Hodge theory and representation theory

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1、HODGETHEORYANDREPRESENTATIONTHEORYPHILLIPGRIFFITHSTenlecturestobegivenduringtheNSF/CBMSRegionalConferenceintheMathematicalSciencesatTCU,June18{22,2012.12TableofContentsIntroductionLecture1:Theclassicaltheory:PartILecture2:Theclassicaltheory:PartIILecture3:PolarizedHodgestructuresandMumford-Tategrou

2、psanddomainsLecture4:HodgerepresentationsandHodgedomainsLecture5:Discreteseriesandn-cohomologyAppendixtoLecture5:TheBorel-Weil-Bott(BWB)theoremLecture6:Geometryof agdomains:PartIAppendixtoLecture6:TheIwasawadecompositionandapplicationsLecture7:Geometryof agdomains:PartIIAppendixtoLecture7:TheBWBthe

3、oremrevisitedLecture8:PenrosetransformsinthetwomainexamplesAppendixtoLecture8:ProofsoftheresultsonPenrosetransformsforDandD0Lecture9:AutomorphiccohomologyAppendixItoLecture9:TheK-typesofTDLDSforSU(2;1)andSp(4)AppendixIItoLecture9:Schmid'sproofofthedegeneracyoftheHSSSforTDLDSintheSU(2;1)andSp(4)case

4、sLecture10:SelectedtopicsandpotentialareasforresearchAppendixtoLecture10:BoundarycomponentsandCarayol'sresultSelectedreferencesIndexNotationsusedinthetalksIntroduction3TheselecturesarecenteredaroundthesubjectsofHodgetheoryandrepresentationtheoryandtheirrelationship.Aunifyingthemeisthegeometryofhomo

5、geneouscomplexmanifolds.Oneobjectiveistopresent,inageneralcontext,someoftherecentworkofCarayol[C1],[C2],[C3].FinitedimensionalrepresentationtheoryinteractswithHodgetheorythroughtheuseofHodgerepresentationstoclassifythepossiblerealizationsofareductive,Q-algebraicgroupasaMumford-Tategroup.Thegeometry

6、ofhomogeneouscomplexmanifoldsentersthroughthestudyofMumford-TatedomainsandHodgedomains.InnitedimensionalrepresentationtheoryandthegeometryofhomogeneouscomplexmanifoldsinteractthroughtherealizationoftheHarish-Chandramodulesassociatedtodiscreteseriesrepresentations,especiallytheirlimits,ascohomology

7、groupsassociatedtohomogeneouslinebundles(workofSchmid).ItalsoentersthroughtheworkofCarayolonautomorphiccohomology,themostrecentofwhichinvolvestheHodgetheoryassociatedtoboundarycomponentsofMumford-Tatedomain