Table Of ContentLecture Notes in Mathematics 2325
Dubravka Ban
p-adic
Banach Space
Representations
With Applications to Principal Series
Lecture Notes in Mathematics
Volume 2325
Editors-in-Chief
Jean-MichelMorel,CMLA,ENS,Cachan,France
BernardTeissier,IMJ-PRG,Paris,France
SeriesEditors
KarinBaur,UniversityofLeeds,Leeds,UK
MichelBrion,UGA,Grenoble,France
AnnetteHuber,AlbertLudwigUniversity,Freiburg,Germany
DavarKhoshnevisan,TheUniversityofUtah,SaltLakeCity,UT,USA
IoannisKontoyiannis,UniversityofCambridge,Cambridge,UK
AngelaKunoth,UniversityofCologne,Cologne,Germany
ArianeMézard,IMJ-PRG,Paris,France
MarkPodolskij,UniversityofLuxembourg,Esch-sur-Alzette,Luxembourg
MarkPolicott,MathematicsInstitute,UniversityofWarwick,Coventry,UK
SylviaSerfaty,NYUCourant,NewYork,NY,USA
László Székelyhidi , Institute of Mathematics, Leipzig University, Leipzig,
Germany
GabrieleVezzosi,UniFI,Florence,Italy
AnnaWienhard,RuprechtKarlUniversity,Heidelberg,Germany
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Dubravka Ban
p-adic Banach Space
Representations
With Applications to Principal Series
DubravkaBan
SchoolofMathematicalandStatistical
Sciences
SouthernIllinoisUniversity
Carbondale,IL,USA
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-031-22683-0 ISBN978-3-031-22684-7 (eBook)
https://doi.org/10.1007/978-3-031-22684-7
MathematicsSubjectClassification:22E50,20G25,11F70,11F85,46S10
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland
AG2022
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To thememoryof myparents,
Anka andNebomir
Preface
ThisbookgrewoutofacoursetaughtinSpring2021atSouthernIllinoisUniversity.
Its purposeis to lay the foundationsof the representationtheoryof p-adic groups
onp-adicBanachspaces,explainthe dualitytheoryofSchneiderandTeitelbaum,
anddemonstrateitsapplicationstocontinuousprincipalseries.Thismonographis
intendedtoservebothasareferencebookandasanintroductorytextforstudents
entering the area. In addition, it could be of interest to mathematicians who are
working in the representation theory on complex vector spaces and would like to
learnmoreaboutp-adicBanachspacerepresentations.
The participants in the course were Devjani Basu, Jeremiah Roberts, Layla
Sorkatti, Oneal Summers, An Tran, Manisha Varahagiri, and Menake Wijerathne.
Theypreparedandpresentedlecturesbased onthe first draftof thebook.I would
liketothankthemfortheirpatienceinnavigatingthroughahalf-finishedbookand
fortheircorrectionsandcomments.
Followingthe suggestionsof the threereferees, this monographincludesmany
improvements,broadeningthescopeofexposition.Iwouldliketothankthereferees
for their detailed reviews and invaluablecomments.Finally, I would like to thank
BrianConrad,MatthiasStrauch,andMarie-FranceVignérasfortheircontributions
tothefinalversionofthebook.
Carterville,IL,USA DubravkaBan
September2022
vii
Contents
1 Introduction .................................................................. 1
1.1 AdmissibleBanachSpaceRepresentations........................... 2
1.2 PrincipalSeriesRepresentations....................................... 3
1.3 SomeQuestionsandFurtherReading................................. 5
1.4 Prerequisites............................................................ 6
1.5 Notation................................................................. 7
1.6 Groups .................................................................. 7
PartI BanachSpaceRepresentationsofp-adicLieGroups
2 IwasawaAlgebras............................................................ 11
2.1 ProjectiveLimits ....................................................... 11
2.1.1 UniversalPropertyofProjectiveLimits...................... 13
2.1.2 ProjectiveLimitTopology .................................... 15
2.2 ProjectiveLimitsofTopologicalGroupsandoK-Modules .......... 19
2.2.1 ProfiniteGroups............................................... 21
2.3 IwasawaRings.......................................................... 24
2.3.1 Linear-TopologicaloK-Modules.............................. 25
2.3.2 AnotherProjectiveLimitRealizationofoK[[G0]] .......... 30
2.3.3 SomePropertiesofIwasawaAlgebras ....................... 32
3 Distributions.................................................................. 35
3.1 LocallyConvexVectorSpaces......................................... 35
3.1.1 BanachSpaces................................................. 37
3.1.2 ContinuousLinearOperators ................................. 37
3.1.3 ExamplesofBanachSpaces .................................. 40
3.1.4 DoubleDualsofaBanachSpace............................. 41
3.2 Distributions............................................................ 43
3.2.1 TheWeakTopologyonDc(G0,oK) ......................... 43
3.2.2 DistributionsandIwasawaRings............................. 46
3.2.3 TheCanonicalPairing......................................... 50
ix
x Contents
3.3 TheBounded-WeakTopology......................................... 50
3.3.1 TheBounded-WeakTopologyisStrictlyFinerthan
theWeakTopology............................................ 53
3.4 LocallyConvexTopologyonK[[G ]]................................ 55
0
3.4.1 TheCanonicalPairing ........................................ 56
3.4.2 p-adicHaarMeasure .......................................... 57
3.4.3 TheRingStructureonDc(G ,K) ........................... 59
0
4 BanachSpaceRepresentations............................................. 63
4.1 p-adicLieGroups...................................................... 63
4.2 LinearOperatorsonBanachSpaces................................... 64
4.2.1 SphericallyCompleteSpaces................................. 64
4.2.2 SomeFundamentalTheoremsinFunctionalAnalysis....... 65
4.2.3 BanachSpaceRepresentations:Definitionand
BasicProperties................................................ 68
4.3 Schneider-TeitelbaumDuality......................................... 73
4.3.1 Schikhof’sDuality............................................. 73
4.3.2 Duality for Banach Space Representations:
IwasawaModules.............................................. 78
4.4 AdmissibleBanachSpaceRepresentations........................... 81
4.4.1 Locally AnalyticVectors:Representationsin
Characteristicp................................................ 84
4.4.2 Dualityforp-adicLieGroups................................. 85
PartII PrincipalSeriesRepresentationsofReductiveGroups
5 ReductiveGroups............................................................ 91
5.1 LinearAlgebraicGroups............................................... 91
5.1.1 BasicPropertiesofLinearAlgebraicGroups................ 92
5.1.2 LieAlgebraofanAlgebraicGroup........................... 95
5.2 ReductiveGroupsOverAlgebraicallyClosedFields................. 96
5.2.1 RationalCharacters............................................ 97
5.2.2 RootsofaReductiveGroup................................... 98
5.2.3 ClassificationofIrreducibleRootSystems................... 103
5.2.4 ClassificationofReductiveGroups........................... 105
5.2.5 StructureofReductiveGroups................................ 107
5.3 F-ReductiveGroups ................................................... 109
5.4 Z-Groups................................................................ 111
5.4.1 AlgebraicR-Groups........................................... 112
5.4.2 SplitZ-Groups................................................. 113
5.5 TheStructureofG(L).................................................. 114
5.5.1 oL-PointsofAlgebraicZ-Groups............................. 114
5.5.2 oL-PointsofSplitZ-Groups .................................. 115
5.5.3 CosetRepresentativesforG/P ............................... 118
5.6 GeneralLinearGroups................................................. 119
Contents xi
6 AlgebraicandSmoothRepresentations................................... 123
6.1 AlgebraicRepresentations............................................. 123
6.1.1 DefinitionandBasicProperties............................... 124
6.1.2 ClassificationofSimpleModulesofReductiveGroups..... 124
6.2 SmoothRepresentations................................................ 132
6.2.1 AbsoluteValue................................................. 133
6.2.2 SmoothRepresentationsandCharacters ..................... 134
6.2.3 BasicProperties................................................ 135
6.2.4 Admissible-SmoothRepresentations......................... 137
6.2.5 SmoothPrincipalSeries....................................... 138
6.2.6 SmoothPrincipalSeriesofGL (L)andSL (L)............ 143
2 2
7 ContinuousPrincipalSeries ................................................ 147
7.1 ContinuousPrincipalSeriesAreBanach.............................. 148
7.1.1 DirectSumDecompositionofIndG0(χ−1) .................. 149
P0 0
7.1.2 UnitaryPrincipalSeries....................................... 153
7.1.3 AlgebraicandSmoothVectors................................ 154
7.1.4 UnitaryPrincipalSeriesofGL2(Qp) ........................ 155
7.2 DualsofPrincipalSeries............................................... 156
7.2.1 ModuleM(χ) .................................................. 156
0
7.3 ProjectiveLimitRealizationofM(χ).................................. 164
0
7.4 DirectSumDecompositionofM(χ)................................... 168
7.4.1 TheCaseG0 =GL2(Zp)..................................... 168
7.4.2 GeneralCase................................................... 169
8 IntertwiningOperators...................................................... 173
8.1 InvariantDistributions.................................................. 174
8.1.1 InvariantDistributionsonVectorGroups .................... 174
8.1.2 “PartiallyInvariant”DistributionsonUnipotentGroups.... 175
8.1.3 T -EquivariantDistributionsonUnipotentGroups.......... 177
0
8.2 IntertwiningAlgebra................................................... 181
8.2.1 OrdinaryRepresentationsofGL2(Qp)....................... 183
8.3 FiniteDimensionalG -InvariantSubspaces.......................... 184
0
8.3.1 InductionfromtheTrivialCharacter:Intertwiners .......... 185
8.4 ReducibilityofPrincipalSeries........................................ 186
8.4.1 LocallyAnalyticVectors...................................... 187
8.4.2 ACriterionforIrreducibility................................. 189
A NonarchimedeanFieldsandSpaces ....................................... 193
A.1 UltrametricSpaces ..................................................... 193
A.2 NonarchimedeanLocalFields......................................... 195
A.2.1 p-AdicNumbers............................................... 195
A.2.2 FiniteExtensionsofQp ....................................... 197
A.2.3 AlgebraicClosureQp ......................................... 199
A.3 NormedVectorSpaces................................................. 199
