M12, 11--122,
!vf23, !V!z4 arc the unique transitive extensions of Aho, Aill.' 121' M22,.J.123
respectively, and that Ahz, !vf24 do not possess transitive extensIOns.
Apart from the Mathieu groups, all the other known sporadic simple
groups have been discovered within the last ten years. The Feit- Thompson
theorem [1] shows that every non-cyclic finite simple group contains an
element of order 2, i.e. an involution. It was shown by Brauer and Fowler
[1] that if C is any finite- group, there are only finitely many finite simple
groups G which have an involutionj whose centralizer C(j) is isomorphic
to C. Brauer initiated the programme of trying to characterize all finite
simple groups in which the centralizer of an involution is a given group,
and very many such characterizations have been obtained. This programme
has in fact led to the discovery of several sporadic simple groups. A typical
result of this type asserts that if G is a simple group which has an involu-
tion j whose centralizer C(j) is isomorphic to a fixed group C, then G
is either one of the previously known simple groups or (say) a group
determined to within isomorphism whose order and character table are
both known. The existence or non-existence of this hypothetical new
simple group then has to be decided. In most of the cases which have
arisen, the existence of the new group has been checked with the aid of a
computer.
An involution in G is called a central involution if it is contained in
the centre of some Sylow 2-s11bgroup of G. Every non-cyclic simple
group contains some central involution, and the characterizations of
simple groups are often carried out in terms of centralizers of central
involutions.
Five sporadic simple groups, denoted by la, Hal, H1M, HHM, LyS
have been discovered by these methods. Janko [1] characterized simple
groups with a central involutionj sLlch that C(j) is isomorphic to C2 x As,
and showed that there is at most one such simple groLlp and that its
order must be 175560. He also succeeded in constructing such °a" group
as a subgroup of the orthogonal group 07(11). This group is denoted by
la. It is known that fa is a subgroup of the Chevalley group G2(11)
(Janko [1]).
Janko then considered simple groups with a central involution j such
that C(j) is an extension of a group of order 2s by A5. He showed that-
..-....
-
!

306
SIMPLE GROUPS OF LIE TYPE
SPORADIC SIMPLE GROUPS
30?
i
there are at most two such groups, of orders 604,800 and 50,232,960 but
did not prove their existence. The existence of the smaller group, denoted
by Hal, was proved by M. Hall using a computer, although It can now
be described by other methods. The existence of the larger group, called
HiM, was proved by G. Higman and MacKay, also using a computer.
The existence of a further sporadic simple group was predicted by
Held. Let Go=PSL5(2) and jo be the involution Xt{l) in Go, where r is
some root. Let C be the centralizer of jo in Go. Held investigated simple
groups G with an involution j such that C(j) is isomorphic to C, an.d
was able to show that there are at most three such groups. The first IS
PSL5(2), the second is M24, and the third would be a spoadic. group of
order 4,030,387,200. Held did not prove the existence of thIs third group,
denoted by HHM, but this was later checked by G. Higman and MacKay
by computer. .. .
Lyons has recently investigated simple groups m. which there I a
central involution j whose centralizer CU) is a non-spltt central extension
of C2 by An. There is at most one such group, of order
51,765,179,004,000,000,
and its existence has recently been proved by Sims using a computer.
This group is denoted by LyS.
',-
t. .
16.3 Towers of Permutation Groups
graph with the elements of A as vertices. The problem thus becomes to
construct a graph with vertex set A on which G operates as a group of
autornorphisms, such that the full automorphism group of the graph is
transitive on A. This can be clone in a number of special cases to give
further sporadic simple groups.
The first new group to be discovered by this method was obtained by
D. Higman and C. Sims. They startcd with the J\ifathieu group lvf22 ,
taking the two transitive permutational representations of degrees 22,
77 obtained from the 22 points and 77 blocks of the Steiner system
S(3, 6, 22). Adding one additional point, they showed that M22 could
be extended to a group HiS operating transitively on these 100 symbols.
HiS is a simple group of order 44,352,000.
A similar construction was carried out by McLaughlin starting from
the group PSU4(32). He took two permutational representations of this
group of degrees 112 and 162. The representation of <.1L:gree 112 is obtained
from the operation OfPSU'I(32) on the 112 isotropic lines on the 'Hermitian
quadric' in 3-dimensional projective space over GF(32) left invariant by
PSU4(32). The representation of degree 162 is obtained from the fact that
PSU4(32) contains a subgroup PSL3(4) of index J 62. Taking these two
representations together and adding an extra point we obtain an intransi-
tive representation of PSU'I(32) of degree 275. It is possible, by con-
structing a suitable graph with these vertices, to find a transitive extension
lI-fcL. McL turns out to be a simple group of order 898,128,000.
The next group of this type was obtained by Suzuki, beginning with
G2(4). This group has permutational representations of degree 416 and
1365. The stabilizer of a symbol in the first representation is isomorphic
to JIa} and the stabilizer of a symbol in the second representation is the
centralizer of the centre of a Sylow 2-subgroup of G2(4). Adding one
extra point we obtain an intransitive representation of degree 1782, and
by constructing a suitable graph with this vertex set the existence of a
transitive extension can be proved. This is a simple group, denoted by
Suz, of order 448,345,497,600.
L
t -


,->"

!
t-.,.
--
 y
t _
A further method of constructing sporadic simple groups was suggested
by a property of the group JIal. This group, whose exs.tence had ben
conjectured by Janko, was exhibited by M. HaIl as a tansltve permutation
group of degree 100. The stabilizer of one symbol IS a sllpe subgr.oup
isomorphic to PSU3(32), and this subgroup has three orbIts III the gIven
representation of degrees 1,36,63. Thus Hal can be obtined as a permuta-
tion group by taking two permutational representatIOns of PSU33.2),
adjoining one additional point, and enlarging the group to a transItive
extension on the resulting set of 100 elements.
The question then arose as to whether a similar proce?ure could be
adopted beginning with two permutationa represe.natlOns of some
other classical group. If G is a group operat1l1g transitively on two sets
A A we introduce an additional element ii and seek to construct a
1, 2, . .
transitive extension G of G on A =A1 UA2 u{a}. Such a transltJ.ve exten-
sion is usually found, when it exists, as a group of automorplllsms of a
16.4 The Leech Lattice
..
We mention next three sporadic simple groups discovered by Conway.
These can be described as groups of isometries of positive definite quad-
ratic forms over the integers. Let 7l.n be a free Abelian group of rank 11.

308
SIMPLE GROUPS OF LIE TYPE
SPORADIC SIMPLE GROUPS
309
---
A positive definite quadratic form of degree Il over 7L is a map Q : 7Ln-+7L
given by
16.5 Groups Generated by a Class of 3- Transpositions
Q(Xl, . . . , xn) = 2:: af,jXf,Xj,
i, j
We conclude by describing three further sporadic simple groups dis-
covered by B. Fischer [1]. Fischer investigated groups generated by a
class of conjugate involutions such that the product of two distinct
involutions in the class has order 2 or 3. Such a conjugacy class of involu-
tions is called by Fischer a class of 3-transpositions. For example, each
symmetric group is generated in this way by its class of transpositions.
More generalIy, the Weyl groups of type Az, Dz, E6, £7, Es are generated
in this way by their classes of reflections. In addition certain classical
groups over very small fields are also generated in this way. The sym-
plectic groups Sp21(2) are generated in this way by the class of symplectic
transvections, the orthogonal groups 02i(2), 0;(2) by orthogonal trans-
vections and the unitary groups PSUn(22) by unitary transvections.
Moreover, in the orthogonal groups 02H3), 0;(3), 02l+1(3) the reflections
corresponding to the elements a satisfying (a, a) = 1 form a class of
3-transpositions in the subgroup they generate, and so do the reflections
corresponding to the elements satisfying (a, a) = -1.
Fischer considered finite groups G generated by a class of 3-transposi-
tions which satisfy the further conditions:
(i) Every normal subgroup of G of order a power of 2 or a power of 3
lies in the centre of G.
(ii) The derived group G' of G is its own derived group, viz., G' = Gn.
He showed that if G is such a group with centre Z(G) then GjZ(G) is
isomorphic to either a symmetric group, or one of the lassical groups
mentioned above (factored by the centre in the case of the groups over
GF(3)), or one of three further groups which we shall denote by Fi22,
Fi23, Fi24. Moreover, there is just one class of 3-transpositions generating
each of these groups, except in the group So,;;;;Spi2),;;;;PO:t(3) which
has two such classes. We observe that the Weyl groups of type Eo, E7, Es
are included in this classification in view of the isomorphisms
-""'.......
where aijElL and ajf,=Gf,j, such that Q(x)O for all xElLn and Q(x)=O
only if x =0. The group lLn together with the map Q : lLnlL is called a
lattice.
We restrict attention to positive definite integral quadratic forms with
two further properties. We assume Q is unimodular, i.e. det (o.[j) = 1,
and that Q is even, i.e. Q(x) is an even integer for all x. it is known that
there is an even unimodular, positive definite lL-form of degree 11 if and
only if n is divisible by 8 (cf. O'Meara [1]). If n=8 there is, to within
isomorphism, just one such form. H 11 = 16 there are two such forms.
If 11 = 24 the forms of this type have recently been classified by Niemeier
[1]. There are twenty-four such forms. One of them is distinguished from
all the others by the fact that there is no x for which Q(x) = 2. The cor-
responding lattice was discovered indcpcndently by Leech, as the set
of centres of spheres in a sphere-packing of greatest density in twenty-four
dimensions [1]. It is called the Leech lattice. Let G be the group of integral
unimodular matrices which are isometries of this form. The centre Z
of G consists of the two elements :!:: I, and G/Z can be shown to be a
simple group. It was first investigated by Conway and its simplicity was
proved by Thompson. It is denoted by CO! and its order is
4,157,776,806,543,360,000.
Since G is represented geometrically as the group of automorphisms
of the Leech lattice, subgroups of G can readily be obtained as stabilizers
of sublattices. Consider the set of points x ElL24 such that Q(x) =4. These
points are permuted transitively by G. The stabilizer of such a point in
G can also be shown to be a simple group. This group is denoted by
C02 and its order is 42,305,421,312,000.
We next consider points xElL2'1 such that Q(x) =6. These points are
also permuted transitively by G, and the stabilizer of one of thcm is also
a simple group. 1t is denoted by C03 and its order is 495,766,656,000.
In addition to the three new sporadic simple groups CO!, C02, C03
contained in G, several of the other sporadic groups occur in G as stabilizers
of sub lattices or as composition factors of such stabilizers. The groups
HiS and JvlcL and all five Mathieu groups can be obtained in this way.
The group SliZ also occurs inside G, although not (so far as is known)
as the stabilizer of a sublattice.
W(E{!) ';;;; 06(2),
W(E7)/Z,;;;; Sp6(2),
W(Es)jZ,;;;; 08(2).
If d is an element in the class of 3-transpositions of G it is shown in
Fischer [1] that the set D d of elements of this class distinct from d but
commuting with it forms a conjugacy class of 3-tral1spositions in the
group it generates. Thus the classification of groups generated by 3-trans- -
positions proceeds by induction. In particular Fischer is able to show
,

[:
l
[
'L
[.,
[
r
t. _'_'-
f
1
L__.
,
"--
.l1i;kI--Joi:."",-...'
r
L_
310
SIMPLE GROUPS OF LIE TYPE
that if the group <Dcl)/Z<Dd) is a symmetric group or a classical group
not isomorphic to PSUo(22) then G is either a symmetric group or one
of the classical groups described above. If <Dd)/Z<Dd) is isomorphic to
PSUo(22) then the class D has 3510 elements and G is shown to be a
simple group Fi22 of order 217.39.52.7.11.13. A maximal commuting
subset L of D has twenty-two elements and the group JVu(L)/<L) of
permutations induced by G on L is isomorphic to the Mathieu group
M22.
If <Dd)/Z<Dd) is isomorphic to Fi22 then the class D has 31,671
elements and G is a simple group Fi23 of order 218.313.52.7.11 . 13.17.23.
A maximal commuting subset L of D has twenty-three elements and the
group JV o(L)/<L) of permutations induced by G on L is isomorphic to
the Mathieu group M23.
If <Dd)/Z<Dd) is isomorphic to Fi23 then the class D has 306,936
elements and G is a group Fi24 of order 222.316.52.73.11.13 .17.23.29.
Fi24 is not simple, but contains a simple subgroup Fi4 of index 2. A
maximal commuting subset L of D has twenty-four elements and the
group JV o(L)/<L) of permutations induced by G on L is isomorphic to
the Mathieu group .A124.
FinaIty it can be shown that there can be no group G in which
<Dd)/Z<Dd) is isomorphic to Fi24.
We conclude our brief survey of the sporadic simple groups with a
table comparing their orders.
Mu
.A11 2
A122
M23
!vf24
Ja
HaJ
J/JM
HiS
!vI cL
Suz
HHM
C01
C02
C03
Fi22
Fi23
Fi24 f
LyS
24.32.5.11
26.33.5.11
27.32.5.7.11
27 . 32 . 5 . 7 . 11 .23
21°.33.5.7.11.23
23.3.5.7.11.19
27.33.52.7
27.35.5.17.19
29 . 32 . 53 .7 . 11
27.36.53.7.11
213.37.52.7.11.13
21°.33.52.73.17
221.39.54.72.11.13.23
218.36.53.7.11.23
21°.37.53.7.11.23
217.39.52.7.11.23
218.313.52.7.11.13.17.23
221.316.52.73.11.13.17.23.29
28.37.56.7.11.31.37.67
BibliogralJhy
Mathieu
Mathieu
Ma thieu
Mathieu
Mathieu
Janko
M. Hall, Janko
G. Higman, Janko, MacKay
D. Higman, Sims
MacLaughlin
Suzuki
Held, G. Higman, MacKay
Conway, Thompson
Conway, Thompson
Conway, Thompson
Fischer
Fischer
Fischer
Lyons, Sims
E. Abe
1. On the groups of C. ChevalJcy, J. Math. Soc. Japan, 11 (1959), 15-4J.
2. Oeon:lCtry in certain simple groups, 7ljllOku !l4ath. J., 14 (I 962), 64-72.
3. On sImple groups associuled with the real simple Lie algebras, T6110ku
lvJath. J., J4 (1962), 244-262.
4. Finite groups admitting Bruhat decompositions of type An, T6hoku l\1ath.
J., 16 (1964), 130-141. ,
J. F. Adams
1. Lectures all Lie Groups, Benjamin, New York (1969).
K. Aomoto
1. On some double coset decompositions of complex semi-simple Lie groups,
J. Math. Soc. Japan, 18 (1966), 1-44.
E. Artin
1. Geometric Algebra, Interscience Publishers, New York (/957).
2. The orders of the linear groups, Comm. Pure Appl.l\1ath., 8 (1955),355-365.
3. The orders of the classical simple groups, Comm. Pure Appl. lvlath.
8 (I955), 455-472. '
H. -Asano
1. A remark on the Coxeter-Killing transformations of finite reflection
groups, Yokohama Math. J., 15 (1967), 45-49.
A. Borel
1. Linear Algebraic Groups, Benjamin, New York (1969).
2. Sur la cohomologie des espaces librcs principaux ct des espaces homogcnes
de groupes de Lie compacts, Ann. of Aluth., 57 (1953), 115-207.
3. Oroupes lineaires algcbriques, AI/I/. of lvlath., 64 (1956), 20-80.
A. Borel and T. A. Springer
1. Rationality properties of linear algebraic groups, (T) Proceedings of Sym-
posia in Pure lv/athematics, Vol. 9, Algebraic groups and discontinuous
subgroups, A.M.S. (1966), 26-32; (I f) TcJllOku Alath. J., 20 (1968), 443-497.
A. Borel and J. de Siebenthal
1. Les sOlls-groupes fermcs de rang maximum des groupcs de Iie clos,
Comment. Jvfath. He/v., 23 (1949),200-221.
A. Borel and J. Tits
1. Oroures reductifs, I.H.E.S. Pub/. Math., 27 (1965), 55-151.
A. Borel, R. Carter, C. W. Curtis, N. Iwahori, T. A. Springer and R. Steinberg
1. Seminar on algebraic groups and related finite groups, Lecture Notes ill
Mathematics, 131 (1970), Springer.
R.Bott
1. An application of the Morse theory to the topology of Lie groups, Bull.
Soc. !vfatlr. France, 84 (1956), 251-282.
311

Griess as a group of automorphisms of a certain algebra of dimension
196883. Remarkably, no computational work was needed in Griess' proof
despite the size of the group involved!
Before the existence of the monster had been proved it was realised
that further sporadic simple groups would be contained in it. It was
shown by Thompson that the monster would contain an element a
of order 3 such that C(a)/(a) is a new sporadic simple group of order
215.310.53.72.13.19.31. This group Th was proved to exist subsequently by
Thompson. He showed it had a 248-dimensional representation over IR
which, on reduction mod 3, give a 248-dimensional representation over
GF(3). Thompson obtained the group as a subgroup of Es(3).
The monster also contains an element a of order 5 such that C(a)/(a)
is a new sporadic simple group of order 214.36.56.7.11.19. This group was
investigated by Harada before the existence of the monster had been
proved. The existence of Harada's group was proved by computational
methods by Norton and P .E. Smith. This group is known as the Harada-
Norton group HN.
The classification of the finite simple groups, completed in 1981, showed
that the sporadic simple groups we have mentioned are the only ones.
Information about the classification can be found in a number of works
by D. Gorenstein, including a survey of the classification in the book 'Finite
Simple Groups-An introduction to their classification', Plenum Press,
New York, 1982.
We conclude by giving a list of the sporadic simple groups together with
their orders.
. "
312
...
- .... I,
.i
" ,
4 ·
. .
MIl
M12
M22
M23
M24
J1
J2
J3
HS
McL
Suz
He
Ru
Co.
CO2
C03
SIMPLE GROUPS OF LIE TYPE
Fi22
Fi23
Fi'24
O'N
Ly
J4
HN
Th
B
M
SPORADIC SIMPLE GROUPS
217.39.52.7.11.23
218.313.52.7.11.13.17.23
221.316.52.73.11.13.17.23.29
29.34.5.73.11.19.31
28.37.56.7.11.31.37.67
221.33.5.7.113.23.29.31.37.43
214.36.56.7.11.19
2i5 .310 .53.72.1_3.19.31
241.313 .56.72.11.13.17 .19.23.31.47
246.32°.59.76.112.133.17.19.23.29
.31.41.47.59.71
Supplement
Hartley's Lemma
313
Fischer
Fischer
Fischer
O'Nans, Sims
Lyons, Sims
Janko, Norton
Harada, Norton, Smith
Thompson, Smith
Fischer, Sims, Leon
Fischer, Griess
/
(See page 263).
(a) Let Gl be a finite twisted group of type 2Ae, 2De, 2E6 or 3D4 and
let n:<P.
Suppose r=r. Then given any tEK* with t=lthere exists h(X)EHI
with X(r) = t.
Suppose r"#r. Then given any tEK* there exists h(X)EHl with X(r) = t
except when Gl=2A3 or 2De or when Gl=2A2 and q==- -1 mod 3. If
Gl=2A3 or 2Dethere exists h(X)EHl with x(r)=t2. If Gl=2A2 and
q==- -1 mod 3 there exists h(X)EHl with x(r)=t3.
(b) Let Gl be a finite twisted group of type 2B2, 2P4 or 202 and let TE<P.
Then given any tEK* there exists h(X)EHI with x(r) = t except when
Gl = 2G2 and r= a + b or 3a + b. In these cases there exists h (X)EHI with
x(r) = t2.
24.32.5.11
26.33.5.11
27.32.5.7.11
27.32.5.7.11.23
210.33.5.7.11.23
23.3.5.7.11.19
27.33.52.7
27.35.5.17.19
29.32.53.7.11
27.36.53.7.11
213 .37 .52. 7 .11.13
210.33.52.73.17
214.33.53.7.13.29
221.39.54.72.11.13.23
218.36.53.7.11.23
210.37.53.7.11.23
Mathieu
Mathieu
Mathieu
Mathieu
Mathieu
Janko
Hall, Janko
Janko, Higman, McKay
Higman, Sims
McLaughlin
Suzuki
Held, Higman, McKay
Rudvalis, Conway, Wales
Conway, Leech
Conway
Conway
"
\
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I!
!
J

!
;
,..,.,
II1H,\
.--"

E. Cartan
1. Oeuvres Compleres, Gauthier-Villars, Paris (1952).
R. W. Carter
1. Simple groups and simple Lie algebras, J. London Math. Soc., 40 (1965),
193-240.
2. Weyl groups and finite Chevalley groups, Proc. Cambridge Phi/os. Soc.,
67 (1970), 269-276.
3. Conjugacy classes in the Weyl groups, To appear.
B. Chang
1. The conjugate classes of Chevalley groups of type (G2), J. Algebra, 9 (1968),
190-211.
C. Chevalley
1. Theory of Lie Groups, Princeton University Press (1946).
2. Theorie des groupes de Lie, Hennann, Paris (1968).
3. Seminaire Chevalley Vols. J, II; Classifications des grollpes de Lie algebriques,
Paris (1956-8).
4. Sur certain groupes simples, T6hoku Math. J., 7 (1955), 14-66.
5. Invariants of finite groups generated by reflections, Amer. J. Math., 77
(1955), 778-782. .
6. Certain schemas de groupes semi-simples, Seminaire Bourbaki, 13 (1960).
Expose 219.
7. La theorie des groupes algebriques, Proceedings of the International
Congress of Mathematicians, Edinburgh, Cambridge University Press
(1960).
P. M. Cohn
1. Lie Groups, Cambridge University Press (1957).
A. J. Coleman
1. The Betti numbers of the simple Lie groups, Canad. J. Math. 10 (1958)
349-356. - , ,
J. H. Conway
1. A group of order 8,315,553,613,086,720,000, Bull. London Math. Soc.,
1 (1969), 79-88.
2. A characterisation of Leech's lattice, Invent. Math., 7 (1969), 137-142.
H. S. M. Coxeter
1. Regular polytopes, 2nd edition, Macmillan, New York (1963).
2. Discrete groups generated by reflections, Ann. of Math., 35 (1934), 588-621.
3. The product of the generators of a finite group generated by reflections,
Duke Math. J., 18 (1951), 765-782.
4. Extreme fonns, Canad. J. Math., 3 (1951), 391-441.
H. S. M. Coxeter and W. Moser
1. Generators and relations for discrete groups, 2nd edition, Ergebnisse del'
Mathematik und ihrer Grenzgebiete, 14 (1965), Springer.
C. W. Curtis
1. Representations of Lie algbras of classical type with applications to
linear groups, J. Math. Mech., 9 (1960), 307-326.
2. On projective representations of certain finite groups, Proc. Amer. Malh.
Soc., 11 (1960), 852-860.
3. An isomorphism theorem for certain finite groups, Illinois J. Math.,
7 (1963), 279-304.
4. Groups with a Bruhat decomposition, Bull. Amer. Math. Soc., 70 (1964),
357-360.
5. Irreducible representations of finite groups of Lie type, J. Reine Ang.
Math., 219 (1965), 180-199.
6. Central extensions of groups of Lie type, J. Reine Ang. Math., 220 (1965),
174-185.
7. The Steinberg character of a finite group with a (B, N)-pair, J. Algebra,
4 (1966), 433-441.
8. A character-theoretic criterion fqr a coset geometry to be a generalized
polygon, J. Algebra, 7 (1967); 208-217.
C. W. Curtis and 1. Reiner
1. Representation Theory of Finite Groups and Associative Algebras, Inter-
science Publishers, New York (1962).
''':''':.2_''-.:..y
N. Bourbaki
1. Grollpes et algebres de Lie, IV, V, VI, Hermann, Paris (1968).
R. Brauer and K. A. Fowler
1. On groups of even order, Ann. of Math., 62 (1955), 565-583.
F. Bruhat
1. Representations induites des groupes de Lie semi-simples connexes, C. R.
A cad. Sci. Paris, 238 (1954), 437-439.
2. Sous-groupes compacts maximaux des groupes semi-simples .9'-adiques,
Seminaire Bourbaki, (1963), Expose 271.
3. Sur une c1asse de sous-groupes compacts maximaux des groupes de Che-
valley sur un corps .9'-adique, I.H.E.S. Publ. Math., 23 (1964), 45-74.
4. .9'-adic groups, Proceedings of ymposia in Pure Mathematics, Vol. 9,
Algebraic groups and discontinuolls subgroups, A.M.S. (1966).
F. Bruhat and J. Tits
1. Groupes algebriques simples sur un corps local, Proceedings of the Con-
ference on Local Fields (Driebergen), 23-36, Springer (1967).
2. C. R. A cad. Sci. Paris, 263 (1966), 598-601, 766-768, 822-825, 867-869.
W. Burnside
1. Theory of Groups of Finite Order, 2nd edition, Dover, New York (1955).
........:..1»>,....
-,
-
Constance Davis
1. A Bibliographical Survey of Simple Groups of Finite Order 1900-1965,
Courant Institute of Mathematical Sciences, New York (1969).
M. Demazure and A. Grothendieck
1. Schemas. en Groupes, I, II, III, Lecture Notes in Mathematics, 151, 152,
153, Spnnger.
L. E. Dickson
1. Linear Groups with an Exposition of the Galois Field Theory, Dover, New
York (1958).
2. Linear groups in an arbitrary field, Trans. Amer. Math. Soc. (1901),
363-394.
'\
1
_ ...-r

'-
........
L
r '

L
r -

L
"
L
I
L
r"
l
-...........-
e
i
\ 
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1
\.-
314
SIMplE GROUPS OF LIE TYPE

f'
L
3.. A new ystem of simple groups, Math. Allll., 60 (1905), 137-150.
J. Dleudonne
1. Sur les grol-lpes classiques, Actualites scientifiques et industrielIes 1040,
Hermann, Paris (1948).
2. Le geometrie des groupes cJassiques, Ergebnisse del' Mathematik und ilzrer
Grenzgebiete, 5 (1955), Springer.
3. Sur 1es generateurs des groupes cIassiques, Summa Brasil. Math. 3 (1955)
149-179. ' ,
4. On simple groups of type Bn, ArneI'. J. Math., 79 (1957), 922-923.
5. Les algebres de Lie sn:pl.es assoc1ees aux groupes simples algebriques
sur un corps de caractenstIque p> 0, Rend. Circ. Mat. Palermo 6 (1957)
198-204. ' ,
Anne Duncan
1. An automorphism of the symplectic group Sp4(2n), Proc. Cambridge
Phi/os. Soc., 64 (1968), 5-9.
E. B. Dynkin
1. The structure of semi-simple Lie algebras, Amer. Math. Soc. Transl. (1),
9 (1955), 328-469.
2. Semisimple subalgebras of semisimple Lie algebras, ArneI'. Math. Soc.
Trans!. (2), 6 (1957), 111-244.
M. Eichler
1. Quadratische Formen ll/1d orthogonale Gruppen, Springer (1952).
V. Ennola
1. On the characters of the finite unitary groups, Ann. A cad. Sci. Fenn. 323
(1963). '
W. Feit
1. Characters of Finite Groups, Benjamin, New York (1967).
2. The Current Situation in the Theory of Finite Simple Groups, Yale University
(1971).
W. Feit and G. Higman '
1. The non-existence of certain generalised polygons, J. Algebra 1 (1964)
114-131. ' ,
W. Feit and J. G. Thompson
1. Solvability of groups of odd order, Pacific J. Math., 13 (1963), 755-1029.
B. Fischer
1. Finite groups generated by 3-transpositions, Invent. Math., to appear.
L. Flatto
1. Basic sets of invariants for finite reflections groups, Bull. ArneI'. Math
Soc., 74 (1968), 730-734.
L. Flatto and Margaret Weiner
1. Invariants of finite reflection groups and mean value problems, ArneI'. J.
Math., 91 (1969), 591-598.
H. Freudenthal
1. Zur Klassifikation der einfachen Lie-Gruppen, Indag. Math., 20 (1958),
379-383.
2. Lie groups in the foundations of geometry, Advances in Math., 1 (1964),
145-190.
H. Freudenthal and H, de Vries
1. Linear Lie Groups, Academic Press (1969).
G. Frobenius
1. Dber Matrizen aus positiven Elementen, Preuss. Akad. JYiss. 'Sifzullgsber
(1908), 471-476.
1. M. Gel'fand and M. 1. Graev
1. Construction of irreducible representations of simple algebraic groups
over a finite field, Dokl. Alcad. Nauk. SSSR, 147 (\ 962), 529-532.
J. A. Gibbs
1. Automorphisms of certain unipotent groups, J. Algebra, 14 -(1970), 203-
228.
D. Gorenstein
1. Finite Groups, Harper and Row, New York (1968).
J. A. Green
1. The characters of the finite general linear groups, Trans. Amer. Math.
Soc., 80 (1955), 402-447.
2. On the Steinberg characters of finite ChevaUey groups, Math. z., 117
(1970), 272-288.
r
lj
M. Hall
1. Theory of Groups, Macmillan, New York (1959).
M. Hall and D. Wales
1. The simple group of order 604,800, J. Algebra, 9 (1968), 417-450.
G. Harder
1. Dber die Galoiskohomologie halbeinfacher Matrizengruppen I, Math. z.,
90 (1965), 404--428.
Harish-Chandra
1. On a lemma of F. Bruhat, J. f,1ath. Pures Appl., 35 (1956), 203-210.
2. Automorphic forms on semisimple Lie groups, Lecture Notes ill Mathematics,
62 (1968), Springer.
3. Eisenstein series over finite fields, Functional Analysis and Related Fields,
edited by F. E. Browder, Springer (1970).
M. Hausner and J. T. Schwartz
1. Lie Groups. Lie Algebras, Nelson (1968).
D. Held
1. The simple group related to MZ4, J. Algebra, 13 (1969), 253-296.
C. Hering, W. M. Kantor and G. M. Seitz
1. Finite groups with a split BN-pair of rank 1, to appear.
D. Hertzig
1. On simple algebraic groups. Proceedings of the International Congress of
Mathematicians, Edinburgh, Cambridge University Press (1960).
2. Forms of algebraic groups, Proc. ArneI'. Math. Soc., 12 (1961), 657-660.
D. G. Higman and C. C. Sims
1. A simple group of order 44,352,000, Math. Z., 105 (1968), 110--113.
G. Higman
1. On the simple group of D. G. Higman and C. C..Sims, Illinois Math. J.,
13 (1969), 74-80.

316
5>IMPLE GROUPS OF LIE TYPE
BmLIOGRAPHY
317
G. Higman and J. McKay
1. On Janko's simple group of order 50,232,960, Bull. London Math. Soc.,
1 (1969), 89-94.
H. Hopf
1. Ober die Topologie der Gruppen-Mannigfaltigkeiten und ihre VeraJJge-
meinerungen, Ann. of Math., 42 (1941), 22-52.
J. E. Humphreys
1. Algebraic groups and modular Lie algebras, Mem. Amer. Math. Soc.,
71 (1967).
2. On the automorphisms of infinite Chevalley groups, Canad. J. Math.,
21 (1969), 908-911.
B. Huppert
1. Endliche Gruppen I, Springer (1967).
S. Ihara and T. Yokonuma
1. On the second cohomology groups (Schur-multipliers) of finite reflection
groups, J. Fac. Sci. Univ. Tokyo, 11 (1965), 155-171.
N. Iwahori
1. On the structure of a Hecke ring of a ChevaUey group over a finite field,
J. Fac. Sci. Univ. Tokyo, 10 (1964), 215-236.
2. Generalized Tits system (Bruhat decomposition) on .9'-adic semisimple
groups, Proceedings of Symposia in Pure Mathematics, Vo/. 9, Algebraic
groups and discontinuous subgroups, A.M.S. (1966), 71-83.
N. Iwahori and H. Matsumoto
1. On some Bruhat decomposition, I.H.E.S. Pub I. Math., 25 (1965), 5-48.
W. Killing
1. Die Zusammensetzung der stetigen endlichen Transformationsgruppen,
I-IV, Math. Ann., 31 (1888), 252-290, 33 (1889), 1--48, 34 (1889): 57-122,
36 (1890), 161-189.
M. Kneser
1. Dber die Ausnahme-Isomorphismen zwischen endlichen klassischen
Gruppen, Abh. Math. Sem. Hamburg, 31 (1967), 136--140.
2. Semisimple algebraic groups, Proceedings of the Instructional Conference
(Brighton) (1965), 250-265.
T. Kondo
1. The characters of the Weyl group of type F4, J. Fac. Sci. Univ. Tokyo,
11 (1965), 145-153.
B. Kostant
1. The principal three-dimensional subgroup and the Betti numbers of a
complex simple Lie Group, Amer. Jour. Math., 81 (1959), 973-1032.
2. Lie group representations on polynomial rings, Bull. Amer. Math. Soc.,
69 (1963), 518-526.
3. Lie group representations on polynomial rings, Amer. J. Math., 85 (1963),
327-404.
4. Groups over Z, Proceedings of Symposia in Pure Mathematics, Vol. 9,
Algebraic groups and discontinuous subgroups, A.M.S. (1966).
N. Jacobson
1. Lie Algebras, Interscience Publishers, New York (1962).
2. Lectures in abstract algebra III Theory of fields, Van Nostrand (1964).
3. Cayley numbers and simple Lie algebras of type G, Duke Math. J., 5 (1939),
775-783.
4. Some groups of transformations defined by Jordan algebras, I, II, III,
J. Reine Ang. Math., 201 (1959), 178-195, 204 (1960), 74-98, 207 (1961),
61-85.
5. A note on automorphisms of Lie algebras, Pacific J. Math., 12 (1962),
303-315.
Z. Janko
1. A new finite simple group with abelian Sylow 2-subgroups, and its charac-
terisation, J. Algebra, 3 (1966), 147-186.
2. Some new simple groups of finite order, Theory of finite groups (Harvard
Symposium), 63-64. Benjamin (1969).
Z. Janko and J. G. Thompson
1. On a class of finite simple groups of Ree, J. Algebra, 4 (1966), 274-292.
C. Jordan
1. TraUt des substitutions et des equations algebriques, Gauthier-Villars, Paris
(1870).
S. Lang
1. Algebraic groups over finite fields, Amer. J. Math., 78 (1956), 555-563.
J. Leech
1. Some sphere packings in higher space, Canad. J. Math., 16 (1964), 657-
682.
S. Lie
1. Seminaire Sophus Lie. Theorie des algebres de Lie. Topologie des grollpes
de Lie, Paris (1955).
D. Livingstone
1. On a permutation representation of the Janko group, J. Algebra, 6 (1967),
43-55.
L. H. Loomis and S. Sternberg
1. Advanced Calculus, Addison-Wesley (1968).
B. Lou
1. The centralizer of a regular unipotent element in a semisimple algebraic
group, Bull. Amer. Math. Soc., 74 (1968), 1144-1146.
H. Luneburg
1. Die Suzukigruppen und ihre Geometrien, Lecture Notes in Mathematics,
10 (1965), Springer.
2. Transitive Erweiterungen endlicher Permutationsgruppen, Lecture Notes
in Mathematics, 84 (1969), Springer.
3. Some remarks concerning the Ree groups of type (G2), J. Algebra, 3 (1966),
256--259.
4. Ober die Gruppen yon Mathieu, J. Algebra, 10 (1968), 194--210.
-',

./


""'- '-
i!:L"'- j
If
l1
t
L
'--
e...
l
-
1. G. Macdonald
1. Spherical functions on a &'-adic Chevalley group, Bull. Amer. Math. Soc.,
74 (1968), 520-525.
2. The Hall algebra, Jber. Deutsch. Math.- Verein, to appear.
J. MacLaughlin
1. Some groups generated by transvections, Arch. Math. (Basel), 18 (1967),
364-368.
2. A simple group of order 898,128,000, Theory of /inite groups (Harvard
Symposium), Benjamin (1969).
E. Mathieu
1. Memoire sur l'etude des fonctions de plusieurs quantites, J. Math. Pures
Appl., 6 (1861), 241-243.
2. Sur les fonctions cinq fois transitives de 24 quantites. J. Math. Pures
Appl., 18 (1873), 25-46.
H. Matsumoto
1. Quelques remarques sur les groupes de Lie algebriques reels, J. Math.
Soc. Japan, 16 (1964), 419-446.
2. Generateurs et relations des groupes de Weyl generalises, c. R. A cad.
Sci. Paris, 258 (1964), 3419-3422.
3. Un theoreme de Sylow pour les groupes semi-simples &'-adiques, C. R.
Acad. Sci. Paris, 262 (1966), 425-427.
W. H. Mills and G. B. Seligman
1. Lie algebras of classical type, J. Math. Mech., 6 (1957), 519-548.
H. V. Niemeyer
1. Definite quadratische Formen der Dimension 24 und Diskriminante 1 (These),
Gottingen (1968).
\-"
O. T. O'Meara
1. Introduction to Quadratic Forms, Springer (1963).
T.Ono
1. Sur les groupes de Chevalley, J. Math. Soc. Japan, 10 (1958), 307-313.
 .
R.Ree
1. On some simple groups defined by C. Chevalley, Trans. Amer. Math.
Soc., 84 (1957), 392-400.
2. A family of simple groups associated with the simple Lie algebra of type
(F4), Amer. J. Math., 83 (1961), 401-420.
3. A family of simple groups associated with the simple Lie algebra of type
(Gz) , Amer. J. Math., 83 (1961).
4. Construction of certain semisimple groups, Canad. J. Math., 16 (1964),
490--508.
5. Sur une famille de groupes de permutations doublement transitifs, Canad.
J. Math., 16 (1964), 797-820.
6. Classification of involutions and centralizers of involutions in finite Che-
valley groups of exceptional types, Proceedings of the International Con-
ference on the Theory of Groups (Canberra). Gordon and Breach (1967).
1.-
\-- .
 -
\.-.
DlBLIOGRAPHY
319
-,
R. Richardson
1. Conjugacy classes in Lie algebras and algebraic groups, Anll. 9f Math.,
86 (1967), 1-15.
F. Richen
1. Modular representations of split (B, N)-pairs, Trans. Amer. Math. Soc.,
140 (1969), 435-460.
B. A. Rozenfel'd
1. Forms of simplicity and semi-simplicity, Trudy Sem. Vektor. Tenzor.
Anal., 12 (1963), 269-285.
H. Samelson
1. Notes on Lie Algebras, Vao Nostrand Reinhold (1969).
W. R. Scott
1. Group Theory, Prentice-Hall (1964).
G. B. Seligman
1. Modular Lie algebras, Ergebni sse der Mathematik lllld ihrer Grenzgebiete,
40 (1967), Springer.
2. Some remarks on classical Lie algebras, J. Math. Mech., 6 (1957), 549-558.
3. On automorphisms of Lie algebras of classical type, I-III, Trans. Amer.
Math. Soc., 92 (1959),430-448, 94 (1960), 452-481, 97 (1960), 286-316.
J-P. Serre
1. Cohomologie galoisienne, Lecture Notes in Mathematics, 5 (1964), Springer.
2. Lie Algebras and Lie Groups, Benjamin, New York (1965).
3. Algebras de Lie semi-simples complexes, Benjamin, New York (1966).
G. C. Shephard
1. Regular complex polytopes, Proc. London Math. Soc., 2 (1952), 82-97.
2. Some problems on finite reflection groups, Ellseignement Math., 2 (1956),
42-48.
D. Soda
1. Some groups of type D4 defined by Jordan algebras. J. Reine Ang. Math.,
223 (1966), 150-163.
L. Solomon
1. Invariants of finite reflection groups, Nagoya Math. J., 22 (1963), 57-64.
2. Invariants of Euclidean reflection groups, Trans. Amer. Math. Soc., 113
(1964), 274-286.
3. A fixed-point formula for the classical groups over a finite field, Trans.
Amer. Math. Soc., 117 (1965), 423-440.
4. The orders of the finite Chevalley groups, J. Algebra, 3 (1966), 376-393.
5. A decomposition of the group algebra of a finite Coxeter group, J. Algebra,
9 (1968), 220--239.
6. The Steinberg character of a finite group with BN-pair, Theory of Finite
Groups (Harvard Symposium), 213-221, Benjamin (1969).
E. L. Spitznagel
1. Hall subgroups of certain families of finite groups, Math. z., 97 (1967),
259-290.

JkV
SIMPLE GROUPS OF LIE TYPE
BIBLIOGRAPHY
321
2. Terminality of the maximal unipotent subgroups of Chevalley groups,
Math. z., 103 (1968), 112-116.
T. A. Springer
1. Some arithmetic results on semi-simple Lie algebras, I.H.E.S. Pub/. Math.,
30 (1966), 115-141.
2. A note on centralizers in semisimple groups, Indag. Math., 28 (1966),
75-77.
3. The unipotent variety of a semisimple group, Proceedings of the Colloquium
in Algebraic Geometry. (Tata Institute), (1969), 373-391.
B. Srinivasan
1. The characters of the finite symplectic group Sp(4, q), Trans. Amer. Math.
Soc., 131 (1968), 488-525.
R. Steinberg
1. A geometric approach to the representations of the full linear group over
a Galois field, Trans. Amer. Math. Soc., 71 (1951), 274-282.
2. Prime power representations of finite linear groups, I, II, Canad. J. Math.,
8 (1956), 580-591, 9 (1957), 347-351.
3. Finite reflection groups, Trans. Amer. Math. Soc., 91 (1959), 493-504.
4. Variations on a theme of Chevalley, Pacific J. Math., 9 (1959), 875-891.
5. Invariants of finite reflection groups, Canad. J. Math., 12 (1960), 616-618.
6. Automorphisms of finite linear groups, Can ad. J. Math., 12 (1960), 606-
615.
7. The simplicity of certain groups, Pacific J. Math., 10 (1960), 1039-1041.
8. Automorphisms of classical Lie algebras, Pacific J. Math., 11 (1961),
1119-1129.
9. A closure property of a set of vectors, Tran{. Amer. Math. Soc., 105
(1962), 118-125.
10. Generators for simple groups, Canad. J. Math., 14 (1962), 277-283.
11. Generateurs, relations et revetements de groupes algebriques, Colloquium
sur la tl1(orie des groupes algebriques (Bruxelles), (1962), 113-127.
12. Representations of algebraic groups, Nagoya Math. J., 22 (1963), 33-56.
13. Differential equations invariant under finite reflection g1OUps, Trans.
Amer. Math. Soc., 112 (1964), 392-400.
14. Regular elements of semisimple algebraic groups, I.H.E.S. Pub!. Math.,
25 (1965), 49-80.
15. Lectures on Chevalley Groups, Yale University (1967).
16. Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc.,
80 (1968).
17. Algebraic groups and finite groups, Illinois J. Math., 13 (1969), 81-86.
1. N. Stewart
1. Lie algebras, Lecture Notes in Mathematics, 127 (1970), Springer.
M. Suzuki
1. A new type of simple groups of finite order, Proc. Nat. A cad. Sci. U.S.A.,
46 (1960), 868-870.
2. On a class of doubly transitive groups, Ann. of Math., 75 (1962), 105-145.
3. A simple group of order 448,345,497,600, Theory of finite groups (Harvard
Symposium), Benjamin (1969).
J. G. Thompson
1. Toward a characterisation of E2(q), J. Algebra, 7 (1967), 406-414.
2. Nonsolvable finite groups all of whose local subgroups are solva'ble, (I)
Bull. Amer. Math. Soc., 74 (1968), 383-437; (II) Pacific J. Math., 33 (1970),
451-536.
3. Quadratic pairs, to appear.
J. Tits
1. Sur les analogue algebriques des groupes semi-simples complexes, Collo-
quium d'algebre superieure (Bruxelles), (1956), 261-289. ..---
2. Les groupes de Lie exceptionnels et leur interpretation geometrique, Bul/.
Math. Soc. Be/g., 8 (1956), 48-81.
3. Les "formes reelles' des groupes de type E6, Seminaire Bourbaki (1958),
Expose 162.
4. Sur la trialite et certains groupes qui ;5'en deduisent, I.H.E.S. Publ. Math.,
2 (1959), 14-60.
5. Sur la classification des groupes algebriques semi-simples, C. R. A cad.
Sci. Paris, 249 (1959), 1438-1440.
6. Les groupes simples de Suzuki et de Ree, Seminaire Bourbaki, 13 (1960),
Expose 210.
7. Groupes algebriques semi-simples et geometries associees, Proceedings of
the colloquium on the algebraic and topological foundations of geometry
(Utrecht), 175-192, Pergamon Press, Oxford (1962). .
8. Groupes simples et geometries associees, Proceedings of the InternatIOnal
Congress of Mathematicians (Stockholm), (1962), 197-221.
9. Ovoides et groupes de Suzuki, Arch. Math., 13 (1962), 187-198.
10. Groupes semi-simples isotropes, Colloquium on the 'I11eory of Algebraic
Groups (Paris), 137-147, Gauthier-Vi1lars (1962).
11. Theoreme de Bruhat et sous-groupes paraboliques, C.R. A cad. Sci. Paris,
254 (1962), 2910-2912.
12. Algebraic and abstract simple groups, Ann. of Math., 80 (1964), 313-329.
13. Geometries polyedriques finis, Rend. Mat. e AppL, 23 (1964), 156-16.
14. Sur les systemes de Steiner associees aux trois "grands' groupes de MathIeu,
Rend. Mat. e Appl., 23 (1964), 166-184.
15. Sur les constantes de structure et Ie theoreme d'existence des algebres de
Lie semi-simples, I.H.E.S. Publ. Math., 31 (1966).
16. Normalisateurs de tores 1. Groupes de Coxeter etendus, J. Algebra, 4
(1966), 96-116. . .
17. Algebres alternatives, algebres de Jordan et algebres de LIe exceptlOnnelles,
Indag. Math., 28 (1966), 223-237. ..
18. Classification of algebraic semisimple groups, Proceedmgs of Symposza
in Pure Mathematics, Vol. 9 Algebraic groups and discontinuous subgroups,
A.M.S. (1966) 33-62.
19. Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen, Lecture
Notes in Mathematics, 40 (1967) Springer.
20. Le groupe de Janko dordre 604800, Theory of finite groups (Harvard
Symposium), 91-95, Benjamin (1969). ,
21. Groupes finis simples sporadiques, Seminaire Bourbaki (1969-70), Expose
375.
.........=
'.
,
.......\
'
....,..,..-'

; ,.
( -
-..- '-"
_L.
i.......
r.;:'>'
:a
l,,,,....
t- ,..
t'
f
L_
If ..
f
,"=,l
J: .-...
 .
It...-j..
.-
\. ",,!,.
( ""
1
1
t.
;
1<___,"
i
1..</
f d,
j
I
'-
L.
322
SIMPLE GROUPS OF LIE TY1'E
22. Notes on finite BN-pairs, to appear.
J. A. Todd and G. C. Shephard
1. Finite unitary reflection groups, Can ad. J. Math.. 6 (1954), 274-304.
T. Tsuzuku
1. A characterisation of finite projective linear groups Proc. Japan Acad
40 (1964), 155-156. ,.,
B. J. Veisfeiler
1. A class of unipotent subgroups of semisimple algebraic groups. Uspehi
Mat. Nauk.. 21 (1966), 222-223.
B. L. van der Waerden
1. Gruppen von linearen Transformatiollen, Chelsea Publishing Co., New
York (1948).
G. E. Wall
1. On the conjugacy classes in the unitary, symplectic and orthogonal groups
J. Austral. lv/(Jth. Soc., 3 (1963), 1-62. ·
H. N. Ward
1. On Ree's series of simple groups. Trails. Amer. Math. Soc., 121 (1966),
62-89.
H. Weyl
1. {) ber die Darstellungen halbeinfacher Gruppen durch lineare Trans-
formationen, I. II, lvlath. Z., 23 (1925), 271-309, 24 (1926), 328-395.
2. Te structure and representations of continuous groups, I.A.S. notes.
Pnncelon (1934-5).
3. The Classical Groups. Princeton University Press (1946).
D. J. Winter
1. On groups of automorphisms of Lie algebras, J. Algebra 8 (1968), 131-
142. '
E. Witt
1. Die 5-fach transitive Gruppe von Mathieu, Abh. Math. Sem. U/liv. Ham-
burg, 12 (1938), 256-264.
2. Ober Steinersche Systeme, Abh. Math. Sem. Ultiv. Hamburg. 12 (1938),
265-274.
3. Spiegelungsgruppen und Auf:cihlung halbeinfacher Liescher Ringe, Abh.
lv/ath. Sem. Univ. Hamburg, 14 (1941). 289-337.
Symbol
II
<.. . >
1-
A
Ai
Al(K)
Al(q)
2Al(K)
2At(q2)
An;
ad x
B
Bl
Bl
Bt(K)
EtCq)
2 B'J(K)
2B2(22m+l)
C
C
CJ
Ctjrs
T. Y okonuma
1. Sur la structure des anneaux de Hecke d'un groupe de Chevalley fini,
C. R. Acad. Sci. Paris. 264 (1967), 344-347.
2. Sur la commutant d'une representation d'un groupe de Chevalley fini.
C. R. A cad. Sci. Paris, 264 (1967). 433-436; J. Fac. Sci. Univ. Tokyo.
15 (1968), 115-129.
Ct
Ct(K)
Ct(q)
DJ
d1. . . . . dl
Dt
Di(K)
DI(q)
2 DI(K)
Index of Notation
Melll/jng
The order of a group
The group generated by a given 'set of elcmcnts
The orthogonal subspacc of a givcn subspace
The rational group algebra of e(Q)
Thc type of a simple Lie algebra ovcr C
Thc Chevalley group of type At ovcr K
Thc Chevalley group of type At over GF(q)
The twisted group of type Az over K
The twisted group of type At ovcr GF(q2)
The Cartan intcger associated with a pair of roots
The adjoint map of left multiplication by thc clemcnt
x in a Lic algebra
The subgroup U I-l of a Cheval Icy group
The subgroup n nGL or a twisted group G1
Thc type of a simplc Lie algebra ovcr C
The Chcvalky group of type Ih over K
The Chevalley group of type Bt ovcr GF(q)
The twisted Suzuki group of type B'l over K
The twistcd Suzuki group of type B:!. over GF(2 2111+ 1)
The complex field
A chamber
The elemcnt of the Coxeter comp1cx associated with
a set J of fundamcntal roots
The constants occurring in Chevallcy's commutator
formula
Thc type of a simple Lie algebra over C
The Cheval1ey group of type Ct over K
The Chevalley group of type Clover GF(q)
The set of distinguished coset representatives of WJ
in v
The degrecs of the basic polynomial invariants of W
The type of a simple Lie algebra over C
The Chevallcy group of type Dz over K
The Chevallcy group of typc Dl over GF(q)
The twistcd orthogonal group of type Dl over K
323
Page of
definition
148
43
64
121
251
251
38
34
104
230
43
64
121
251
251
34
21
31
77
43
64
121
29
130
43
64
121
251

324
Symbol
2 Dl(q2)
3 D'l(K)
3 D4(q3)
er
e(Q)
E6
Eo(K)
£O(q)
2Ejj(K)
2£jj(q2)
E7
E7(K)
E7( l{ )
£8
E8(K)
E8(q)
£.1
F4(K)
FL1(q)
2F4(K)
2F4(22m+1)
G
G
G
G'
GI
G2
G2(K)
G2(q)
2Gz(K)
2G2(3 2m+ I)
GF(q)
GLn(K)
H
'j[)
hr
'j[)r
J-Ir
hex)
fI
hr('A)
SIMPLE GROUPS OF LIE TYPE
Meaning
The twistcd orthogonal group of type Dl over GF(q2)
The triality twisted group of type D4 over K
The triality twisted group of type D4 over GF(q3)
A root vcctor in a simple Lic algebra
A multiplicative group isomorphic to the additive
group Q
The type of a simple Lie algebra over C
The Chevalley group of type E6 over K
The Chev.dley group of type Eo ovcr GF(q)
The twistcd group of type Eo ovcr K
The twistcd group of type Eo over GF(q2)
The type of a simple Lic algebra ovcr C
The Chevalley group of type E7 over K
The Chcvalley group of type E7 over GF(q)
The type of a simple Lie algebra over C
The Chcvallcy group of type E8 over K
The Chcvalley group of type £8 over GF(q)
The type of a simple Lie algebra over C
The Chevalley group of type F4 over K
The Chevalley group of type F4 over GF(q)
The twisted Rcc group of type F4 over K
The twisted Ree group of type F4 over GF(22m+1)
A group, usually a Chevallcy group
The universal Chevalley group associated with G
The extension of G by its group of diagonal auto-
morphisms
The commutator subgroup (derived group) of G
A twisted group
The typc of a simple Lie algebra over C
The Chevalley group of type Gz over K
The Chevalley group of type G2 over GF(q)
The twisted Rce group of type G2 over K
The twisted Ree group of type G2 over GF(32m+1)
The Galois field with q elements
The gencrallinear group of degree n over K
Thc diagonal subgroup of a Chcvallcy group G
A Cartan subalgebra of a simple Lie algebra
The co-root associate<.l to a root r
The I-dimensional subs pace ofJ£) containing hr
The hyperplane orthogonal to a root r
The automorphism of the Lie algebra JLK determined
by a K-character X of P
The group of automorphisms hex) of JLK
The demcnt of H associated with a root r and an
elemcnt ,\ of K
Page of
definition
251
251
251
51
148
43
64
121
251
251
43
64
121
43
64
121
43
64
121
251
251
68
190
118
170
226
43
64
121
251
251
2
2
97
35
42
83
20
98
98
92
Symbol
her)
HI
]
]+
]n
It, . . . , /z
jJ
]'1
]J
(] J) Tl
J
K
K*
l
l(w)
JL
JLr
JLx
JL(K)
JL(q)
JL (K)
LJ
Mr,/!. t
/111, . . . , ml
N
N
NT, II
NJ
llw
/lr
I1r(t)
, llJ(W)
Nl
OJ
J
OIl(K, f)
INDEX OF NOTATION
Meaning
The height of a root r
The diagonal subgroup H n Gl of a twisted group Gl
The ring of polynomial invariants of a Weyl group W
The set of polynomials in 3J with no constant term
The set of homogeneous polynomials in 3J of degree n
A set of basic polynomial invariants of the Weyl
group W
The set of alternating polynomials on 1)
The set of homogeneolls altcrnating polynomials of
dcgree n
Thc sct of polynomials on 1) invariant under W J
The set of homogeneous polynomials in ]J of degree n
A subsct of the set n of fundamental roots
A field
Thc multiplicative group of non-zero elements of K
The rank of a simple Lie algebra
Thc length of an clement w of the Weyl group
A Lie algebra, usually simple ovcr C
The I-dimcnsional subspace ofJL corresponding to the
root ,.
The Lie algebra over.K constructed from the simple
Lie algebra JL over C
The Chevalley group of type JL over K
The Chevalley group of type JL over GF(q)
The twisted group constructed from a Chevalley
group JL (K) by mcans of a symmetry of order i
A Levi subgroup of a parabolic subgroup P J
The integer :!,Nr, 8Nr,r+/! . . . Nr. (i-l)r+/!
l.
The exponents of the Weyl group
The number of positive roots
The monomial subgroup of a ChevaUey group G
The structure constant of a simple Lie algebra deter-
mined by a pair of roots
The inverse image in N of the subgroup W J of W
An element of N mapping to W in W
The generator Xr(1)x-r( -l)xr(l) of N
The element Xr(t)X_r( - r1)Xr(t) of N
The number of clements fixed by w in the W-orbit
containi-ng CJ of the Coxeter complex
The monomi:_d subgroup N n GI of a twisted group GI
The number of p-orbits in a p-invariant set J of
fundamental roots
The orthogonal group of degree n over K leaving
invariant the quadratic formf
325
Page of
definitioll
16
226
124
125
132
..--....l
126
139
140
140
140
27
2
97
35
18
33
36
"-"'''1,
62
64
121
""',
251
119
61
155
43
101
52
108
115
93
96
136
226
254

4

i
I
"- ..........


L
.......-
L..
r-

t..-
f
r
if
.
- 'c,,_,..
r
[
L
r
.
'--
L
326
Symbol
Oiz(q)
°2lCq)
PI,..., PZ
P
PJ
Pw(t)
PW1(t)
PGLn(K)
PSpn(K)
PSOn(K. f)
POn(K, f)
PSUn(K, f)
Q
ql. . . . , qz
Q

R
S
S
i1>
n
SLn(K)
Spn(K)
SOn(K. J)
SUn(K, f)
St A
U
Urn
Ur
U+
w
U
SIMPLE GROUPS OF LIE TYPE
Meaning
The orthogonal group of degree 21 over GF(q) leaving
invariant a quadratic form of index I
Thc orthogonal group of degree 2/ ovcr GF(q) leaving
invariant a quadratic form of index /-1
A system of fundamental roots
The additive group generatcd by PI, . . . ,Pz
The parabolic subgroup BNJB associated with a set J
of fundamental roots
The polynomial  tZ(w)
weW
The polynomial  tl (w)
weWl
The projective general linear group of degree II over K
The projcctive symplectic group of degrce n over K
The p ojcctive special orthogonal group of degree 11
over K leaving invariant the quadratic form f
The projective group of the commutator subgroup
Dn(K, f) of On(K, f)
The projective special unitary group of degree II over
K leaving invariant the Hermitian form f
The field of rationals
The fundamental weights of a simple Lie algebra
The additive group generated by ql. . . . , qz
The real number field
The roc: of maximum height
The sum of the fundamental weights
An equivalence class in cD
The algebra of polynomial functions on l'
Thc set of homogeneous polynomials in of degree Il
The special linear group of degrce Il over K
The symplectic group of degree 11 over K
The special orthogonal group of degree 11 over K
leaving invariant the quadratic form f
The special unitary group of degree n over K leaving
invariant the Hermitian form f
The star of an element A in a chamber complex
The unipotent subgroup of a Chevalley group G
generated by the positive root subgroups
The subgroup of U generated by the root subgroups
corresponding to roots of height at least m
The product of the root subgroups corresponding to
positive roots other than r
The product' of the root subgroups corresponding to
positive roots transformed by w into positive roots
The product of the root subgroups corresponding to
positive roots transformed by w into negative roots
Page of
definition
Symbol
V.I
UL
(U) 1
U n(K, f)
1)
1)J
6
6
38
97
108
]35
254
2
4
v
V-r
1)
VI
VI
W
5
5
II'r
7
153
98
99
12
156
145
227
123
132
2
2
IVO
WJ
WI
w
WCE)
Xr(t)
Xr
5
Xs
7
274
Xl
S
7L
Z
68
E"l. . . . , E"Z
78
T} 1, . . . . T}Z
T}r. 6
>.(r)
104
115
lI( f)
II
III
P
]15
INDEX OF NOTATION
Meaning
The unipotent radical of the parabolic subgroup P J
Thc unipotent subgroup Un c; I or a twisted group Gl
Thl.: subgroup U n Gl of a twistcd group Gl
The unitary group of degrcc /1 ovcr K leaving invariant
the Hcrmitian form f
A wctor space
The subspace of 1) spanned by a set J of fundamental
roots
The unipotent subgroup of a ChevaJIey group G
gcnerated by the negative root su bgroups
The product of the root subgroups corresponding to
negative roots othcr than - r
The dual spacc of 1)
Thc sct of clements 01'1) invariant under the isomctry T
The unipotcnt subgroup V n Gl of a twisted group Gl
A Weyl group
Thc reOection in the hypcrplane orthogonal to the
root r
The element of W which transforms cach positive root
to a negative root
The subgroup of W gcneratcd by the fundamental
rcllections w,- for r in J
The subgroup of clements of W commuting with the
isometry T
The clcmcnt of WJ which transforms every positive
root jn (DJ to a ncgative root
The group of type-preserving automorphisms of the
abstract Coxetcr group 2:;
The generator exp (t ad ('r) of a Chcvalley group
The root subgroup of a Chcvallcy group correspond-
ing to the root r
The subgroup of a Chevalley group gcneratcd by the
root subgroups corresponding to roots in S
The root subgroup Xs n Gl of a twisted group Gl
The ring of rational integcrs
Thc ccntrc of the univcrsal Cheva\lcy group (;
The eigenvalues of T with eigenvectors It, . . . , /z
The eigenvalues of the isometry T of 1)
The integer -=!: 1 defincd by fir. es = T}r, sewr(s)
An integer which takes value 1 if r is a short root and
2 or 3 if r is a long root
The Witt index of a formf
A fundamental system of roots
The set of projections on to ))1 of roots in II
A symmetry of the Dynkin diagram
327
Page of
definition
J19
226
229
7
2
27
68
105
123
217
226
13
12
20
27
217
218
284
64
68
227
227
50
190
254
251
93
204. 206
5
J3
219
200

328
Symbol
a
L:
L:( W, IT)
T
<D
(1)+
<1>-
<DJ
(1)*
(})1
X
Xr, A
n
a(G; B, N)
nn(K,f)
>-
SIMPLE GROUPS OF Lffi TYPE
Meaning
An automorphism of a Chcvalley group obtained by
combining a graph and a lield automorphism
An abstract Coxetcr complex
The abstract Coxeter complex associated with a Weyl
group Wand fundamcntal system II
The isometry of V dctermined by a symmetry p of the
Dynkin diagram
A root system
The sct of positivc roots in <D
Thc set of negative roots in <D
The set of roots in QJ which lie in the subspace 19J
The set of co-roots of roots in (D
The set of projections on to 1> 1 of roots in (D
A K-character of P
The K-character of P taking value AAra at the root s
A building
The building associated with a group G with (B. N)-
pair
The commutator subgroup of the orthogonal group
On (K, f)
A total order relation on V
Page of
dejinition
225
281
288
201,217
12
14
16
27
49
219
97
98
292
abstract Coxeter complex 281, 284,
288, 291, 296
adjacent chambers 276
adjoint Chcvalley group 198
alternating elements 148
alternating polynomials 139
apartments 292, 296
automorphisms, of Chevalley groups
199, 211
of simple Lie algebras 60, 63
293
5
13
basic polynomial invariants 128, 129
Bctti numbcrs 169
(B,N)-pair 107-114, 294
in twisted groups 227, 230
operating on a building 299, 302
Borel subgroup 104
Bruhat decomposition 104, 106, 109
building 292, 296, 302
associated with a group with (B,N)-
pair 293, 299
canonical form, for clements of a
Chevallcy group 115, 117
for elements of a unipotcnfsubgroup
78 .
for elements of a twisted group 229
Cartan decomposition 35
Cartan matrix 43-45, 99. 122
ccntral series of unipotent group 78
chamber 21-23, 275
chamber complex 275
Chevalley basis 56
Chcvalley group. definition 64
Chevalley's theorems, on existence of
integral basis 56
on commutator formula 76
on polynomial invariants j 28
-\
Index
classification of simple Lie algcbras 43
closed set of roots 114
Colcman's theorcm 166
commutator formula 76
complex 274
Conway groups 308
co-roots 49
Coxctcr complex 30-32. 136. 137.
255, 288
Coxcter element 156
Coxetcr group 25, 284, 286, 292
Coxeter's thcorem 156
degrees of basic invariants 130-133,
145, 155
derivations 34, 61
diagonal automorphism 200
diagonal subgroup H, of a Chevalley
group 97, 99, 117
of a twisted group 238, 244
dihedral subgroup of Weyl group
158, 161
distinguished coset representatives 30,
138
double coset decomposition 109, 110
duality of exponents 156, 168
dual root system 49, 50
Dynkin diagram. of a simple Lie
algebra 40
of the twisted groups 224

J ..."".
i!1J<
I
f
I 1I!!ltI:.\!1,l
r
f
I
! fI'!Iom'l;-"*
eigenvalues of Coxeter element 165,
166, 168
elementary symmetric polynomials
124
existence theorem for simple Lie
algebras 42
exponential map "60, 66

}
329

330
SIMPLE GROUPS OF LIE TYPE
L \.__-
xponents of Weyl group 169
eXlraspccial pair of roots 58
6.""
factorization of the polynomial fl(w),
in Chcvalley groups 135
in twisted groups 254
field automorphism 200
finite Chevallcy groups 120-122
finite twisted groups 251
Fischer groups 309
folding 277
Frobenius-Pcrron theorem 162
fundamental group 99
fundamental rd1cction 17
fundamental system of roots 13, 19,
21
fundamental weights 98, 146, 148
l.._
1

:......'-"
gallery 276
generators and relations, for Chevalley
group 190
for Weyl group 23, 25
graph automorphism, of the groups
Bz(K), F4(K) 204
of the group G2(K) 206
:.;
l-
height of a root 16,77.153-155
Held group 306
Higman-Sims group 307
homomorphism, from SL2«([:) 87
from SL2(K) 88
ideal of Lie algebra 33
induced character 136
isomorphism theorem_ for simple Lie
algebras 42
Jacobian 134. 166
Jacobi identity 33
Jankl) groups 305
K-character 97,99, 121, 199
self-conjugate 238
Killing form 34
,
Leech lattice 308
Lershetz fixed point formula 169
length function 18, 136
Lcvi decomposition 118
Levi subgroup 119
Lie algebra, definition 33
linear groups 2, 184, 185
Lyons group 306
semi-simple Lie algebra 36
simplex 274
simplicity, of Chevallcy groups 172
of groups with (B.N)-pair 170
of twisted groups 262
SL2 67, 81. 87, 88
Solomon's theorem 135. 143
special pair of roots 58
sporadic simple groups 303
Steinberg's theorems, on automor-
phisms of finite Chevalley groups
211
on generators and relations for
Chevalley groups 190
on generators and relations for the
Weyl group 23, 25
Steiner system 303
structure constants 52, 55. 58
for the Lie algebra G2 211
subalgebras of Lie algcbras 33
Suzuki groups 251, 307
symmetric algebra 123
symmetric group 124
'-symmetric polynomials 124
symmetry of Dynkin diagram 200,
221-223
symplectic groups 3, 184, 186
Macdonald's theorem 151
Mathieu groups 303
Matsumoto's theorem 286
McLaughlin group 307
monomial subgroup N, of Chevalley
group 101
of twisted group 228
relationship to Weyl group 102
morphism of chamber complexes 276
nilpotent derivations 61. 66. 69
nilpotent Lie algebra 35
opposite folding 280
order, of Coxeter elements 163, 168
of finite Chevalley groups 122, 144
of finite twisted groups 253, 259,
262
of sporadic simple groups 3 JO
ordering of vector space 13
orthogonal groups 4-8, 184-188, 271
parabolic subgroups, of a group with
(B,N)-pair 111-113
of a Chevalley group 118, 119
of a twisted group 231
of a Weyl group 27-30, 32
Poincare polynomial 169
polynomial invariants of Weyl group
123
positive systems of roots 13
rank of a Lie algebra 35
Rcc groups 251
Rec's theorem 184
rcllcclions J 2, 284
retraclions 282, 296
roots of a simp1c Lie algebra 36
root subgroups, of Chcvalley groups
68
of twistcd groups 233, 235
root syslcl11s 12, 45-49
",_ __--'-- l__ _
fNDEX
331
"-

t:

Fi.
r
f

t
1
}

Tits' theorems, on finite groups with
(B,N)-pair 302
on parabolic subgroups 112, 113
on simplicity of groups with (B,N)-
pair 170
on the apartmcnts of a building 296
transitive extensions 304, 306
twisted group, definition 226
type, of clement in a building 298
of element in a chamber complex
283
type-preserving automorphisms 284,
290,291,299
unipotent linear transformation 68
unipotent subgroup, of a ChevaJley
group 68. 78, 104, 114
of a twisted group 231
universal Chevalley group 197
Weyl's theorem 149
Weyl group, of a root system 13
of a ChevaJley group 102
of a twisted group 221-224, 226,
228
lXi SM1) M 3 
- :o-"""",,:;,--_.-:--,--.--,r