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ApplicationsofthecategoryoflinearcomplexesoftiltingmodulesassociatedwiththecategoryOVolodymyrMazorchukU.U.D.M.Report2005:3ISSN1101–3591DepartmentofMathematicsUppsalaUniversityApplicationsofthecategoryoflinearcomplexesoftiltingmodulesassociatedwiththecategoryOVolodymyrMazorchukAbstractWeusethecategoryoflinearcomplexesoftiltingmodulesfortheBGGcategoryO,associatedwithasemi-simplecomplexfinite-dimensionalLiealgebrag,toreproveinpurelyalgebraicwayseveralknownresultsaboutOobtainedearlierbydifferentauthorsusinggeometricmethods.WealsoobtainseveralnewresultsabouttheparaboliccategoryO(p,Λ).1IntroductionandpreliminariesLetgbeasemi-simplecomplexfinite-dimensionalLiealgebrawithafixedtriangulardecomposition,g=n−⊕h⊕n+,andπbethecorrespondingbasisoftherootsystemofg.LetfurtherρbethehalfofthesumofallpositiverootsandWbetheWeylgroupofg.Forw∈Wandλ∈h∗setw·λ=w(λ+ρ)−ρ.Denotebyw0thelongestelementofWandbylthelengthfunctiononW.WeconsiderWasthepartiallyorderedsetwithrespecttotheBruhatorderundertheconventionthattheidentityeisthesmallestelement.ConsidertheprincipalblockO0oftheBGGcategoryOforg,associatedwiththetriangulardecompositionabove(see[BGG,So1]).Thiscategoryisahighestweightcategoryinthesenseof[CPS]andhenceisequivalenttothemodulecategoryofsomefinite-dimensionalassociativequasi-hereditarybasicalgebra,whichwedenotebyA(see[BGG,DR]).TheChevalleyinvolutiononginducesanaturalinvolutivecontravariantexactself-equivalenceonO,andhenceonA−mod,whichwewilldenoteby?.TheisomorphismclassesofsimplemodulesinO0(andthusalsoinA−mod)areinnaturalbijectionwiththeelementsofWundertheconventionthatecorrespondstotheone-dimensionalmoduleinO0.Forw∈Wweintroducethefollowingnotation:1•L(w)isthesimpleA-module,whichcorrespondstothesimplehighestweightmoduleinOwiththehighestweightw·0(see[Di,Chapter7]);•Δ(w)isthestandardA-modulecorrespondingtoL(w)(inOthemod-uleΔ(w)istheVermamodulewiththehighestweightw·0,see[Di,Chapter7]);•∇(w)=Δ(w)?isthecorrespondingcostandardmoduleforA(thedualVermamoduleinO);•P(w)istheprojectivecoverofL(w);•I(w)=P(w)?istheinjectiveenvelopeofL(w);•T(w)istheindecomposabletiltingA-module,whichcorrespondstoΔ(w)(see[Ri]).TheprincipalresultaboutthecategoryOistheso-calledKazhdan-Lusztig(or,simply,theKL-)Theorem,provedin[BB,BK],whichdescribesthecompositionmultiplicitiesofΔ(w)intermsoftheKazhdan-Lusztigcom-binatoricsoftheHeckealgebra,see[KL].ThisresultandthegeometricapproachtoOhavebeenusedbySoergelin[So1]toshowthatAisKoszulandevenKoszulself-dual.Thiswasfurtherextendedin[BGS],whereitwasshownthatallassociativealgebrasassociatedwiththeblocksofOareKoszulandthattheKoszuldualofthealgebraofasingularblockofOisthealgebraoftheregularblockofcertainparabolicgeneralizationofO,intro-ducedbyRocha-Caridiin[RC].In[Ba]thisresultwasevenfurtherextendedtoallparabolic-singularblocks.Both[BGS]and[Ba]use“heavy”geometricarguments.ThealgebraAcanbegivenanicecombinatorialdescriptionviathecoinvariantalgebraCassociatedwithW.In[So1]itisshownthatC∼=EndA(P(w0));thatthemoduleP(w0),viewedasaC-module,canbede-scribedcombinatorially;andthatA∼=EndC(P(w0)C)(seealso[KSX]forthelastisomorphism).Furthermore,in[So1]itisshownthatEndA(P(w0))isisomorphictothequotientofthepolynomialalgebraoverahomogeneousideal,inparticular,thatitisZ-graded(howeverthegradingisnotunique).IfweletthegeneratorsofEndA(P(w0))tohavedegree2,weobtainthegrad-ingwhichcoincideswiththegradingonCobtainedviainterpretationofCasthecohomologyalgebraofacertainflagmanifold(seeforexample[Hi]).Further,thecombinatorialconstructionofSoergelimpliesthatP(w0)isaZ-gradedC-module,whichmakesAintoaZ-gradedalgebra.In[BGS]itwasshownthatthisgradingistheKoszulgradingonAusingsomeargumentsfromthe“mixed”geometry.2Thepaper[St]initiatesthestudyofthegradedversionofthecategoryOwithrespecttothe“natural”gradingdescribedabove.ItisimportantthatSoergel’scombinatorialdescriptionandhencethenaturalgradingonAdonotdependontheKL-Theorem.Inparticular,in[St]theauthorobtainsseveralresultsaboutthegradedversionofthecategoryO,whichdonotdependontheKL-Theorem.ItseemsthatthereisahopethatthegradedapproachmightbeawaytogiveanalgebraicproofoftheKL-Theorem.BecauseofVogan’sformulationoftheKL-Theorem,see[Vo],andthegradeddescriptionofthetranslationfunctorsfrom[St],theKL-TheoremwouldfollowifonewouldshowthatthenaturalgradingonAispositiveinthesensethatallnon-zerocomponentshavenon-negativedegrees,andtheradicalofAcoincideswiththesumofallcomponentsofpositivedegrees.However,sofarthisseemstobeverydifficult.LetusnowconsiderAasagradedalgebrawithrespecttothenaturalgrading.Thenitiseasytoseethatallsimple,projective,injective,standardandcostandardmodulesadmitgradedlifts.In[MO,Zh]itwasshownthatinthiscasealltiltingmodulesadmitgradedliftsaswell.Inparticular,theRingeldualR(A)ofA(see[Ri])isautomaticallygraded.FixingnaturalgradedliftsofindecomposabletiltingmodulesonecanconsiderthecategoryT(A)oflinearcomplexesoftiltingmodulesforA.In[MO]itwasshownthatT(A)isequivalenttothecategoryoflocallyfinite-dimensionalgradedmodulesoverthequadraticdualR(A)!ofR(A).UsingtheKoszulself-dualityofA(see[So1])andtheRingelself-dualityofA(see[So2]),o