Representation theory of finite reductive groups:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2004
|
| Edition: | 1. publ. |
| Series: | New mathematical monographs
1 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 436 S. |
| ISBN: | 0521825172 |
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MARC
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| 245 | 1 | 0 | |a Representation theory of finite reductive groups |c Marc Cabanes ; Michel Enguehard |
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| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2004 | |
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Record in the Search Index
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| adam_text | Contents
Preface page xi
List of terminology xv
PART I REPRESENTING FINITE BN PAIRS 1
1 Cuspidality in finite groups 3
1.1 Subquotients and associated restrictions 4
1.2 Cuspidality and induction 6
1.3 Morphisms and an invariance theorem 8
1.4 Endomorphism algebras of induced cuspidal modules 11
1.5 Self injective endomorphism rings and an equivalence
of categories 14
1.6 Structure of induced cuspidal modules and series 17
2 Finite BN pairs 22
2.1 Coxeter groups and root systems 23
2.2 BN pairs 27
2.3 Root subgroups 29
2.4 Levi decompositions 32
2.5 Other properties of split BN pairs 35
3 Modular Hecke algebras for finite BN pairs 41
3.1 Hecke algebras in transversal characteristics 42
3.2 Quotient root system and a presentation of
the Hecke algebra 47
4 The modular duality functor and derived category 55
4.1 Homology 56
4.2 Fixed point coefficient system and cuspidality 59
4.3 The case of finite BN pairs 63
v
vi Contents
4.4 Duality functor as a derived equivalence 67
4.5 A theorem of Curtis type 69
5 Local methods for the transversal characteristics 74
5.1 Local methods and two main theorems
of Brauer s 75
5.2 A model: blocks of symmetric groups 78
5.3 Principal series and the principal block 82
5.4 Hecke algebras and decomposition matrices 84
5.5 A proof of Brauer s third Main Theorem 86
6 Simple modules in the natural characteristic 88
6.1 Modular Hecke algebra associated with a
Sylow p subgroup 88
6.2 Some modules in characteristic p 93
6.3 Alperin s weight conjecture in characteristic p 95
6.4 The/7 blocks 97
PART II DELIGNE LUSZTIG VARIETIES, RATIONAL
SERIES, AND MORITA EQUIVALENCES 101
7 Finite reductive groups and Deligne Lusztig varieties 103
7.1 Reductive groups and Lang s theorem 104
7.2 Varieties defined by the Lang map 105
7.3 Deligne Lusztig varieties 109
7.4 Deligne Lusztig varieties are quasi affine 114
8 Characters of finite reductive groups 118
8.1 Reductive groups, isogenies 119
8.2 Some exact sequences and groups in duality 122
8.3 Twisted induction 125
8.4 Lusztig s series 127
9 Blocks of finite reductive groups and rational series 131
9.1 Blocks and characters 132
9.2 Blocks and rational series 133
9.3 Morita equivalence and ordinary characters 136
10 Jordan decomposition as a Morita equivalence:
the main reductions 141
10.1 The condition i*Rj*f = 0 142
10.2 A first reduction 144
10.3 More notation: smooth compactifications 146
Contents vii
10.4 Ramification and generation 149
10.5 A second reduction 150
11 Jordan decomposition as a Morita equivalence: sheaves 155
11.1 Ramification in Deligne Lusztig varieties 156
11.2 Coroot lattices associated with intervals 162
11.3 Deligne Lusztig varieties associated with intervals 165
11.4 Application: some mapping cones 168
12 Jordan decomposition as a Morita equivalence: modules 173
12.1 Generating perfect complexes 174
12.2 The case of modules induced by Deligne Lusztig varieties 176
12.3 Varieties of minimal dimension inducing a
simple module 177
12.4 Disjunction of series 181
PART III UNIPOTENT CHARACTERS AND
UNIPOTENT BLOCKS 187
13 Levi subgroups and polynomial orders 189
13.1 Polynomial orders of / stable tori 189
13.2 Good primes 193
13.3 Centralizers of ^ subgroups and some Levi
subgroups 194
14 Unipotent characters as a basic set 199
14.1 Dual conjugacy classes for ^ elements 199
14.2 Basic sets in the case of connected center 201
15 Jordan decomposition of characters 205
15.1 From non connected center to connected center and
dual morphism 206
15.2 Jordan decomposition of characters 209
16 On conjugacy classes in type D 219
16.1 Notation; some power series 220
16.2 Orthogonal groups 221
16.3 Special orthogonal groups and their derived subgroup;
Clifford groups 227
16.4 Spin2n(F) 235
16.5 Non semi simple groups, conformal groups 239
16.6 Group with connected center and derived group
Spin2n (F); conjugacy classes 245
viii Contents
16.7 Group with connected center and derived group Spin2n (F);
Jordan decomposition of characters 248
16.8 Last computation, yi,y 2,y4 250
17 Standard isomorphisms for unipotent blocks 259
17.1 The set of unipotent blocks 260
17.2 ^ series and non connected center 261
17.3 A ring isomorphism 264
PART IV DECOMPOSITION NUMBERS AND g SCHUR
ALGEBRAS 269
18 Some integral Hecke algebras 271
18.1 Hecke algebras and sign ideals 272
18.2 Hecke algebras of type A 275
18.3 Hecke algebras of type BC, Hoefsmit s matrices and
Jucys Murphy elements 279
18.4 Hecke algebras of type BC: some computations 281
18.5 Hecke algebras of type BC: a Morita equivalence 285
18.6 Cyclic Clifford theory and decomposition numbers 288
19 Decomposition numbers and g Schur algebras: general
linear groups 297
19.1 Horn functors and decomposition numbers 298
19.2 Cuspidal simple modules and Gelfand Graev lattices 301
19.3 Simple modules and decomposition matrices for unipotent
blocks 305
19.4 Modular Harish Chandra series 309
20 Decomposition numbers and y Schur algebras:
linear primes 318
20.1 Finite classical groups and linear primes 319
20.2 Hecke algebras 322
20.3 TypeBC 326
20.4 TypeD 328
PART V UNIPOTENT BLOCKS AND TWISTED INDUCTION 331
21 Local methods; twisted induction for blocks 333
21.1 Connected subpairs in finite reductive groups 333
21.2 Twisted induction for blocks 334
21.3 A bad prime 341
Contents ix
22 Unipotent blocks and generalized Harish Chandra theory 345
22.1 Local subgroups in finite reductive groups, ^ elements and tori 346
22.2 The theorem 350
22.3 Self centralizing subpairs 352
22.4 The defect groups 354
23 Local structure and ring structure of unipotent blocks 360
23.1 Non unipotent characters in unipotent blocks 361
23.2 Control subgroups 363
23.3 (q l) blocks and abelian defect conjecture 366
APPENDICES 373
Appendix 1 Derived categories and derived functors 374
A 1.1 Abelian categories 374
A 1.2 Complexes and standard constructions 375
A 1.3 The mapping cone 375
A 1.4 Homology 376
A 1.5 The homotopic category 376
A1.6 Derived categories 377
A1.7 Cones and distinguished triangles 378
A 1.8 Derived functors 379
A1.9 Composition of derived functors 379
A1.10 Exact sequences of functors 380
A 1.11 Bi functors 380
A1.12 Module categories 381
A 1.13 Sheaves on topological spaces 382
A 1.14 Locally constant sheaves and the fundamental group 384
A 1.15 Derived operations on sheaves 385
Appendix 2 Varieties and schemes 389
A2.1 Affine F varieties 389
A2.2 Locally ringed spaces and F varieties 390
A2.3 Tangent sheaf, smoothness 392
A2.4 Linear algebraic groups and reductive groups 393
A2.5 Rational structures on affine varieties 395
A2.6 Morphisms and quotients 395
A2.7 Schemes 397
A2.8 Coherent sheaves 399
A2.9 Vector bundles 400
A2.10 A criterion of quasi affinity 401
x Contents
Appendix 3 Etale cohomology 404
A3.1 The etale topology 404
A3.2 Sheaves for the etale topology 405
A3.3 Basic operations on sheaves 406
A3.4 Homology and derived functors 407
A3.5 Base change for a proper morphism 408
A3.6 Homology and direct images with compact support 408
A3.7 Finiteness of cohomology 409
A3.8 Coefficients 409
A3.9 The open closed situation 410
A3.10 Higher direct images and stalks 411
A3.11 Projection and Kiinneth formulae 411
A3.12 Poincare Verdier duality and twisted inverse images 412
A3.13 Purity 413
A3.14 Finite group actions and constant sheaves 414
A3.15 Finite group actions and projectivity 414
A3.16 Locally constant sheaves and the fundamental group 415
A3.17 Tame ramification along a divisor with
normal crossings 417
A3.18 Tame ramification and direct images 418
References 422
Index 431
|
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| id | DE-604.BV019626156 |
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| indexdate | 2024-07-09T20:01:38Z |
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| language | English |
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| spelling | Cabanes, Marc 1959- Verfasser (DE-588)114937419 aut Representation theory of finite reductive groups Marc Cabanes ; Michel Enguehard 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2004 XVII, 436 S. txt rdacontent n rdamedia nc rdacarrier New mathematical monographs 1 Groupes finis Reductieve groepen gtt Representatie (wiskunde) gtt Représentations de groupes Finite groups Representations of groups Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Endliche Gruppe (DE-588)4014651-0 gnd rswk-swf Endliche Gruppe (DE-588)4014651-0 s Darstellungstheorie (DE-588)4148816-7 s DE-604 Enguehard, Michel Verfasser (DE-588)141639679 aut New mathematical monographs 1 (DE-604)BV035420183 1 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012955410&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cabanes, Marc 1959- Enguehard, Michel Representation theory of finite reductive groups New mathematical monographs Groupes finis Reductieve groepen gtt Representatie (wiskunde) gtt Représentations de groupes Finite groups Representations of groups Darstellungstheorie (DE-588)4148816-7 gnd Endliche Gruppe (DE-588)4014651-0 gnd |
| subject_GND | (DE-588)4148816-7 (DE-588)4014651-0 |
| title | Representation theory of finite reductive groups |
| title_auth | Representation theory of finite reductive groups |
| title_exact_search | Representation theory of finite reductive groups |
| title_full | Representation theory of finite reductive groups Marc Cabanes ; Michel Enguehard |
| title_fullStr | Representation theory of finite reductive groups Marc Cabanes ; Michel Enguehard |
| title_full_unstemmed | Representation theory of finite reductive groups Marc Cabanes ; Michel Enguehard |
| title_short | Representation theory of finite reductive groups |
| title_sort | representation theory of finite reductive groups |
| topic | Groupes finis Reductieve groepen gtt Representatie (wiskunde) gtt Représentations de groupes Finite groups Representations of groups Darstellungstheorie (DE-588)4148816-7 gnd Endliche Gruppe (DE-588)4014651-0 gnd |
| topic_facet | Groupes finis Reductieve groepen Representatie (wiskunde) Représentations de groupes Finite groups Representations of groups Darstellungstheorie Endliche Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012955410&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035420183 |
| work_keys_str_mv | AT cabanesmarc representationtheoryoffinitereductivegroups AT enguehardmichel representationtheoryoffinitereductivegroups |