Representation Theory and Algebraic Geometry: A Conference Celebrating the Birthdays of Sasha Beilinson and Victor Ginzburg PDF
Preview Representation Theory and Algebraic Geometry: A Conference Celebrating the Birthdays of Sasha Beilinson and Victor Ginzburg
Trends in Mathematics Vladimir Baranovsky Nicolas Guay Travis Schedler Editors Representation Theory and Algebraic Geometry A Conference Celebrating the Birthdays of Sasha Beilinson and Victor Ginzburg Trends in Mathematics TrendsinMathematicsisaseriesdevotedtothepublicationofvolumesarisingfrom conferences and lecture series focusing on a particular topic from any area of mathematics.Itsaimistomakecurrentdevelopmentsavailabletothecommunityas rapidlyaspossiblewithoutcompromisetoqualityandtoarchivetheseforreference. ProposalsforvolumescanbesubmittedusingtheOnlineBookProjectSubmission Formatourwebsitewww.birkhauser-science.com. Materialsubmittedforpublicationmustbescreenedandpreparedasfollows: Allcontributionsshouldundergoareviewingprocesssimilartothatcarriedoutby journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, shouldberejected.Highqualitysurveypapers,however,arewelcome. 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Vladimir Baranovsky • Nicolas Guay Travis Schedler Editors Representation Theory and Algebraic Geometry A Conference Celebrating the Birthdays of Sasha Beilinson and Victor Ginzburg Editors VladimirBaranovsky NicolasGuay DepartmentofMathematics DepartmentofMath.&Stat.Sciences UniversityofCalifornia,Irvine UniversityofAlberta Irvine,CA,USA Edmonton,AB,Canada TravisSchedler DepartmentofMathematics ImperialCollegeLondon London,UK ISSN2297-0215 ISSN2297-024X (electronic) TrendsinMathematics ISBN978-3-030-82006-0 ISBN978-3-030-82007-7 (eBook) https://doi.org/10.1007/978-3-030-82007-7 MathematicsSubjectClassification:14L30,14M15,20C08,14F10,14F07,18M20,17B37,32G24 ©SpringerNatureSwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Theconference“InteractionsbetweenRepresentationTheoryandAlgebraicGeom- etry” was held at the University of Chicago on August 21–25, 2017. It brought together about 150 participants from several major universities in the USA and abroad,morethanhalfofwhomwerejuniormathematicians.Itfeatured21talksby eminent mathematicians from the USA, Europe, and Asia on topics at the cutting edgeofmathematicalresearch.Somejuniorparticipantsgaveintroductorytalksin theeveningtotrytomakethepresentationsbythemainspeakersmoreaccessible tothegeneralaudience,inadditiontogivingposterpresentationsoftheirresearch. Theresearcharticlesintheseproceedingsarededicated,aswastheconference, to Alexander Beilinson and Victor Ginzburg, two visionaries in the fields of Representation Theory and Algebraic Geometry, in honor of their 60th birthdays. Their work and mentoring has influenced a great number of researchers and will continue to shape the development of those branches of mathematics for many years to come. Their influence can be perceived throughout this volume, for instance where important roles are played by D-modules and perverse sheaves, Grassmannians and their affine and loop group analogues, categorical approaches torepresentationtheory,versionsofHeckealgebras,symplecticalgebraicgeometry, andmanyothertopics. Thechaptershavebeenorganizedthematicallyasfollows:thefirstthreedealwith thesubjectsofgroups,algebras,andcategoriesandtheirrepresentationtheory;the next four deal with D-modules and perverse sheaves, particularly on flag varieties and their generalizations; the final two deal with analogous varieties defined by quivers and their relationships to representation theory, cohomology theories, and symplecticgeometry. Irvine,CA,USA VladimirBaranovsky Edmonton,AB,Canada NicolasGuay London,UK TravisSchedler v Contents PartI Groups,Algebras,Categories,andTheirRepresentation Theory OnSemisimplificationofTensorCategories .................................. 3 PavelEtingofandVictorOstrik TotallyAsphericalParametersforCherednikAlgebras..................... 37 IvanLosev MicrolocalApproachtoLusztig’sSymmetries ............................... 57 MichaelFinkelbergandVadimSchechtman PartII D-ModulesandPerverseSheaves,ParticularlyonFlag VarietiesandTheirGeneralizations Fourier-SatoTransformonHyperplaneArrangements..................... 87 MichaelFinkelberg,MikhailKapranov,andVadimSchechtman AQuasi-CoherentDescriptionoftheCategoryD-mod(Gr ).......... 133 GL(n) AlexanderBravermanandMichaelFinkelberg TheSemi-infiniteIntersectionCohomologySheaf-II:TheRan SpaceVersion..................................................................... 151 DennisGaitsgory ATopologicalApproachtoSoergelTheory................................... 267 RomanBezrukavnikovandSimonRiche PartIII Varieties Associated to Quivers and Relations to RepresentationTheoryandSymplecticGeometry LoopGrassmanniansofQuiversandAffineQuantumGroups ............ 347 IvanMirkovic´,YapingYang,andGufangZhao vii viii Contents SymplecticResolutionsforMultiplicativeQuiverVarietiesand CharacterVarietiesforPuncturedSurfaces .................................. 393 TravisSchedlerandAndreaTirelli Part I Groups, Algebras, Categories, and Their Representation Theory On Semisimplification of Tensor Categories PavelEtingofandVictorOstrik Contents 1 Introduction..................................................................................... 4 2 Preliminaries ................................................................................... 5 2.1 TensorIdeals............................................................................. 5 2.2 SemisimplificationofaSphericalTensorCategory.................................... 6 2.3 GeneralizationtoPivotalKaroubianCategories........................................ 7 3 GeneralResultsonSemisimplificationofTensorCategories................................ 10 3.1 SplittingoftheSemisimplificationFunctorforTannakianCategoriesin CharacteristicZero,ReductiveEnvelopes,andtheJacobson-MorozovLemma...... 10 3.2 CompatibilityofSemisimplificationwithEquivariantization ......................... 14 3.3 CompatibilityofNegligibleMorphismswithSurjectiveTensorFunctors............ 15 4 SemisimplificationofRepresentationCategoriesofFiniteGroupsinCharacteristicp.... 16 4.1 TheResult................................................................................ 16 4.2 ProofofTheorem4.2.................................................................... 17 4.3 TheCaseofSylowSubgroupofPrimeOrder.......................................... 18 4.4 TheCaseoftheSymmetricGroupSp+n,wheren<p................................ 19 4.5 Application:TheSemisimplificationoftheDeligneCategoryRepabSn .............. 21 5 SemisimplificationofSomeNon-SymmetricCategories.................................... 22 5.1 Genericq................................................................................. 22 5.2 RootsofUnity............................................................................ 23 6 SurjectiveSymmetricTensorFunctorsBetweenVerlindeCategoriesVerp(G) ............ 25 7 ObjectsofFiniteTypeinSemisimplifications................................................ 27 8 SemisimplificationofTilt(GL(n))whenchar(k)=2....................................... 29 AppendixA:CategorificationsofBasedRingsAttachedtoSO(3)............................. 31 References.......................................................................................... 34 ToSashaBeilinsonandVityaGinzburgontheir60thbirthdayswithadmiration P.Etingof((cid:2)) DepartmentofMathematics,MassachusettsInstituteofTechnology,Cambridge,MA,USA e-mail:[email protected] V.Ostrik DepartmentofMathematics,UniversityofOregon,Eugene,OR,USA LaboratoryofAlgebraicGeometry,NationalResearchUniversityHigherSchoolofEconomics, Moscow,Russia e-mail:[email protected] ©SpringerNatureSwitzerlandAG2022 3 V.Baranovskyetal.(eds.),RepresentationTheoryandAlgebraicGeometry,Trends inMathematics,https://doi.org/10.1007/978-3-030-82007-7_1
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