Lie Groups and Lie Algebras: Chapters 4-6

314 Pages·2002·16.33 MB·English

by  N. Bourbaki

ELEMENTS OF MATHEMATICS
Lie Groups and lbie Algebras
Chapters 4-6
Springer
ELEMENTS OF MATHEMATICS
Springer
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NICOLAS BOURBAKI
ELEMENTS OF MATHEMATICS
Lie Groups and
Lie Algebras
Chapters 4-6
' Springer
Originally published as
GROUPES ET ALGEBRES DE LIE
Hermann, Paris, i968
Translator
Andrew Pressley
Department of Mathematics
King's College London
Strand, London WC2R 2LS
United Kingdom
e-mail: [email protected]
Mathematics Subject Classification (2000 ):
22E10;22E15;22E60;22-02
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Die Deutsche Bibliothek · CIP-Einheitsaufnahme
Bourbaki, Nicolas:
Elements of mathematics IN. Bourbaki. -Berlin; Heidelberg; New York;
London; Paris; Tokyo; Hong Kong: Springer
Einheitssacht.: £1ements de mathematique <engl.>
Lie groups and Lie algebras
Chapters 4/6. -2002
ISBN 3-540-42650-7
ISBN 3-540-42650-7 Springer-Verlag Berlin Heidelberg New York
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INTRODl.JCTION TO CHAPTERS IV, V
AND VI
The study of semi-simple (analytic or algebraic) groups and their Lie algebras
leads to the consideration of root systems, Co:r:eter groups and Tits systems.
Chapters IV, V and VI are devoted to these structures.
To orient the reader, we give several examples below.
I. Let g be a complex semi-simple Lie algebra and IJ a Cartan subalgebra
of g1. A root of g with respect to 1J is a non-zero linear form a on 1J such
that there exists a non-zero element x of g with [h, x] = a(h)x for all h E IJ.
These roots form a reduced root system R in the vector space IJ* dual to
IJ. Giving R determines g up to isomorphism and every reduced root system
is isomorphic to a root system obtained in this way. An automorphism of g
leaving IJ stable defines an automorphism of IJ* leaving R invariant, and every
automorphism of R is obtained in this way. The Wey! group of R consists of
all the automorphisms of IJ" defined by the inner automorphisms of g leaving
1J stable; this is a Coxeter group.
Let G be a connected complex Lie group with Lie algebra g, and let I'
be the subgroup of G consisting of the elements h such that expc(27rih) = 1.
Let K be the root system in IJ inverse to R, let Q(K) be the subgroup of IJ
generated by R- and let P(K) be the subgroup associated to the subgroup
Q(R) of I)* generated by R (i.e. the set of h. E IJ such that >..(h) is an integer
for all ).. E Q(R)). Then P(K) ~ I' ~ Q(R-). Moreover, the centre of G
is canonically isomorphic to P(K)/I ' and the fundamental group of G to
I'/Q(K). In particular, I' is equal to P(K) if G is the adjoint group and I'
is equal to Q(K) if G is simply-connected. Finally, the weights of the finite
dimensional linear representations of G are the elements of the subgroup of
r.
IJ* associated to
II. Let G be a semi-simple connected compact real Lie group, and let g be its
Lie algebra. Let T be a maximal torus of G, with Lie algebra t, and let X be
the group of characters of T. Let R be the set of non-zero elements a of X
such that there exists a non-zero element .r of g with (Ad t).x = o(t)x for all
t E T. Identify X with a lattice in the real vector space V = X @z R; then R
is a reduced root system in V. Let N be the normaliser of T in G; the ac~ion
1 In this Introduction, we use freely the traditional terminology as well as the
notions defined in Chapters IV, V and VI.
VI TO CHAPTERS IV, V AND Vl
of Non T defines an isomorphism from the group N/T to the Weyl group of
R. We have P(R) ::> X ::> Q(R); moreover, X = P(R) if G is simply-connected
and X = Q(R) if the centre of G reduces to the identity element.
The complexified Lie algebra 9(C) of g is semi-simple and t(c) is a Cartan
subalgebra of it. There exists a canonical isomorphism from V(c) to the dual
of t(c) that transforms R into the root system of 9(C) with respect to t(c).
III. Let G be a connected semi-simple algebraic group over a commutative
field k. Let T be a maximal element of the set of tori of G split over k and
let X be the group of characters of T (the homomorphisms from T to the
multiplicative group). We identify X with a lattice in the real vector space
V = X ®z R. The roots of G with respect to T are the non-zero elements a of
X such that there exists a non-zero element x of the Lie algebra g of G with
(Ad t).x = a(t)x for all t E T. This gives a root system R in V, which is not
necessarily reduced. Let N be the normaliser and Z the centraliser of T in G
and let N(k) and Z(k) be their groups of rational points over k. The action
of N(k) on T defines an isomorphism from N(k)/Z(k) to the Wey! group of
R.
Let U be a maximal element of the set of unipotent subgroups of G, de
=
fined over k and normalised by Z. Put P = Z.U. Then P(k) Z(k).U(k) and
P(k) n N(k) = Z(k). Moreover, there exists a basis (a1, ... , O'.n) of R such
that the weights of T in U are the positive roots of R for this basis; if S de
notes the set of elements of N(k)/Z(k) that correspond, via the isomorphism
defined above, to the symmetries s0 , E W(R) associated to the roots ai, the
quadruple (G(k), P(k), N(k), S) is a Tits system.
IV. In the theory of semi-simple algebraic groups over a local field, Tits
systems are encountered whose Wey! group is the affine Wey! group of a.
root system. For example, let G = SL(n + 1, Qp) (with n 2: 1). Let B be
the group of matrices (aij) E SL(n + 1, Zp) such that a;,; E pZP for i < j,
and let N be the subgroup of G consisting of the matric~s having only one
non-zero element in each row and column. Then there exists a subset S of
N/(BnN) such that the quadruple (G,B,N,S) is a Tits system. The group
W = N/(B n N) is the affine Wey! group of a root system of type An; this is
an infinite Coxeter group.
Numerous conversations with J. Tits have been of invaluable assistance to ·ns
in the preparation of these chapters. We thank him ver·y cordially.
CONTENTS
INTRODUCTION TO CHAPTERS IV, V AND VI . . . . . . . . V
CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
CHAPTER IV COXETER GROUPS AND TITS SYSTEMS
§ 1. Coxeter Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I. Length and reduced decompositions . . . . . . . . . . . . . . . . . . . . . 1
2. Dihedral groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3. First properties of Coxeter groups . . . . . . . . . . . . . . . . . . . . . . 4
4. Reduced decompositions in a Coxeter group . . . . . . . . . . . . . . 5
5. The exchange condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
6. Characterisation of Coxcter groups . . . . . . . . . . . . . . . . . . . . . . 10
7. Families of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
8. Subgroups of Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
9. Coxeter matrices and Coxeter graphs . . . . . . . . . . . . . . . . . . . . 13
§ 2. Tits Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1. Definitions and first properties . . . . . . . . . . . . . . . . . . . . . . . . . 15
2. An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3. Decomposition of G into double cosets . . . . . . . . . . . . . . . . . . 18
4. Relations with Coxeter systems . . . . . . . . . . . . . . . . . . . . . . . . . 19
5. Subgroups of G containing B . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6. Parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7. The simplicity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Appendix. Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
L Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2. The connected components of a graph . . . . . . . . . . . . . . . . . . . 27
3. Forests and trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Exercises for § 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Exercises for § 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
VIII CONTENTS
CHAPTER V GROUPS GENERATED BY REFLECTIONS
§ 1. Hyperplanes, chambers and facets . . . . . . . . . . . . . . . . . . . . . . 61
1. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2. Facets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3. Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4. Walls and faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5. Intersecting hyperplanes ................... _· . . . . . . . . . . . 67
6. Simplicial cones and simplices . . . . . . . . . . . . . . . . . . . . . . . . . . 68
§ 2. Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1. Pseudo-reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2. Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3. Orthogonal reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4. Orthogonal reflections in a euclidean affine space . . . . . . . . . 73
5. Complements on plane rotations . . . . . . . . . . . . . . . . . . . . . . . . 74
§ 3. Groups of displacements generated by reflections . . . . . . . 76
1. Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2. Relation with Coxeter systems . . . . . . . . . . . . . . . . . . . . . . . . . 78
3. Fundamental domain, stabilisers . . . . . . . . . . . . . . . . . . . . . . . . 79
4. Coxeter matrix and Coxeter graph of HI . . . . . . . . . . . . . . . . . 81
5. Systems of vectors with negative scalar products . . . . . . . . . . 82
6. Finiteness theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7. Decomposition of the linear representation of W on T . . . . 86
8. Product decomposition of the affine space E . . . . . . . . . . . . . . 88
9. The structure of chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10. Special points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
§ 4. The geometric representation of a Coxeter group . . . . . . . 94
1. The form associated to a Coxeter group . . . . . . . . . . . . 94
2. The plane Es,s' and the group generated by (J3 and (J's' 95
3. The group and representation associated to a Coxeter
matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4. The contragredient representation . . . . . . . . . . . . . . . . . . . . . . 97
5. Proof of lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6. The fundamental domain of W in the union of the
chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7. Irreducibility of the geometric representation of a Coxeter
group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8. Finiteness criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
9. The case in which BM is positive and degenerate . . . . . . . 105
CONTENTS IX
§ 5. Invariants in the symmetric algebra .................... . 108
1. Poincare series of graded algebras ...................... . 108
2. Invariants of a finite linear group: modular properties 110
3. Invariants of a finite linear group: ring-theoretic properties . 112
4. Anti-invariant elements ............................... . 117
5. Complements 119
§ 6. The Coxeter transformation ........................... . 121
1. Definition of Coxeter transformations .................. . 121
2. Eigenvalues of a Coxeter transformation: exponents . . . . . . . 122
Appendix: Complements on linear representations . . . . . . . . . . 129
Exercises for § 2. 133
Exercises for § 3. 134
Exercises for § 4. 137
Exercises for § 5. 144
Exercises for § 6. 150
CHAPTER VI ROOT SYSTEMS
§ 1. Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
1. Definition of a root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
2. Direct sum of root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
3. Relation between two roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4. Reduced root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5. Chambers and bases of root systems . . . . . . . . . . . . . . . . . . . . 166
6. Positive roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7. Closed sets of roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8. Highest root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
9. Weights, radical weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
10. Fundamental weights, dominant. weights . . . . . . . . . . . . . . . . . 180
11. Coxeter transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
12. Canonical bilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
§ 2. Affine Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
1. Affine Wey! group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
2. Weights and special weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
3. The normaliser of \Va . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4. Application: order of the Weyl group . . . . . . . . . . . . . . . . . . . . 190
5. Root systems and groups generated by reflections . . . . . . . . . 191