The representation theory of reductive algebraic groups and related finite reductive groups has many applications. The articles in this volume provide introductions to various aspects of the subject, including algebraic groups and Lie algebras, reflection groups, abelian and derived categories, the Deligne-Lusztig representation theory of finite reductive groups, Harish-Chandra theory and its generalizations, quantum groups, subgroup structure of algebraic groups, intersection cohomology, and Lusztig's conjectured character formula for irreducible representations in prime characteristic. The articles are carefully designed to reinforce one another, and are written by a team of distinguished authors: M. Brou?, R. W. Carter, S. Donkin, M. Geck, J. C. Jantzen, B. Keller, M. W. Liebeck, G. Malle, J. C. Rickard and R. Rouquier. This volume as a whole should provide a very accessible introduction to an important, though technical, subject.
Author(s): Roger W. Carter, Meinolf Geck
Series: Publications of the Newton Institute
Publisher: CUP
Year: 1998
Language: English
Pages: 200
Cover......Page 1
About......Page 2
Representations of Reductive Groups......Page 4
9780521643252......Page 5
Contents......Page 6
Preface......Page 8
1 Basic concepts......Page 10
2 Linear algebraic groups......Page 11
3 Maximal tori and Borel subgroups......Page 13
4 Roots and coroot......Page 16
5 Classification of simple algebraic groups......Page 19
6 Representations of simple algebraic groups......Page 22
7 The Lie algebra of a linear algebraic group......Page 25
8 Hopf algebra structures......Page 27
References......Page 29
2 Coxeter groups......Page 30
3 Real reflection groups......Page 33
4 Coxeter groups as reflection groups......Page 34
5 Finite Coxeter groups......Page 35
6 Root systems and Weyl groups......Page 36
7 Affine Weyl groups......Page 38
8 Braid groups and Iwahori-Hecke algebras......Page 40
9 Pseudo-reflection groups......Page 43
10 Topological construction of braid groups and Iwahori-Hecke algebras......Page 46
References......Page 47
1 Abelian categories......Page 50
2 Derived categories and derived functors......Page 55
3 Triangulated categories......Page 61
4 Morita theory for derived categories......Page 65
References......Page 67
1 Introduction......Page 72
2 Fields of definition......Page 73
3 Frobenius maps......Page 74
4 Applications of Lang's Theorem......Page 76
5 Finite reductive groups......Page 77
6 Characters of finite Weyl groups......Page 79
7 Twisted induction......Page 81
8 The dual group......Page 83
9 The Jordan decomposition of characters......Page 84
10 The multiplicity formula......Page 86
11 Computing character values......Page 89
References......Page 90
1 Introduction......Page 94
2 Generic finite reductive groups......Page 95
3 The polynomial order......Page 98
4 d-Sylow theorems......Page 100
5 Ordinary Harish-Chandra theory......Page 102
6 Generic unipotent characters......Page 103
7 d-Harish-Chandra theories......Page 105
8 Generic blocks......Page 107
9 Relative Weyl groups......Page 108
References......Page 111
Lecture 1: Quantum sl_2......Page 114
Lecture 2: The general case......Page 122
Lecture 3: Bases......Page 128
References......Page 134
1 Generalities......Page 138
2 Subgroups containing a maximal torus......Page 140
3 Unipotent elements......Page 144
4 Classical groups......Page 145
5 Exceptional groups......Page 150
References......Page 155
1 Introduction......Page 160
2 Simplicial homology......Page 161
3 Simplicial intersection homology......Page 166
4 Sheaf theory......Page 171
5 Sheaf-theoretic intersection cohomology......Page 176
6 Applications in representation theory......Page 179
References......Page 180
An Introduction to the Lusztig Conjecture - Stephen Donkin......Page 182
1 The general framework......Page 183
2 An Example......Page 184
3 The Chevalley Construction......Page 185
4 Weyl's Character Formula......Page 188
5 Some Fundamental Results......Page 189
7 Infinitesimal Theory......Page 192
References......Page 195
Index......Page 198
