Representations of Finite Groups of Lie Type (London Mathematical Society Student Texts)

267 Pages·2020·1.317 MB·English

by  François Digne, Jean Michel

LONDONMATHEMATICALSOCIETYSTUDENTTEXTS
ManagingEditor:IanJ.Leary,
MathematicalSciences,UniversityofSouthampton,UK
57 IntroductiontoBanachalgebras,operatorsandharmonicanalysis,GARTHDALESetal
58 Computationalalgebraicgeometry,HALSCHENCK
59 Frobeniusalgebrasand2-Dtopologicalquantumfieldtheories,JOACHIMKOCK
60 Linearoperatorsandlinearsystems,JONATHANR.PARTINGTON
61 AnintroductiontononcommutativeNoetherianrings(2ndEdition),K.R.GOODEARL&
R.B.WARFIELD,JR
62 Topicsfromone-dimensionaldynamics,KARENM.BRUCKS&HENKBRUIN
63 Singularpointsofplanecurves,C.T.C.WALL
64 AshortcourseonBanachspacetheory,N.L.CAROTHERS
65 ElementsoftherepresentationtheoryofassociativealgebrasI,IBRAHIMASSEM,
DANIELSIMSON&ANDRZEJSKOWRON´SKI
66 Anintroductiontosievemethodsandtheirapplications,ALINACARMENCOJOCARU&
M.RAMMURTY
67 Ellipticfunctions,J.V.ARMITAGE&W.F.EBERLEIN
68 Hyperbolicgeometryfromalocalviewpoint,LINDAKEEN&NIKOLALAKIC
69 LecturesonKählergeometry,ANDREIMOROIANU
70 Dependencelogic,JOUKUVÄÄNÄNEN
71 ElementsoftherepresentationtheoryofassociativealgebrasII,DANIELSIMSON&
ANDRZEJSKOWRON´SKI
72 ElementsoftherepresentationtheoryofassociativealgebrasIII,DANIELSIMSON&
ANDRZEJSKOWRON´SKI
73 Groups,graphsandtrees,JOHNMEIER
74 RepresentationtheoremsinHardyspaces,JAVADMASHREGHI
75 Anintroductiontothetheoryofgraphspectra,DRAGOŠCVETKOVIC´,
PETERROWLINSON&SLOBODANSIMIC´
76 NumbertheoryinthespiritofLiouville,KENNETHS.WILLIAMS
77 Lecturesonprofinitetopicsingrouptheory,BENJAMINKLOPSCH,NIKOLAYNIKOLOV&
CHRISTOPHERVOLL
78 Cliffordalgebras:Anintroduction,D.J.H.GARLING
79 IntroductiontocompactRiemannsurfacesanddessinsd’enfants,ERNESTOGIRONDO&
GABINOGONZÁLEZ-DIEZ
80 TheRiemannhypothesisforfunctionfields,MACHIELVANFRANKENHUIJSEN
81 Numbertheory,Fourieranalysisandgeometricdiscrepancy,GIANCARLOTRAVAGLINI
82 Finitegeometryandcombinatorialapplications,SIMEONBALL
83 Thegeometryofcelestialmechanics,HANSJÖRGGEIGES
84 Randomgraphs,geometryandasymptoticstructure,MICHAELKRIVELEVICHetal
85 Fourieranalysis:PartI-Theory,ADRIANCONSTANTIN
86 Dispersivepartialdifferentialequations,M.BURAKERDOGˇAN&NIKOLAOSTZIRAKIS
87 Riemannsurfacesandalgebraiccurves,R.CAVALIERI&E.MILES
88 Groups,languagesandautomata,DEREKF.HOLT,SARAHREES&CLAASE.RÖVER
89 AnalysisonPolishspacesandanintroductiontooptimaltransportation,D.J.H.GARLING
90 Thehomotopytheoryof(∞,1)-categories,JULIAE.BERGNER
91 TheblocktheoryoffinitegroupalgebrasI,M.LINCKELMANN
92 TheblocktheoryoffinitegroupalgebrasII,M.LINCKELMANN
93 Semigroupsoflinearoperators,D.APPLEBAUM
94 Introductiontoapproximategroups,M.C.H.TOINTON
95 RepresentationsoffinitegroupsofLietype(2ndEdition),F.DIGNE&J.MICHEL
96 TensorproductsofC*-algebrasandoperatorspaces,G.PISIER

LondonMathematicalSocietyStudentTexts95
Representations of Finite Groups
of Lie Type
secondedition
FRANÇOIS DIGNE
UniversitédePicardieJulesVerne,Amiens
JEAN MICHEL
CentreNationaldelaRechercheScientifique(CNRS),Paris
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108481489
DOI:10.1017/9781108673655
Firstedition©CambridgeUniversityPress1991
Secondedition©FrançoisDigneandJeanMichel2020
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished1991
Secondedition2020
PrintedintheUnitedKingdombyTJInternationalLtd.PadstowCornwall
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Digne,François,author.|Michel,Jean,author.
Title:RepresentationsoffinitegroupsofLietype/FrançoisDigne,
UniversitédePicardieJulesVerne,Amiens,JeanMichel,Centre
NationaldelaRechercheScientifique(CNRS),Paris.
Description:Secondedition.|Cambridge,UnitedKingdom;NewYork,NY:
CambridgeUniversityPress,2020.|Series:Londonmathematicalsociety
studenttexts;95|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2019035714|ISBN9781108481489(hardback)|ISBN
9781108673655(ebook)
Subjects:LCSH:Liegroups.|Representationsofgroups.
Classification:LCCQA387.D542020|DDC512/.482--dc23
LCrecordavailableathttps://lccn.loc.gov/2019035714
ISBN978-1-108-48148-9Hardback
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Contents
IntroductiontotheSecondEdition page1
FromtheIntroductiontotheFirstEdition 2
1 BasicResultsonAlgebraicGroups 5
1.1 BasicResultsonAlgebraicGroups 5
1.2 DiagonalisableGroups,Tori,X(T),Y(T) 8
1.3 SolvableGroups,BorelSubgroups 11
1.4 UnipotentGroups,Radical,ReductiveandSemi-Simple
Groups 13
1.5 ExamplesofReductiveGroups 15
2 StructureTheoremsforReductiveGroups 19
2.1 CoxeterGroups 19
2.2 FiniteRootSystems 24
2.3 StructureofReductiveGroups 29
2.4 RootData,Isogenies,PresentationofG 35
3 (B,N)-Pairs;Parabolic,Levi,andReductiveSubgroups;
CentralisersofSemi-SimpleElements 39
3.1 (B,N)-Pairs 39
3.2 ParabolicSubgroupsofCoxeterGroups
andof(B,N)-Pairs 42
3.3 ClosedSubsetsofaCrystallographicRootSystem 45
3.4 ParabolicSubgroupsandLeviSubgroups 51
3.5 CentralisersofSemi-SimpleElements 56
v
vi Contents
4 Rationality,theFrobeniusEndomorphism,
theLang–SteinbergTheorem 59
4.1 k -Varieties,FrobeniusEndomorphisms 59
0
4.2 TheLang–SteinbergTheorem;GaloisCohomology 63
4.3 ClassificationofFiniteGroupsofLieType 70
4.4 TheRelative(B,N)-Pair 75
5 Harish-ChandraTheory 79
5.1 Harish-ChandraInductionandRestriction 79
5.2 TheMackeyFormula 83
5.3 Harish-ChandraTheory 86
6 Iwahori–HeckeAlgebras 91
6.1 EndomorphismAlgebras 91
6.2 Iwahori–HeckeAlgebras 97
6.3 SchurElementsandGenericDegrees 104
6.4 TheExampleofG 108
2
7 TheDualityFunctorandtheSteinbergCharacter 113
7.1 F-rank 113
7.2 TheDualityFunctor 116
7.3 RestrictiontoCentralisersofSemi-SimpleElements 123
7.4 TheSteinbergCharacter 126
8 (cid:2)-AdicCohomology 130
8.1 (cid:2)-AdicCohomology 130
9 Deligne–LusztigInduction:TheMackeyFormula 137
9.1 Deligne–LusztigInduction 137
9.2 MackeyFormulaforLusztigFunctors 140
9.3 Consequences:ScalarProducts 146
10 TheCharacterFormulaandOtherResults
onDeligne–LusztigInduction 148
10.1 TheCharacterFormula 148
10.2 UniformFunctions 153
10.3 TheCharacteristicFunctionofaSemi-SimpleClass 157
11 GeometricConjugacyandtheLusztigSeries 161
11.1 GeometricConjugacy 161
11.2 MoreonCentralisersofSemi-SimpleElements 167
11.3 TheLusztigSeries 170
Contents vii
11.4 Lusztig’sJordanDecompositionofCharacters:
TheLeviCase 175
11.5 Lusztig’sJordanDecompositionofCharacters:
TheGeneralCase 183
11.6 MoreaboutUnipotentCharacters 188
11.7 TheIrreducibleCharactersofGLF andUF 191
n n
12 RegularElements;Gelfand–GraevRepresentations;
RegularandSemi-SimpleCharacters 196
12.1 RegularElements 196
12.2 RegularUnipotentElements 201
12.3 Gelfand–GraevRepresentations 207
12.4 RegularandSemi-SimpleCharacters 214
12.5 TheCharacterTableofSL (F ) 220
2 q
13 GreenFunctions 225
13.1 Invariants 225
13.2 GreenFunctionsandtheSpringerCorrespondence 231
13.3 TheLusztig–ShojiAlgorithm 236
14 TheDecompositionofDeligne–LusztigCharacters 242
14.1 LusztigFamiliesandSpecialUnipotentClasses 242
14.2 SplitGroups 244
14.3 TwistedGroups 247
References 249
Index 255

Introduction to the Second Edition
Wehadtwomainaimsinwritingthisedition:
• Be more self-contained where possible. For instance, we have added brief
overviewsofCoxetergroupsandrootsystems,andgivensomemoredetails
aboutthetheoryofalgebraicgroups.
• Whileretainingthesamelevelofexpositionasinthefirstedition,wehave
givenamorecompleteaccountoftherepresentationtheoryoffinitegroups
ofLietype.
Inviewofthesecondaim,wehaveaddedthefollowingtopicstoourexpo-
sition:
• WecoverReeandSuzukigroupsextendingourexpositionofFrobeniusmor-
phismstothemoregeneralcaseofFrobeniusroots.
• We have added to Harish-Chandra theory the topic of Hecke algebras and
givenasmanyresultsaswecouldeasilydoforfieldsofarbitrarycharacter-
isticprimetoq,inviewofapplicationstomodularrepresentations.
• We have added a chapter on the computation of Green functions, with a
brief review of invariant theory of reflection groups, and a chapter on the
decompositionofunipotentDeligne–Lusztigcharacters.
Acknowledgements
In addition to the people we thank in the introduction to the first edition, we
havebenefittedfromtheinputofyoungercolleaguessuchasCédricBonnafé,
Olivier Dudas, Meinolf Geck, Daniel Juteau, and Raphaël Rouquier, and we
givespecialthankstoGunterMalleforathoroughproofreadingofthisedition.
We also thank the many people who pointed out to us misprints and other
errorsinthefirstedition.
1