Bruhat–Tits Theory: A New Approach

749 Pages·2023·8.578 MB·English

by  Tasho Kaletha, Gopal Prasad

Bruhat–TitsTheory
Bruhat–Tits theory is an important topic in number theory, representation theory,
harmonic analysis, and algebraic geometry. This book gives the first comprehensive
treatment of this theory over discretely valued Henselian fields. It can serve both as
a reference for researchers in the field and as a thorough introduction for graduate
studentsandearlycareermathematicians.
PartIofthebookgivesareviewoftherelevantbackgroundmaterial,touchingupon
Lie theory, metric geometry, algebraic groups, and integral models. Part II gives a
complete,detailed,andmotivatedtreatmentofthecoretheory.Formoreexperienced
readers looking to learn the essentials for use in their own work, there is also an
axiomaticsummaryofBruhat–Titstheorythatsufficesforthemainapplications.PartIII
treatsmoderntopicsthathavebecomeimportantincurrentresearch.PartIVprovides
afewsampleapplicationsofthetheory.Theappendicescontainfurtherdetailsonthe
topicofintegralmodels,includingadetailedstudyoftheintegralmodelsoftori.
Tasho Kaletha isProfessorofMathematicsattheUniversityofMichigan.He
isanexpertontheLanglandsprogram,andhasstudiedarithmeticandrepresentation-
theoreticaspectsofthelocalLanglandscorrespondenceforp-adicgroups.
Gopal Prasad isRaoulBottProfessorEmeritusofMathematicsattheUniversity
ofMichigan.Heisaleadingexpertonrealandp-adicLiegroupsandalgebraicgroups.
TogetherwithOferGabberandBrianConrad,hepublishedthecompleteclassification
andstructuretheoryofpseudo-reductivegroupsinthebooksPseudo-reductiveGroups
(2010,2015)andClassificationofPseudo-reductiveGroups(2015).
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6. S.Berhanu,P.D.CordaroandJ.HounieAnIntroductiontoInvolutiveStructures
7. A.ShlapentokhHilbert’sTenthProblem
8. G.MichlerTheoryofFiniteSimpleGroupsI
9. A.BakerandG.Wu¨stholzLogarithmicFormsandDiophantineGeometry
10. P.KronheimerandT.MrowkaMonopolesandThree-Manifolds
11. B.Bekka,P.delaHarpeandA.ValetteKazhdan’sProperty(T)
12. J.NeisendorferAlgebraicMethodsinUnstableHomotopyTheory
13. M.GrandisDirectedAlgebraicTopology
14. G.MichlerTheoryofFiniteSimpleGroupsII
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16. S.BlochLecturesonAlgebraicCycles(2ndEdition)
17. B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups
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22. J.Goubault-LarrecqNon-HausdorffTopologyandDomainTheory
23. J.S´niatyckiDifferentialGeometryofSingularSpacesandReductionofSymmetry
24. E.RiehlCategoricalHomotopyTheory
25. B.A.MunsonandI.Volic´CubicalHomotopyTheory
26. B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups(2ndEdition)
27. J.Heinonen,P.Koskela,N.ShanmugalingamandJ.T.TysonSobolevSpacesonMetric
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30. A.BobrowskiConvergenceofOne-ParameterOperatorSemigroups
31. K.CostelloandO.GwilliamFactorizationAlgebrasinQuantumFieldTheoryI
32. J.-H.EvertseandK.Gyo˜ryDiscriminantEquationsinDiophantineNumberTheory
33. G.FriedmanSingularIntersectionHomology
34. S.SchwedeGlobalHomotopyTheory
35. M.Dickmann,N.SchwartzandM.TresslSpectralSpaces
36. A.BaernsteinIISymmetrizationinAnalysis
37. A.Defant,D.Garc´ıa,M.MaestreandP.Sevilla-PerisDirichletSeriesandHolomorphic
FunctionsinHighDimensions
38. N.Th.VaropoulosPotentialTheoryandGeometryonLieGroups
39. D.ArnalandB.CurreyRepresentationsofSolvableLieGroups
40. M.A.Hill,M.J.HopkinsandD.C.RavenelEquivariantStableHomotopyTheoryandthe
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41. K.CostelloandO.GwilliamFactorizationAlgebrasinQuantumFieldTheoryII
42. S.KumarConformalBlocks,GeneralizedThetaFunctionsandtheVerlindeFormula
43. P.F.X.Mu¨llerHardyMartingales
44. T.KalethaandG.PrasadBruhat–TitsTheory
45. J.SchwermerReductionTheoryandArithmeticGroups
Bruhat–Tits Theory
A New Approach
TASHO KALETHA
UniversityofMichigan,AnnArbor
GOPAL PRASAD
UniversityofMichigan,AnnArbor
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Informationonthistitle:www.cambridge.org/9781108831963
DOI:10.1017/9781108933049
©TashoKalethaandGopalPrasad2023
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Contents
Listofillustrations pagexii
Listoftables xiii
Introduction xv
PARTONE BACKGROUNDANDREVIEW 1
1 AffineRootSystemsandAbstractBuildings 3
1.1 MetricSpaces 3
1.2 AffineSpaces 8
1.3 AffineRootSystems 14
1.4 TitsSystems 47
1.5 AbstractBuildings 54
(cid:2)
1.6 TheMonoidR 66
2 AlgebraicGroups 68
2.1 HenselianFields 68
2.2 BoundedSubgroupsofReductiveGroups 70
2.3 FieldsofDimension(cid:2)178
2.4 AffineGroupSchemesoverPerfectFields 81
2.5 Tori 86
2.6 ReductiveGroups 92
2.7 TheGroupSU 108
3
2.8 SeparableQuadraticExtensionsofLocalFields 110
2.9 ChevalleySystems 112
2.10 IntegralModels 121
2.11 GroupSchemeActionsandtheDynamicMethod 130
vii
viii Contents
PARTTWO BRUHAT–TITSTHEORY 139
3 Examples:Quasi-splitSimpleGroupsofRank1 141
3.1 TheExampleofSL 141
2
3.2 TheExampleofSU 147
3
4 OverviewandSummaryofBruhat–TitsTheory 157
4.1 AxiomatizationofBruhat–TitsTheory 157
4.2 Metric 168
4.3 TheEnlargedBuilding 178
4.4 UniquenessoftheApartmentandtheBuilding 180
5 Bruhat,Cartan,andIwasawaDecompositions 186
5.1 The(affine)BruhatDecomposition 186
5.2 TheCartanDecomposition 187
5.3 TheIwasawaDecomposition 188
5.4 TheIntersectionofCartanandIwasawaDoubleCosets 191
6 TheApartment 195
6.1 TheApartmentofaQuasi-splitReductiveGroup 196
6.2 AffineReflectionsandUniquenessofValuations 212
6.3 AffineRootsandAffineRootGroups 217
6.4 TheAffineRootSystemofaQuasi-splitGroup 223
6.5 ChangeofValuation 229
6.6 TheAffineWeylGroup 230
6.7 ProjectiontoaLeviSubgroup 233
7 TheBruhat–TitsBuildingforaValuationoftheRootDatum 235
7.1 CommutatorComputations 236
7.2 AFiltrationofZ(k) 245
7.3 ConcaveFunctions 246
7.4 ParahoricSubgroups 255
7.5 TheIwahori–TitsSystem 259
7.6 The(Reduced)Building 263
7.7 DisconnectedParahoricSubgroups 267
7.8 TheIwahori–WeylGroup 272
7.9 ChangeofBaseFieldandAutomorphisms 273
7.10 PassagetoCompletion 276
7.11 AbsolutelySpecialPoints 278
8 IntegralModels 283
8.1 Preliminaries 283
8.2 GeneralPropertiesofSmoothModelsofG 286
8.3 ParahoricIntegralModels 295
Contents ix
8.4 TheStructureoftheSpecialFiberofG0 302
Ω
8.5 IntegralModelsAssociatedtoConcaveFunctions 310
8.6 PassagetoCompletion 321
9 UnramifiedDescent 323
9.1 Preliminaries 324
9.2 StatementoftheMainResult 325
9.3 TheBuildinganditsApartments 330
9.4 TheAffineRootSystem 345
9.5 CompletionoftheProofoftheMainResult 363
9.6 ValuationofRootDatum 364
9.7 LeviSubgroups 368
9.8 ConcaveFunctionGroups 374
9.9 Special,Superspecial,andHyperspecialPoints 379
9.10 ResiduallySplitandResiduallyQuasi-splitGroups 381
9.11 RestrictionofScalars 385
PARTTHREE ADDITIONALDEVELOPMENTS 387
10 ResidueFieldfofDimension(cid:2)1 389
10.1 ConjugacyofSpecialTori 389
10.2 SuperspecialPoints 391
10.3 AnisotropicGroups 391
10.4 FixedPointsofLargeSubgroupsofTori 393
10.5 ExistenceofAnisotropicTori 395
10.6 CohomologicalResults 397
10.7 ClassificationofConnectedReductivek-Groups 401
11 ComponentGroupsofIntegralModels 420
11.1 TheKottwitzHomomorphismforTori 421
11.2 TheComponentGroupsofTftandTlft 426
11.3 TheAlgebraicFundamentalGroup 427
11.4 z-Extensions 428
11.5 TheKottwitzHomomorphismforReductiveGroups 429
11.6 TheComponentGroupsofParahoricIntegralModels 432
11.7 TheCaseofdim(f)(cid:2)1 433
12 FiniteGroupActionsandTamelyRamifiedDescent 436
12.1 Preliminaries 437
12.2 CertainGroupSchemesAssociatedtoH andG 439
12.3 AReduction 443
12.4 ApartmentsofB 445