Representation theory of finite reductive groups

454 Pages·2004·2.12 MB·english

by  Cabanes M.

RepresentationTheoryofFiniteReductiveGroups
Atthecrossroadsofrepresentationtheory,algebraicgeometryandfinitegrouptheory,
thisbookblendstogethermanyofthemainconcernsofmodernalgebra,synthesizing
thepast25yearsofresearch,withfullproofsofsomeofthemostremarkable
achievementsinthearea.
CabanesandEnguehardfollowthreemainthemes:first,applicationsofe´tale
cohomology,leading,vianotionsoftwistedinduction,unipotentcharactersand
Lusztig’sapproachtotheJordandecompositionofcharacters,totheproofofthe
recentBonnafe´–Rouquiertheorems.Thesecondisastraightforwardandsimplified
accountoftheDipper–Jamestheoremsrelatingirreduciblecharactersandmodular
representations,whileintroducingmodularHeckeandSchuralgebras.Thefinaltheme
islocalrepresentationtheory.Oneofthemainresultshereistheauthors’versionof
Fong–Srinivasantheoremsshowingtherelationsbetweentwistedinductionandblocks
ofmodularrepresentations.
Throughout,thetextisillustratedbymanyexamples;backgroundisprovidedby
severalintroductorychaptersonbasicresults,andappendicesonalgebraicgeometry
andderivedcategories.Theresultisanessentialintroductionforgraduatestudentsand
referenceforallalgebraists.
marc cabanes isCharge´deRechercheatUniversite´Paris7
michel enguehard isProfesseurEme´rite,Universite´Paris8.
NewMathematicalMonographs
EditorialBoard
Be´laBolloba´s,UniversityofMemphis
WilliamFulton,UniversityofMichigan
FrancesKirwan,MathematicalInstitute,UniversityofOxford
PeterSarnak,PrincetonUniversity
BarrySimon,CaliforniaInstituteofTechnology
For information about Cambridge University Press mathematics publications visit
http://publishing.cambridge.org/stm/mathematics/
Representation Theory of Finite
Reductive Groups
MARC CABANES, MICHEL ENGUEHARD
Universite´ Paris
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521825177
© Cambridge University Press 2004
This publication is in copyright. Subject to statutory exception and to the provision of
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without the written permission of Cambridge University Press.
First published in print format 2004
isbn-13 978-0-511-16566-5 eBook (NetLibrary)
isbn-10 0-511-16566-8
eBook (NetLibrary)
isbn-13 978-0-521-82517-7 hardback
isbn-10 0-521-82517-2 hardback
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Contents
Preface pagexi
Listofterminology xv
PARTI REPRESENTINGFINITEBN-PAIRS 1
1 Cuspidalityinfinitegroups 3
1.1 Subquotientsandassociatedrestrictions 4
1.2 Cuspidalityandinduction 6
1.3 Morphismsandaninvariancetheorem 8
1.4 Endomorphismalgebrasofinducedcuspidalmodules 11
1.5 Self-injectiveendomorphismringsandanequivalence
ofcategories 14
1.6 Structureofinducedcuspidalmodulesandseries 17
2 FiniteBN-pairs 22
2.1 Coxetergroupsandrootsystems 23
2.2 BN-pairs 27
2.3 Rootsubgroups 29
2.4 Levidecompositions 32
2.5 OtherpropertiesofsplitBN-pairs 35
3 ModularHeckealgebrasforfiniteBN-pairs 41
3.1 Heckealgebrasintransversalcharacteristics 42
3.2 Quotientrootsystemandapresentationof
theHeckealgebra 47
4 Themodulardualityfunctorandderivedcategory 55
4.1 Homology 56
4.2 Fixedpointcoefficientsystemandcuspidality 59
4.3 ThecaseoffiniteBN-pairs 63
v
vi Contents
4.4 Dualityfunctorasaderivedequivalence 67
4.5 AtheoremofCurtistype 69
5 Localmethodsforthetransversalcharacteristics 74
5.1 Localmethodsandtwomaintheorems
ofBrauer’s 75
5.2 Amodel:blocksofsymmetricgroups 78
5.3 Principalseriesandtheprincipalblock 82
5.4 Heckealgebrasanddecompositionmatrices 84
5.5 AproofofBrauer’sthirdMainTheorem 86
6 Simplemodulesinthenaturalcharacteristic 88
6.1 ModularHeckealgebraassociatedwitha
Sylowp-subgroup 88
6.2 Somemodulesincharacteristicp 93
6.3 Alperin’sweightconjectureincharacteristicp 95
6.4 Thep-blocks 97
PARTII DELIGNE–LUSZTIGVARIETIES,RATIONAL
SERIES,ANDMORITAEQUIVALENCES 101
7 FinitereductivegroupsandDeligne–Lusztigvarieties 103
7.1 ReductivegroupsandLang’stheorem 104
7.2 VarietiesdefinedbytheLangmap 105
7.3 Deligne–Lusztigvarieties 109
7.4 Deligne–Lusztigvarietiesarequasi-affine 114
8 Charactersoffinitereductivegroups 118
8.1 Reductivegroups,isogenies 119
8.2 Someexactsequencesandgroupsinduality 122
8.3 Twistedinduction 125
8.4 Lusztig’sseries 127
9 Blocksoffinitereductivegroupsandrationalseries 131
9.1 Blocksandcharacters 132
9.2 Blocksandrationalseries 133
9.3 Moritaequivalenceandordinarycharacters 136
10 JordandecompositionasaMoritaequivalence:
themainreductions 141
10.1 Theconditioni∗Rj∗F =0 142
10.2 Afirstreduction 144
10.3 Morenotation:smoothcompactifications 146
Contents vii
10.4 Ramificationandgeneration 149
10.5 Asecondreduction 150
11 JordandecompositionasaMoritaequivalence:sheaves 155
11.1 RamificationinDeligne–Lusztigvarieties 156
11.2 Corootlatticesassociatedwithintervals 162
11.3 Deligne–Lusztigvarietiesassociatedwithintervals 165
11.4 Application:somemappingcones 168
12 JordandecompositionasaMoritaequivalence:modules 173
12.1 Generatingperfectcomplexes 174
12.2 ThecaseofmodulesinducedbyDeligne–Lusztigvarieties 176
12.3 Varietiesofminimaldimensioninducinga
simplemodule 177
12.4 Disjunctionofseries 181
PARTIII UNIPOTENTCHARACTERSAND
UNIPOTENTBLOCKS 187
13 Levisubgroupsandpolynomialorders 189
13.1 Polynomialordersof F-stabletori 189
13.2 Goodprimes 193
13.3 Centralizersof(cid:2)-subgroupsandsomeLevi
subgroups 194
14 Unipotentcharactersasabasicset 199
14.1 Dualconjugacyclassesfor(cid:2)-elements 199
14.2 Basicsetsinthecaseofconnectedcenter 201
15 Jordandecompositionofcharacters 205
15.1 Fromnon-connectedcentertoconnectedcenterand
dualmorphism 206
15.2 Jordandecompositionofcharacters 209
16 OnconjugacyclassesintypeD 219
16.1 Notation;somepowerseries 220
16.2 Orthogonalgroups 221
16.3 Specialorthogonalgroupsandtheirderivedsubgroup;
Cliffordgroups 227
16.4 Spin (F) 235
2n
16.5 Non-semi-simplegroups,conformalgroups 239
16.6 Groupwithconnectedcenterandderivedgroup
Spin (F);conjugacyclasses 245
2n
viii Contents
16.7 GroupwithconnectedcenterandderivedgroupSpin (F);
2n
Jordandecompositionofcharacters 248
16.8 Lastcomputation,y ,y ,y 250
1 2 4
17 Standardisomorphismsforunipotentblocks 259
17.1 Thesetofunipotentblocks 260
17.2 (cid:2)-seriesandnon-connectedcenter 261
17.3 Aringisomorphism 264
PARTIV DECOMPOSITIONNUMBERSANDq-SCHUR
ALGEBRAS 269
18 SomeintegralHeckealgebras 271
18.1 Heckealgebrasandsignideals 272
18.2 HeckealgebrasoftypeA 275
18.3 HeckealgebrasoftypeBC,Hoefsmit’smatricesand
Jucys–Murphyelements 279
18.4 HeckealgebrasoftypeBC:somecomputations 281
18.5 HeckealgebrasoftypeBC:aMoritaequivalence 285
18.6 CyclicCliffordtheoryanddecompositionnumbers 288
19 Decompositionnumbersandq-Schuralgebras:general
lineargroups 297
19.1 Homfunctorsanddecompositionnumbers 298
19.2 CuspidalsimplemodulesandGelfand–Graevlattices 301
19.3 Simplemodulesanddecompositionmatricesforunipotent
blocks 305
19.4 ModularHarish-Chandraseries 309
20 Decompositionnumbersandq-Schuralgebras:
linearprimes 318
20.1 Finiteclassicalgroupsandlinearprimes 319
20.2 Heckealgebras 322
20.3 TypeBC 326
20.4 TypeD 328
PARTV UNIPOTENTBLOCKSANDTWISTEDINDUCTION 331
21 Localmethods;twistedinductionforblocks 333
21.1 “Connected”subpairsinfinitereductivegroups 333
21.2 Twistedinductionforblocks 334
21.3 Abadprime 341
Contents ix
22 UnipotentblocksandgeneralizedHarish-Chandratheory 345
22.1 Localsubgroupsinfinitereductivegroups,(cid:2)-elementsandtori 346
22.2 Thetheorem 350
22.3 Self-centralizingsubpairs 352
22.4 Thedefectgroups 354
23 Localstructureandringstructureofunipotentblocks 360
23.1 Non-unipotentcharactersinunipotentblocks 361
23.2 Controlsubgroups 363
23.3 (q–1)-blocksandabeliandefectconjecture 366
APPENDICES 373
Appendix1 Derivedcategoriesandderivedfunctors 374
A1.1 Abeliancategories 374
A1.2 Complexesandstandardconstructions 375
A1.3 Themappingcone 375
A1.4 Homology 376
A1.5 Thehomotopiccategory 376
A1.6 Derivedcategories 377
A1.7 Conesanddistinguishedtriangles 378
A1.8 Derivedfunctors 379
A1.9 Compositionofderivedfunctors 379
A1.10 Exactsequencesoffunctors 380
A1.11 Bi-functors 380
A1.12 Modulecategories 381
A1.13 Sheavesontopologicalspaces 382
A1.14 Locallyconstantsheavesandthefundamentalgroup 384
A1.15 Derivedoperationsonsheaves 385
Appendix2 Varietiesandschemes 389
A2.1 AffineF-varieties 389
A2.2 LocallyringedspacesandF-varieties 390
A2.3 Tangentsheaf,smoothness 392
A2.4 Linearalgebraicgroupsandreductivegroups 393
A2.5 Rationalstructuresonaffinevarieties 395
A2.6 Morphismsandquotients 395
A2.7 Schemes 397
A2.8 Coherentsheaves 399
A2.9 Vectorbundles 400
A2.10 Acriterionofquasi-affinity 401