On a generic Verma module at the critical level ov

arXiv:math/0504253v2[math.RT]23Jan2007ONAGENERICVERMAMODULEATTHECRITICALLEVELOVERAFFINELIESUPERALGEBRASMARIAGORELIKAbstract.WedescribethestructureofaVermamodulewithagenerichighestweightatthecriticalleveloverasymmetrizableaffineLiesuperalgebraˆg6=A(2k,2l)(4).WeobtainthecharacterformulaforasimplemodulewithagenerichighestweightatthecriticallevelconjecturedbyV.G.KacandD.A.Kazhdan.1.IntroductionItiswell-knownthattherepresentationtheoryofacomplexaffineLiealgebrachangesdrasticallyatthecriticallevel.Inparticular,Vermamodulescontaininfinitenumberofsingularvectorsofimaginarydegrees.Asitisshownin[Ku],aVermamodulewitha“generic”highestweightatthecriticallevellookslikeapolynomialalgebrainacount-ablenumberofvariables:submodulescorrespondtotheidealsinthepolynomialal-gebraandtheJantzenfiltrationcorrespondstotheadicfiltration.Here“genericity”meansthatallsingularvectorslieintheimaginarydegrees.Asaconsequence,J.-M.KuobtainsthecharacterformulaconjecturedbyV.G.KacandD.A.Kazhdanin[KK]:chL(λ)=eλQα∈ˆΔ+re(1−e−α)−1,whereλisagenerichighestweightatthecriticallevelandL(λ)isthecorrespondingsimplemodule.Inthispaperweextendtheresultsof[Ku]tosymmetrizableaffineLiesuperalgebrasˆg6=A(2k,2l)(4)(see1.3).Fornon-twistedaffineLiealgebrastheKac-Kazhdancharacterformulawasprovenbydifferentmethods:forˆsl(2)byM.Wakimoto[Wk1],N.Wallach[Wl];fortheaffiniza-tionsofclassicalalgebrasbyT.Hayashi[H]andR.Goodman,N.Wallach[GW];fortheaffinizationofageneralsimpleLiealgebrabyB.FeiginandE.Frenkel[FF],[F]andrecentlybyT.Arakawa[Ar];infinitecharacteristicbyO.Mathieu[M].ForanarbitraryaffineLiealgebras(includingthetwistedcase)theformulawasprovenbyJ.-M.Ku[Ku]andrecentlyreprovenbyM.Szczesny[Sz].TheapproachofB.Feigin,E.FrenkelandM.Szczesnyisbasedontheexplicitrealiza-tionofL(λ):theyshowthatifλisagenerichighestweightatthecriticallevelthenL(λ)isisomorphictoaWakimotomodule,whichisarepresentationofˆginaFockmoduleoversomeinfinite-dimensionalHeisenbergalgebra;theconstructionofWakimotomodulesusesatechniqueofvertexalgebras.TheHeisenbergalgebraherecorrespondstothesetofreal1991MathematicsSubjectClassification.17B67.TheauthorwaspartiallysupportedbyTMRGrantNo.FMRX-CT97-0100.12MARIAGORELIKrootsofˆg.ThemethodofJ.-M.Kuismuchmore“elementary”:itisbasedonastudyofsingularvectorsintheVermamoduleM(λ).Itcanbeinterpreted(see1.2)intermsofaninfinite-dimensionalHeisenbergalgebrawhichcorrespondstothesetofimaginaryrootsofˆg(thisHeisenbergalgebraisasubalgebraofˆg).OurapproachisclosetooneofJ.-M.Ku.1.1.Mainresult.Letˆg=ˆn−⊕ˆh⊕ˆnbeanaffineLiesuperalgebrawithasymmetrizableindecomposableCartanmatrix(see1.3).LetM(λ)beaVermamoduleofthehighestweightλandvλbeitscanonicalgenerator,whichweassumetobeeven.TheLiesuperalgebraˆn−admitsatriangulardecompositionˆn−=N−−⊕H−⊕N+−,whereH−consistsoftheelementsofimaginaryweights(foranon-twistedcase,H−=Lh∩ˆn−andN±−=Ln±∩ˆn−,whereLstandsfortheloopspaceofagivensubalgebraofg).SetS:=U(H−).IntroducetheprojectionsHC±:U(ˆn−)→S,whereKerHC+=U(ˆn−)N−−+N+−U(ˆn−),KerHC−=U(ˆn−)N+−+N−−U(ˆn−).Foreachλ∈ˆh∗defineHC±:M(λ)→SviathenaturalidentificationofM(λ)withU(ˆn−);theseprojectionsplayacentralroleinourdescriptionofM(λ).TheprojectionHC+appearedin[Ku]and[Ch].Wecallv∈M(λ)singularifvisaweightvectorandv∈M(λ)ˆn(thesevectorsarealsocalledprimitive).Wesaythatλisacriticalweightorλhasthecriticallevelif(λ+ˆρ,δ)=0foranimaginaryrootδ.IfλisacriticalweightthenM(λ)λ−δcontainsasingularvector;wesaythatλisagenericcriticalweight(λ∈Λcrit)ifM(λ)λ−αhasnosingularvectorsunlessαisproportionaltoδ.Ifλisagenericcriticalweight,thespaceM(λ)ˆnhasanaturalstructureofanassociative(super)algebra.Indeed,themapψ7→ψ(vλ)givesanembeddingofEnd[ˆg,ˆg](M(λ))intothespaceofsingularvectorsM(λ)ˆn.ThegenericityconditiononλmeansthatthisembeddingisbijectiveandthisprovidesM(λ)ˆnwiththealgebrastructure.In1.1.1–1.1.3weassumethatˆgisnotoftypeA(2k,2l)(4)andthatλisagenericcriticalweight.1.1.1.Theorem.Letλbeagenericcriticalweight.(i)Onehas[M(λ):L(λ−sδ)]=dimM(λ)ˆnλ−sδ;thusanysubmoduleofM(λ)isgeneratedbysingularvectors.(ii)TherestrictionsofHC+:M(λ)→SandofHC−:M(λ)→StothespaceofsingularvectorsM(λ)ˆngivealgebraisomorphismsM(λ)ˆn∼−→S,wheretheimageofvλis1∈S.(iii)AnysingularvectorgeneratesasubmoduleisomorphictoM(λ−sδ)forsomes≥0.3NoticethatH−isevenandcommutativesoSisthealgebraofpolynomialsincountablymanyvariables.1.1.2.ForasubmoduleNofM(λ)setH(N):=HC+(Nˆn)⊂S.FromTheorem1.1.1weseethatHprovidesaone-to-onecorrespondencebetweenthesubmodulesofM(λ)andadˆh-invariantidealsofS.IfλisagenericcriticalweightthenallsimplesubquotientsofM(λ)areoftheformL(λ−sδ);notethatL(λ)∼=L(λ−sδ)as[ˆg,ˆg]-modulessochL(λ−sδ)=e−sδchL(λ).Asaresult,thecharactersofNandofH(N)areconnectedbythefollowingformula:(1)chN=chL(λ)·chH(N).ApplyingthisformulatoN=M(λ)wegettheKac-Kazhdancharacterformula:chM(λ)=chL(λ)·chS,thatischL(λ)=eλYα∈ˆΔ+re;0(1−e−α)−1Yα∈ˆΔ+re;1(1+e−α).NotethatchL(λ)=eλchU(N+−)chU(N−−).1.1.3.Jantzenfiltration.RecallthatH−iscommutativesoS=U(H−)isthesymmetricalgebra:S=P∞j=0Sj.ThespacesS≥k:=P∞j=kSjformtheadicfiltrationonS.Theorem.IfλisagenericcriticalweightthenHmapstheJantzenfiltration{M(λ)k}totheadicfiltrationonS,i.e.H(M(λ)k)