Representation theory and projective geometry

arXiv:math/0203260v2[math.AG]2Jul2002REPRESENTATIONTHEORYANDPROJECTIVEGEOMETRYJ.M.LANDSBERGANDL.MANIVELContents1.Overview12.ConstructionofcomplexsimpleLiealgebrasviageometry22.1.Differentialgeometry,algebraicgeometryandrepresentationtheory22.2.Firstalgorithm62.3.Secondalgorithm92.4.Outlineoftheproofs102.5.Applications,generalizationsandrelatedwork112.6.Whysecantandtangentlines?122.7.Titscorrespondencesandapplications133.TrialityandexceptionalLiealgebras153.1.Divisionalgebras153.2.Derivations153.3.Triality163.4.Trialityandso4,4:4-ality173.5.Themagicsquare193.6.Inclusions203.7.Automorphisms,symmetricandtrisymmetricspaces223.8.Dualpairs243.9.Thequaternionicform254.SeriesofLiealgebrasviaknottheoryandgeometry274.1.FromknottheorytotheuniversalLiealgebra274.2.Vogel’sdecompositionsandTitscorrespondences294.3.Theexceptionalseries294.4.Freudenthalgeometries324.5.Ageometricmagicsquare334.6.Thesubexceptionalseries33References351.OverviewThisarticlehastwopurposes.Thefirstistoprovideanelementaryintroductiontopapers[59,60,61,62,63],relatedworksandtheirhistoricalcontext.Sections2.1–2.3,2.5–2.6,3.1–3.5,4.1–4.3shouldbeaccessibletoageneralaudienceofmathematicians.Thesecondistoprovidegeneralizations,newperspectives,andcomplementstoresultsinthesepapers,i.e.,thingswethoughtofafterthepaperswerepublished.Inparticular,wemention2.5,2.7,3.4and3.6–3.9.Eachsectionbeginswithadescriptionofitscontents.Simplyput,ourgoalsaretousegeometrytosolvequestionsinrepresentationtheoryandtouserepresentationtheorytosolvequestionsingeometry.Onthegeometricside,theobjectsofinterestarehomogeneousvarietiesX=G/P⊂PV.HereGisacomplexsemi-simpleLiegroup,VisanirreducibleG-module,Xistheuniqueclosedorbit(projectivizationoftheorbitofahighestweightvector)andPisthestabilizerofapoint.Date:July2002.12J.M.LANDSBERGANDL.MANIVELForexample,letW=CmandletX=G(k,W)⊂P(ΛkW),theGrassmannianofk-planesthroughtheorigininW.HereG/P=SL(W)/P,V=ΛkW.WearemoregenerallyinterestedinthegeometryoforbitclosuresinPV.Basicquestionsonecanaskaboutavarietyareitsdimension,itsdegree,andmoregenerallyitsHilbertfunction.WemayaskthesamequestionsforvarietiesassociatedtoX.ForexamplethedegreesofthedualvarietiesofGrassmanniansarestillunknowningeneral(see[67,26,78]).Othertypesofproblemsincluderecognitionquestions.Forexample,givenavarietywithcertaingeometricproperties,arethosepropertiesenoughtocharacterizethevariety?Foranexampleofthis,seeZak’stheoremonSeverivarietiesbelowin§2.6.Foranotherexample,HwangandMokcharacterizerationalhomogeneousvarietiesbythevarietyoftangentdirectionstominimaldegreerationalcurvespassingthroughageneralpoint,see[44]foranoverview.Someresultsalongthoselinesaredescribedin§2.5below.WealsomentiontheLeBrun-Salamonconjecture[68,69]whichstatesthatanyFanovarietyequippedwithaholomorphiccontactstructuremustbetheclosedorbitintheadjointrepresentationXad⊂PgforacomplexsimpleLiealgebra.Inthiscontextalsosee[8,52].Ontherepresentationtheoryside,thebasicobjectsareg,acomplexsemi-simpleLiealgebraandV,anirreducibleg-module(e.g.,g=sl(W),V=ΛkW).ProblemsincludetheclassificationoforbitclosuresinPV,toconstructexplicitmodelsforthegroupaction,togeometricallyinter-pretthedecompositionofV⊗kintoirreducibleg-modules.Wediscusstheseclassicalquestionsbelow,primarilyforalgebrasoccuringin“series”.VassilievtheorypointstotheneedfordefiningobjectsmoregeneralthanLiealgebras.Wehavenothingtoaddaboutthissubjectatthemoment,buttheresultsof[62,63]werepartlyinspiredbyworkofDeligne[24]andVogel[82,83]inthisdirection.Forthemysticallyinclined,therearemanystrangeformulasrelatedtotheexceptionalgroups.Wepresentsomesuchformulasin§4.3,4.6below.ProctorandGelfand-Zelevinskifilledin“holes”intheclassicalformulasfortheospnseriesusingthenon-reductiveoddsymplecticgroups.Ourformulasledustoexceptionalanaloguesoftheoddsymplecticgroups.Theseanaloguesarecurrentlyunderinvestigation(see[65]).Whennototherwisespecified,weusetheorderingofrootsasin[9].Wenowturntodetails.Webeginwithsomeobservationsthatleadtointerestingrationalmapsofprojectivespaces.2.ConstructionofcomplexsimpleLiealgebrasviageometryWebeginin§2.1withthreeingredientsthatgointoourstudy:localdifferentialgeometry(asymptoticdirections),elementaryalgebraicgeometry(rationalmapsofprojectivespace)andhomogeneousvarieties(thecorrespondencebetweenrationalhomogeneousvarietiesandmarkedDynkindiagrams).Wethen,in§2.2-2.4describetwoalgorithmsthatconstructnewvarietiesfromoldthatleadtonewproofsoftheclassificationofcompactHermitiansymmetricspacesandtheCartan-KillingclassificationofcomplexsimpleLiealgebras.Theproofsareconstructive,viaexplicitrationalmapsandin§2.5wedescribeapplicationsandgeneralizationsofthesemaps.OurmapsgeneralizemapsusedbyZakinhisclassificationofSeverivarietiesandin§2.6wedescribehisinfluenceonourwork.In§2.7wereturntoatopicraisedin§2.1,wherewedeterminedtheparameterspaceoflinesthroughapointofahomogeneousvarietyX=G/P.WeexplainTits’correspondenceswhichallowonetodeterminetheparameterspaceofalllinesonXandinfactparameterspacesforallG-homogeneousvarietiesonX.WeexplainhowtouseTitscorrespondencestoexplicitlyconstructcertain