Eisenstein Series and Automorphic Representations: With Applications in String Theory

588 Pages·2018·4.376 MB·English

by  Philipp Fleig, Henrik P. A. Gustafsson, Axel Kleinschmidt, Daniel Persson

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS176
EditorialBoard
B. BOLLOBA´S, W. FULTON, F. KIRWAN,
P. SARNAK, B. SIMON, B. TOTARO
EISENSTEINSERIESANDAUTOMORPHIC
REPRESENTATIONS
ThisintroductiontoautomorphicformsonadelicgroupsG(A)emphasisestheroleof
representationtheory.Theexpositionisdrivenbyexamplesandcollectsandextends
many results scattered throughout the literature, in particular the Langlands constant
termformulaforEisensteinseriesonG(A)aswellastheCasselman–Shalikaformula
forthep-adicsphericalWhittakerfunction.Thisbookalsocoversmoreadvancedtopics
suchassphericalHeckealgebrasandautomorphicL-functions.
Many of these mathematical results have natural interpretations in string theory,
and so some basic concepts of string theory are introduced with an emphasis on
connections with automorphic forms. Throughout the book special attention is paid
to small automorphic representations which are of particular importance in string
theory, but are also of independent mathematical interest. Numerous open questions
and conjectures, partially motivated by physics, are included to prompt the reader’s
ownresearch.
PhilippFleigisaPostdoctoralResearcherattheMaxPlanckInstituteforDynamics
andSelf-Organization,Germany.
HenrikP.A.GustafssonisaPostdoctoralResearcherintheDepartmentofMathemat-
icsatStanfordUniversity,California.
AxelKleinschmidtisaSeniorScientistattheMaxPlanckInstituteforGravitational
Physics(AlbertEinsteinInstitute),Germany,andattheInternationalSolvayInstitutes,
Brussels.
DanielPerssonisanAssociateProfessorintheDepartmentofMathematicalSciences
atChalmersUniversityofTechnology,Gothenburg.
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS
EditorialBoard:
B.Bolloba´s,W.Fulton,F.Kirwan,P.Sarnak,B.Simon,B.Totaro
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138 C.Muscalu&W.SchlagClassicalandmultilinearharmonicanalysis,II
139 B.HelfferSpectraltheoryanditsapplications
140 R.Pemantle&M.C.WilsonAnalyticcombinatoricsinseveralvariables
141 B.Branner&N.FagellaQuasiconformalsurgeryinholomorphicdynamics
142 R.M.DudleyUniformcentrallimittheorems (2ndEdition)
143 T.LeinsterBasiccategorytheory
144 I.Arzhantsev,U.Derenthal,J.Hausen&A.LafaceCoxrings
145 M.VianaLecturesonLyapunovexponents
146 J.-H.Evertse&K.Gyo˝ryUnitequationsinDiophantinenumbertheory
147 A.PrasadRepresentationtheory
148 S.R.Garcia,J.Mashreghi&W.T.RossIntroductiontomodelspacesandtheiroperators
149 C.Godsil&K.MeagherErdo˝s–Ko–Radotheorems:Algebraicapproaches
150 P.MattilaFourieranalysisandHausdorffdimension
151 M.Viana&K.OliveiraFoundationsofergodictheory
152 V.I.Paulsen&M.RaghupathiAnintroductiontothetheoryofreproducingkernelHilbertspaces
153 R.Beals&R.WongSpecialfunctionsandorthogonalpolynomials
154 V.JurdjevicOptimalcontrolandgeometry:Integrablesystems
155 G.PisierMartingalesinBanachspaces
156 C.T.C.WallDifferentialtopology
157 J.C.Robinson,J.L.Rodrigo&W.SadowskiThethree-dimensionalNavier–Stokesequations
158 D.HuybrechtsLecturesonK3surfaces
159 H.Matsumoto&S.TaniguchiStochasticanalysis
160 A.Borodin&G.OlshanskiRepresentationsoftheinfinitesymmetricgroup
161 P.WebbFinitegrouprepresentationsforthepuremathematician
162 C.J.Bishop&Y.PeresFractalsinprobabilityandanalysis
163 A.BovierGaussianprocessesontrees
164 P.SchneiderGaloisrepresentationsand(ϕ,(cid:3))-modules
165 P.Gille&T.SzamuelyCentralsimplealgebrasandGaloiscohomology (2ndEdition)
166 D.Li&H.QueffelecIntroductiontoBanachspaces,I
167 D.Li&H.QueffelecIntroductiontoBanachspaces,II
168 J.Carlson,S.Mu¨ller-Stach&C.PetersPeriodmappingsandperioddomains (2ndEdition)
169 J.M.LandsbergGeometryandcomplexitytheory
170 J.S.MilneAlgebraicgroups
171 J.Gough&J.KupschQuantumfieldsandprocesses
172 T.Ceccherini-Silberstein,F.Scarabotti&F.TolliDiscreteharmonicanalysis
173 P.GarrettModernanalysisofautomorphicformsbyexample,I
174 P.GarrettModernanalysisofautomorphicformsbyexample,II
175 G.NavarroCharactertheoryandtheMcKayconjecture
176 P.Fleig,H.P.A.Gustafsson,A.Kleinschmidt&D.PerssonEisensteinseriesandautomorphic
representations

Eisenstein Series and
Automorphic Representations
with Applications in String Theory
PHILIPP FLEIG
Max-Planck-Institutfu¨rDynamikundSelbstorganisation,Germany
HENRIK P. A. GUSTAFSSON
StanfordUniversity,California
AXEL KLEINSCHMIDT
Max-Planck-Institutfu¨rGravitationsphysik,Germany
DANIEL PERSSON
ChalmersUniversityofTechnology,Gothenburg
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Informationonthistitle:www.cambridge.org/9781107189928
DOI:10.1017/9781316995860
©PhilippFleig,HenrikP.A.Gustafsson,AxelKleinschmidtandDanielPersson2018
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Title:Eisensteinseriesandautomorphicrepresentationswith
applicationsinstringtheory/PhilippFleig,Max-Planck-Institutfu¨r
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Description:Cambridge:CambridgeUniversityPress,2018.|
Series:Cambridgestudiesinadvancedmathematics;176|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2018018863|ISBN9781107189928(hardback:alk.paper)
Subjects:LCSH:Eisensteinseries.|Automorphicfunctions.|Stringmodels.
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Contents
ListofDefinitionsandTheorems pagexi
ListofExamples xiv
Preface xvii
1 MotivationandBackground 1
1.1 AutomorphicFormsandEisensteinSeries 1
1.2 WhyEisensteinSeriesandAutomorphicForms? 5
1.3 AnalysingAutomorphicFormsandAdelisation 6
1.4 Reader’sGuideandMainTheorems 9
PARTONE AUTOMORPHICREPRESENTATIONS 17
2 Preliminariesonp-adicandAdelicTechnology 19
2.1 p-adicNumbers 19
2.2 p-adicIntegration 23
2.3 p-adicCharactersandtheFourierTransform 25
2.4 p-adicGaussianandBesselFunctions 30
2.5 Adeles 32
2.6 Adelisation 34
2.7 AdelicAnalysisoftheRiemannZetaFunction 35
3 BasicNotionsfromLieAlgebrasandLieGroups 39
3.1 RealLieAlgebrasandRealLieGroups 39
3.2 p-adicandAdelicGroups 49
4 AutomorphicForms 58
4.1 PreliminariesonSL(2,R) 58
4.2 ClassicalModularForms 61
4.3 FromClassicalModularFormstoAutomorphicForms 67
4.4 AdelicAutomorphicForms 75
4.5 EisensteinSeries 81
vii
Contents
5 AutomorphicRepresentationsandEisensteinSeries 87
5.1 AFirstGlimpseatAutomorphicRepresentations 87
5.2 AutomorphicRepresentations 93
5.3 PrincipalSeriesRepresentations 96
5.4 EisensteinSeriesandInducedRepresentations 97
5.5 ClassifyingAutomorphicRepresentations 98
5.6 EmbeddingoftheDiscreteSeriesinthePrincipalSeries 100
5.7 EisensteinSeriesforNon-minimalParabolics 106
5.8 InducedRepresentationsandSphericalVectors* 115
6 WhittakerFunctionsandFourierCoefficients 123
6.1 PreliminaryExample:SL(2,R)WhittakerFunctions 123
6.2 FourierExpansionsandUnitaryCharacters 128
6.3 InducedRepresentationsandWhittakerModels 136
6.4 WavefrontSetandSmallRepresentations 141
6.5 MethodofPiatetski-ShapiroandShalika* 151
7 FourierCoefficientsofEisensteinSeriesonSL(2,A) 155
7.1 StatementofTheorem 155
7.2 ConstantTerm 158
7.3 TheNon-constantFourierCoefficients 165
8 LanglandsConstantTermFormula 170
8.1 StatementofTheorem 170
8.2 BruhatDecomposition 171
8.3 ParametrisingtheIntegral 172
8.4 Obtainingthea-dependenceoftheIntegral 173
8.5 SolvingtheRemainingIntegralbyInduction 174
8.6 TheGindikin–KarpelevichFormula 175
8.7 AssemblingtheConstantTerm 178
8.8 FunctionalRelationsforEisensteinSeries 179
8.9 ExpansioninMaximalParabolics* 181
9 WhittakerCoefficientsofEisensteinSeries 186
9.1 ReductionoftheIntegralandtheLongestWeylWord 186
9.2 UnramifiedLocalWhittakerFunctions 189
9.3 TheCasselman–ShalikaFormula 191
9.4 WhittakerFunctionsforGenericCharactersψ 197
9.5 DegenerateWhittakerCoefficients 199
9.6 TheCasselman–ShalikaFormulaandLanglandsDuality* 204
9.7 QuantumWhittakerFunctions* 207
9.8 WhittakerCoefficientsonSL(3,A)* 209
viii