Modern analysis of automorphic forms by example, vol.1,2

775 Pages·2018·5.346 MB·English

by  Garrett, Paul B

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 173
EditorialBoard
B. BOLLOBÁS, W. FULTON, F. KIRWAN,
P. SARNAK, B. SIMON, B. TOTARO
MODERNANALYSISOF
AUTOMORPHIC FORMSBYEXAMPLE,
VOLUME 1
ThisisVolume1ofatwo-volumebookthatprovidesaself-containedintroductiontothe
theoryandapplicationofautomorphicforms,usingexamplestoillustrateseveralcritical
analyticalconceptssurroundingandsupportingthetheoryofautomorphicforms.The
two-volumebooktreatsthreeinstances,startingwithsomesmallunimodularexamples,
followedbyadelicGL2,andfinallyGLn.Volume1featurescriticalresults,whichare
provencarefullyandindetail,includingdiscretedecompositionofcuspforms,mero-
morphiccontinuationofEisensteinseries,spectraldecompositionofpseudo-Eisenstein
series,andautomorphicPlanchereltheorem.Volume2featuresautomorphicGreen’s
functions, metrics and topologies on natural function spaces, unbounded operators,
vector-valued integrals, vector-valued holomorphic functions, and asymptotics. With
numerous proofs and extensive examples, this classroom-tested introductory text is
meantforasecond-yearoradvancedgraduatecourseinautomorphicforms,andalsoas
aresourceforresearchersworkinginautomorphicforms,analyticnumbertheory,and
relatedfields.
PaulGarrettisProfessorofMathematicsattheUniversityofMinnesota–Minneapolis.
Hisresearchfocusesonanalyticalissuesinthetheoryofautomorphicforms.Hehas
publishednumerousjournalarticlesaswellasfivebooks.
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS
EditorialBoard:
B.Bollobás,W.Fulton,F.Kirwan,P.Sarnak,B.Simon,B.Totaro
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Modern Analysis of
Automorphic Forms by Example,
Volume 1
PAUL GARRETT
UniversityofMinnesota–Minneapolis
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Informationonthistitle:www.cambridge.org/9781107154001
DOI:10.1017/9781316650332
©PaulGarrett2018
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Contents
IntroductionandHistoricalNotes pageix
1 FourSmallExamples 1
1.1 GroupsG=SL (R),SL (C),Sp∗ ,andSL (H) 2
2 2 1,1 2
1.2 CompactSubgroupsK ⊂G,CartanDecompositions 5
1.3 IwasawaDecompositionG=PK =NA+K 9
1.4 SomeConvenientEuclideanRings 13
1.5 DiscreteSubgroups(cid:3) ⊂G,ReductionTheory 15
1.6 InvariantMeasures,InvariantLaplacians 19
1.7 DiscreteDecompositionofL2((cid:3)\G/K)Cuspforms 22
1.8 Pseudo-EisensteinSeries 24
1.9 EisensteinSeries 27
1.10 MeromorphicContinuationofEisensteinSeries 32
1.11 TruncationandMaaß-SelbergRelations 35
1.12 DecompositionofPseudo-EisensteinSeries 44
1.13 PlancherelforPseudo-EisensteinSeries 49
1.14 AutomorphicSpectralExpansionandPlancherel
Theorem 52
1.15 ExoticEigenfunctions,Discretenessof
Pseudo-Cuspforms 53
2 TheQuotientZ+GL (k)\GL (A) 55
2 2
2.1 GroupsKv =GL2(ov)⊂Gv =GL2(kv) 56
2.2 DiscreteSubgroupGL (k)⊂GL (A),Reduction
2 2
Theory 59
2.3 InvariantMeasures 67
2.4 HeckeOperators,IntegralOperators 70
v
vi Contents
2.5 DecompositionbyCentralCharacters 74
2.6 DiscreteDecompositionofCuspforms 76
2.7 Pseudo-EisensteinSeries 79
2.8 EisensteinSeries 85
2.9 MeromorphicContinuationofEisensteinSeries 96
2.10 TruncationandMaaß-SelbergRelations 100
2.11 DecompositionofPseudo-EisensteinSeries:
LevelOne 105
2.12 DecompositionofPseudo-EisensteinSeries:
HigherLevel 114
2.13 PlancherelforPseudo-EisensteinSeries:LevelOne 119
2.14 SpectralExpansion,PlancherelTheorem:LevelOne 124
2.15 ExoticEigenfunctions,Discretenessof
Pseudo-Cuspforms 125
2.A Appendix:CompactnessofJ1/k× 127
2.B Appendix:MeromorphicContinuation 128
2.C Appendix:Hecke-MaaßPeriodsofEisensteinSeries 134
3 SL (Z),SL (Z),SL (Z),... 138
3 4 5
3.1 ParabolicSubgroupsofGL 139
r
3.2 GroupsKv =GLr(øv)⊂Gv =GLr(kv) 142
3.3 DiscreteSubgroupG =GL (k),ReductionTheory 145
k r
3.4 InvariantDifferentialOperatorsandIntegralOperators 148
3.5 HeckeOperatorsandIntegralOperators 150
3.6 DecompositionbyCentralCharacters 152
3.7 DiscreteDecompositionofCuspforms 152
3.8 Pseudo-EisensteinSeries 155
3.9 Cuspidal-DataPseudo-EisensteinSeries 158
3.10 Minimal-ParabolicEisensteinSeries 160
3.11 Cuspidal-DataEisensteinSeries 167
3.12 ContinuationofMinimal-ParabolicEisensteinSeries 181
3.13 ContinuationofCuspidal-DataEisensteinSeries 193
3.14 TruncationandMaaß-SelbergRelations 195
3.15 Minimal-ParabolicDecomposition 204
3.16 Cuspidal-DataDecomposition 208
3.17 PlancherelforPseudo-EisensteinSeries 214
3.18 AutomorphicSpectralExpansions 218
3.A Appendix:Bochner’sLemma 220
3.B Appendix:Phragmén-LindelöfTheorem 222
Contents vii
4 InvariantDifferentialOperators 224
4.1 DerivativesofGroupActions:LieAlgebras 225
4.2 LaplaciansandCasimirOperators 229
4.3 DetailsaboutUniversalAlgebras 232
4.4 DescendingtoG/K 235
4.5 ExampleComputation:SL (R)andH 236
2
4.6 ExampleComputation:SL (C) 239
2
4.7 ExampleComputation:Sp∗ 242
1,1
4.8 ExampleComputation:SL (H) 244
2
4.A Appendix:Brackets 246
4.B Appendix:ExistenceandUniqueness 250
5 IntegrationonQuotients 253
5.1 SurjectivityofAveragingMaps 254
5.2 InvariantMeasuresandIntegralsonQuotientsH\G 256
5.A Appendix:ApocryphalLemmaX ≈G/G 258
x
5.B Appendix:TopologyonQuotientsH\GorG/H 262
6 ActionofGonFunctionSpacesonG 263
6.1 ActionofGonL2((cid:3)\G) 263
6.2 ActionofGonCo((cid:3)\G) 267
c
6.3 TestFunctionsonZ+Gk\GA 270
6.4 ActionofGAonCc∞(Z+Gk\GA) 274
6.5 SymmetryofInvariantLaplacians 277
6.6 AnInstanceofSchur’sLemma 279
6.7 DualityofInducedRepresentations 285
6.8 AnInstanceofFrobeniusReciprocity 289
6.9 InductioninStages 290
6.10 RepresentationsofCompactG/Z 295
6.11 Gelfand-KazhdanCriterion 296
6.A Appendix:ActionofCompactAbelianGroups 301
7 DiscreteDecompositionofCuspforms 306
7.1 TheFourSimplestExamples 307
7.2 Z+GL (k)\GL (A) 320
2 2
7.3 Z+GL (k)\GL (A) 333
r r
7.A Appendix:Dualities 345
7.B Appendix:CompactQuotients(cid:3)\G 351
8 ModerateGrowthFunctions,TheoryoftheConstantTerm 354
8.1 TheFourSmallExamples 354
8.2 GL (A) 360
2
viii Contents
8.3 SL (Z),SL (Z),SL (Z),... 362
3 4 5
8.4 ModerateGrowthofConvergentEisensteinSeries 367
8.5 IntegralOperatorsonCuspidal-DataEisensteinSeries 371
8.A Appendix:JointContinuityofBilinearFunctionals 372
Bibliography 373
Index 381