AdvancedStudiesinPureMathematics3?,200?RepresentationTheoryofAlgebraic - PDF document

AdvancedStudiesinPureMathematics3?,200?RepresentationTheoryofAlgebraicGroupsandQuantumGroupspp.1{44RepresentationsofLiealgebrasinpositivecharacteristicJensCarstenJantzenAbout50yearsagoitwasdiscoveredthatnitedimensionalLieal-gebrasinpositivecharacteristiconlyhavenitedimensionalirreduciblerepresentations.About15yearsagotheirreduciblerepresentationsfortheLiealgebraglnwereclassied.About5yearsagoaconjecturewasformulatedthatshouldleadtoacalculationofthedimensionsofthesesimplegln{modulesifp�n.ForLiealgebrasofotherreductivegroupsourknowledgeismorerestricted,buttherehasbeensomeremarkableprogressinthisareaoverthelastyears.Thepurposeofthissurveyistoreportonthesedevelopmentsandtoupdatetheearliersurveys[H3]and[J3].ThroughoutthispaperletKbeanalgebraicallyclosedeldofprimecharacteristicp.AllLiealgebrasoverKwillbeassumedtobenitedimensional.AGeneralTheoryA.1.IfgisaLiealgebraoverK,thenwedenotebyU(g)theuniversalenvelopingalgebraofgandbyZ(g)thecentreofU(g).ArestrictedLiealgebraoverKisaLiealgebragoverKtogetherwithamapg!g,X7!X[p],oftencalledthep{thpowermap,providedcertainconditionsaresatised.TherstconditionsaysthatforeachX2gtheelement(X)=XpX[p]2U(g)actuallybelongstothecentreZ(g)ofU(g).(HereXpisthep{thpowerofXtakeninU(g).)Theotherconditionsaysthat:g!Z(g)issemi-linearinthefollowingsense:Wehave(X+Y)=(X)+(Y)and(aX)=ap(X) ReceivedJuly24,2002.RevisedNovember18,2002. 2J.C.JantzenforallX;Y2ganda2K.A.2.Forexample,letAbeanassociativealgebraoverKconsideredasaLiealgebravia[X;Y]=XYYX.ThenAbecomesarestrictedLiealgebraifwesetX[p]equaltothep{thpowerofXtakeninA.Indeed,ifwewritelXandrXforleftandwritemultiplicationbyXinA(sowehave,e.g.,lX(Y)=XYforallY2A),thenad(X)=lXrX.SincelXandrXcommuteandsinceweareincharacteristicp,wegetad(X)p=(lX)p(rX)p.Now(lX)pisclearlyleftmultiplicationbythep{thpowerofXthatwehavedecidedtodenotebyX[p];similarlyfor(rX)p.Wegetthusad(X)p=ad(X[p]).NowthesameargumentusedinU(A)insteadofAshowsthatalsoad(X)p=ad(Xp),hencethatXpX[p]commuteswitheachelementofg.Therefore(X)=XpX[p]belongstoZ(A).Itremainstocheckthesemi-linearityof:Theidentity(aX)=ap(X)isobvious.TheproofoftheadditivityofrequiresaformuladuetoJacobsonexpressing(X+Y)pXpYpintermsofcommutators,see[Ja],xV.7.Ontheotherhand,ifGisanalgebraicgroupoverK,thenLie(G)hasanaturalstructureasarestrictedLiealgebra:OnecanthinkofLie(G)astheLiealgebraofcertaininvariantderivations,cf.[H2],9.1;ThenonedenesX[p]asthep{thpowerofXtakenasderivation.IncaseG=GLn(K),thenonegetsthusonLie(G)thesamestructureasfromtheidenticationofLie(G)withthespaceMn(k)ofall(nn){matricesoverKandfromtheconstructionintheprecedingparagraph.IfGisaclosedsubgroupofGLn(K),thenwecanidentifyLie(G)withaLiesubalgebraofMn(k)andthep{thpowermapofLie(G)istherestrictionofthatonMn(k).A.3.LetnowgbeanarbitraryrestrictedLiealgebraoverK.DenotebyZ0(g)thesubalgebraofZ(g)generatedbyall(X)=XpX[p]withX2g.Thissubalgebraisoftencalledthep{centreofU(g).LetX1,X2;:::;Xnbeabasisforg.UsingthePBW-theoremonechecksnoweasily:ThealgebraZ0(g)isgeneratedbyallXpiX[p]iwith1in;theseelementsarealgebraicallyindependentoverK.ConsideredasaZ0(g){moduleunderleft(=right)multiplicationU(g)isfreeofrankpdim(g);allproductsXm(1)1Xm(2)2:::Xm(n)nwith0m(i)pforalliformabasisofU(g)overZ0(g).Therstofthesetwoclaimscanberestatedasfollows:ThemapinducesanisomorphismofalgebrasS(g(1))!Z0(g): Liealgebrasinpositivecharacteristic3Hereweusethefollowingconvention:IfVisavectorspaceoverK,wedenotebyV(1)thevectorspaceoverKthatisequaltoVasanadditivegroup,butwhereanya2KactsonV(1)asa1=pdoesonV.Nowthesemi-linearityofmeansthatisalinearmapg(1)!Z0(g),henceinducesanalgebrahomomorphismfromthesymmetricalgebraS(g(1))tothecommutativealgebraZ0(g).Theclaimintheprecedingparagraphshowsthatthismapisbijective.Itisnoweasytodeducefromtheresultsabove:Proposition:ThecentreZ(g)ofU(g)isanitelygeneratedalgebraoverK.ConsideredasaZ(g){moduleU(g)isnitelygenerated.Thisresultactuallygeneralisestoall(nitedimensional!)Liealge-brasoverK.A.4.Theorem:Eachsimpleg{moduleisnitedimensional.Itsdi-mensionislessthanorequaltopdim(g).Proof:Chooseu1,u2;:::;ur2U(g)suchthatU(g)=Pri=1Z(g)ui.Thisispossiblebytheproposition;infact,wemayassumethatrpdim(g)asU(g)hasthatrankoverthesmallersubalgebraZ0(g).LetEbeasimpleg{module.Pickv2E,v=0.WehavethenE=U(g)v,henceE=Pri=1Z(g)uiv.SoEisnitelygeneratedasamoduleoverZ(g).SinceZ(g)isanitelygeneratedK{algebra,henceaNoetherianring,thereexistsamaximalsubmoduleE0E.ThesimpleZ(g){moduleE=E0isthenisomorphictoZ(g)=mforsomemaximalidealmZ(g).NowxEisag{submoduleofEforallx2Z(g).Ifx2m,thenx(E=E0)=0,hencexEE0isasubmoduledierentfromE.AsEissimple,thisimpliesthatxE=0.WegetthusthatmE=0.AweakversionoftheHilbertNullstellensatzsaysthatmhascodi-mension1inZ(g),hencethatZ(g)=K1+m.SoE=Pri=1Z(g)uivimpliesE=Pri=1Kuiv,hencedim(E)rpdim(g).A.5.ThankstoTheoremA.4wecannowassociatetogthenumberM(g)=maxfdimEjEasimpleg{moduleg:Zassenhausgavein[Za]aringtheoreticinterpretationofM(g).DenotebyF0aeldoffractionsforZ0(g).ThenFrac(U(g))=U(g)\nZ0(g)F0isalocalisationofU(g).Themapu7!u\n1fromU(g)toFrac(U(g))isinjectivebecauseU(g)isfreeoverZ0(g);weuseittoidentifyU(g)withasubringofFrac(U(g)).Eachnon-zeroelementu2U(g)isinvertibleinFrac(U(g))asuisintegraloverZ0(g).ThereforeFrac(U(g))isadivision 4J.C.Jantzenring.ItcontainsaeldoffractionsFforZ(g)andFisthecentreofFrac(U(g)).ItnowturnsoutthatM(g)2=dimFFrac(U(g))cf.[Za],Thms.1and6.Ontheotherhand,wehavedimF0Frac(U(g))=pdim(g)becauseU(g)isfreeofrankpdim(g)overZ0(g).Itfollowsthatpdim(g)=M(g)2
dimF0F,hencethatM(g)isapowerofp.A.6.In[VK],1.2VeisfeilerandKatsmadeaconjectureonthevalueofM(g).Foreachlinearform2gdenotebygitsstabiliseringforthecoadjointaction:g=fX2gjX
=0g=fX2gj([X;g])=0g:ThisisarestrictedLiesubalgebraofg.Foreach2gthebilinearform(X;Y)7!([X;Y])onginducesanon-degeneratealternatingformong=g;thereforethedimensionofthisquotientspaceiseven.Setr(g)=minfdimgj2gg:Thenalsodim(g)r(g)isevenandtheconjecturesays:Conjecture([VK]):M(g)=p(dim(g)r(g))=2.WorkbyMil0nerandbyPremetandSkryabin(see[Mi],Thm.3and[PS],Thm.4.4)shows:Theorem:Ifthereexistsalinearformongsuchthatgisatoralsubalgebraofg,thenthisconjectureholds.(AsubalgebrahofarestrictedLiealgebraiscalledtoralifitiscom-mutativeandifthep{powermapX7!X[p]restrictstoabijectivemaph!h.ThismeansthathisisomorphicasarestrictedLiealgebratotheLiealgebraofatorus.)A.7.LetEbeasimpleg{module.Sincedim(E)1,Schur'slemmaimpliesthateachelementinZ(g)actsasmultiplicationbyascalaronE.Thisappliesinparticulartoall(X)=XpX[p]withX2g.Usingthesemi-linearityofoneshowsnowthatthereexistsalinearformE2gwith(XpX[p])jE=E(X)pidEforallX2g.OnecallsEthep{characterofE.Foreach2gsetU(g)=U(g)=(XpX[p](X)p1jX2g): Liealgebrasinpositivecharacteristic5ThisisanitedimensionalalgebraoverKofdimensionpdim(g).IfX1,X2;:::;Xnisabasisforg,thentheclassesofallXm(1)1Xm(2)2:::Xm(n)nwith0m(i)pforalliareabasisforU(g).OnecallsU(g)areducedenvelopingalgebraofg.(ThespecialcaseU0(g)isusuallycalledtherestrictedenvelopingalgebraofg.)EachsimpleU(g){module(forany)isinanaturalwayasimpleg{module.Thediscussionintherstparagraphofthissubsectionshows:Eachsimpleg{moduleisasimpleU(g){moduleforexactlyone2g.A.8.If\r:g!gisanautomorphismofgasarestrictedLiealgebra(i.e.,aLiealgebraautomorphismwith\r(X[p])=\r(X)[p]forallX2g),then\rinducesanisomorphismU(g)!U\r
(g)where(\r
)(X)=(\r1(X)).Inparticular,ifg=Lie(G)forsomealgebraicgroupGoverK,thenanyg2GactsviatheadjointactionAd(g)ong.EachAd(g)isanautomorphismofgasarestrictedLiealgebra.Sowegetanisomor-phismU(g)!Ug
(g)whereg
referstothecoadjointactionofg.Thisimplies:Ifweknowallsimpleg{moduleswithagivenp{character,thenweknowalsoallsimpleg{moduleswithap{characterinthecoadjointorbitG
.BReductiveLieAlgebrasB.1.AssumefromnowonthatGisaconnectedreductivealgebraicgroupoverKwithamaximaltorusTandaBorelsubgroupB+T.DenotetheLiealgebrasofG,T,B+byg,h,b+respectively.WedenotebyXthecharactergroupofTandsetRXequaltothesetofrootsofGrelativetoT.WedenotebyR+thesetofrootsofB+relativetoT;thisisasystemofpositiverootsinR.Foreach2Rletgdenotethecorrespondingrootsubspaceing.Setn+(resp.n)equaltothesumofallgwith2R+(resp.with2R+).Wehavetheng=nhn+andb+=hn+.FixforeachabasisvectorXforg.TheseelementssatisfyX[p]=0,e.g.,sinceX[p]2gp=0.Ontheotherhand,theLiealgebrahofthetorusThasabasisH1,H2;:::;HmwithH[p]i=Hiforalli.IfaisarestrictedLiesubalgebraofg,thenweshallusuallywriteU(a)=Uja(a)forall2g. 6J.C.JantzenB.2.ForeachY2gonecanndg2GwithAd(g)(Y)2b+,cf.[Bo],Prop.14.25.Oneshouldhaveanalogously:Foreach2gthereexistsg2Gwith(g
)(n+)=0.()Thiswasprovedin[KW],Lemma3.2foralmostsimpleGexceptforthecasewhereG=SO2n+1andp=2.Theirargumentcanbeextendedtoprove()wheneverthederivedgroupofGissimplyconnected.Inmanycasesthereexistsanon-degenerateG{invariantbilinearformong.Wecanuseittoidentifygandg.LetY2gcorrespondto2g.Chooseg2GwithAd(g)(Y)2b+.Thentheimageg
ofAd(g)(Y)vanisheson(b+)?=n+.Soweget()inthiscase.SupposethatGsatises().ItthenfollowsfromA.8thatitsucestodeterminethesimpleU(g){modulesforallwith(n+)=0ifwewanttodescribeallsimpleg{modules.B.3.Let2gwith(n+)=0.TheneachXwith2R+actsnilpotentlyonanyU(n+){modulesince(X)=0andX[p]=0.Thisimplies(usinganinductiveargument)foreachU(n+){moduleMM=0=)Mn+=0:(WewritegenerallyMaforthespaceofxedpointsinamoduleMoveraLiealgebraa.)IfMisaU(b+){module,thenhstabilisesMn+asn+isanidealinb+.Sincehiscommutative,itthenhasacommoneigenvectorinMn+providedM=0.Sowegetinthiscasesomev2M,v=0andsome2hwithHv=(H)vforallH2handwithYv=0forallY2n+.Each2hdenesaonedimensionalb+{moduleKwheren+actsas0andwhereeachH2hactsas(H).ThenKisaU(b+){moduleifandonlyif2whereweset=f2hj(H)p(H[p])=(H)pforallH2hg:NowthelinearfunctionintheprecedingparagraphhastobelongtobecauseMisaU(b+){module.RecallthathhasabasisH1,H2;:::;HmwithH[p]i=Hiforalli.Thesemi-linearityofthemapH7!HpH[p]impliesthatsome2hbelongstoifandonlyif(Hi)p=(Hi)p(H[p]i)=(Hi)p(Hi)foralli.Given,thisshowsthateach(Hi)cantakeexactlypdistinctvalues.Thisimpliesthatjj=pdim(h): Liealgebrasinpositivecharacteristic7B.4.Letagain2gwith(n+)=0.Foreach2wecannowconsidertheU(b+){moduleKandtheinducedU(g){moduleZ()=U(g)\nU(b+)KwhichisoftencalledababyVermamodule.WehavedimZ()=pNwhereN=jR+j.Ifwesetv=1\n1andchooseanumbering1,2;:::;Nofthepositiveroots,thenallXm(1)1Xm(2)2:::Xm(N)Nvwith0m(i)pforalliformabasisforZ().IfMisanon-zeroU(g){module,thenthediscussioninB.3showsthatthereexistssome2withHomb+(K;M)=0.TheuniversalpropertyofthetensorproductimpliesthenHomg(Z();M)=0.Wegettherefore(asobservedin[Ru]):Lemma:IfEisasimpleU(g){module,thenEisahomomorphicimageofsomeZ()with2.B.5.LetusassumethatthederivedgroupofGissimplyconnected.Ifthep{characterofasimpleg{moduleEsatises(n+)=0,thentheresultsinB.4implythatdimEdimZ()forasuitable,hencedimEpN.ByourassumptionGsatisesB.2(),sothisinequalityholdsforallsimpleg{modulesE.Ontheotherhand,oneknowsthatGhasaSteinbergmodulethatisirreducibleofdimensionpNandremainsirreducibleunderrestrictiontog.Thisshows(inthenotationfromA.5)thatM(g)=pN:NotethatthisresultiscompatiblewiththeconjecturementionedinA.6.Sincedim(g)=2N+dim(h),wejusthavetocheckthatdim(h)istheminimaldimensionofallgwith2g.Well,ourassumptiononthederivedgroupofGimpliesthat[g;g]=0forall2R.Thereforewecannd2gwith(n+)=0=(n)and([g;g])=0forall2R.Thensatisesg=h.(Thisshows,bytheway,thattheassumptioninTheoremA.6issatised.)Onecannowusesemi-continuityargumentstoshowthatdimgdim(h)forall.(TheunionoftheorbitsG
withg=hisdenseing.) 8J.C.JantzenB.6.Letusmakefromnowonthefollowingsimplifyingassumptions:(H1)ThederivedgroupofGissimplyconnected.(H2)TheprimepisgoodforG.(H3)Thereexistsanon-degenerateG{invariantbilinearformong.Theassumption(H2)excludesp=2ifRhasacomponentnotoftypeA,itexcludesp=3ifRhasacomponentofexceptionaltype,anditexcludesp=5ifRhasacomponentoftypeE8.IfGisalmostsimpleandandif(H2)holds,then(H3)holdsunlessRhastypeAnwithpjn+1.NotethatG=GLnsatisesallthreeconditionsforallnandp:In(H3)onecantakethebilinearform(Y;Z)=trace(YZ)onLie(G)=Mn(K).Oneniceaspectof(H1){(H3)isthefollowing:WithGalsoeachLevisubgroupofGsatisesthesehypotheses.B.7.Premethasshownin[Pr],Thm.3.10(seealso[PS],Thm.5.6)underourassumptionsorslightlyweakerones(provingaconjecturefrom[VK],3.5):Theorem:Let2g.Thenpdim(G
)=2dividesdim(M)foreachnitedimensionalU(g){moduleM.(ItturnsoutthatunderourassumptioneachorbitG
hasanevendimension;sotheclaimmakessense.)ForanintroductiontoPremet'soriginalproofonemayalsocompare[J3],sections7and8.B.8.Let :g!gbeanisomorphismofG{modulesinducedbyabilinearformasin(H3).Wecanuse totransportnotionslikenilpotentorsemi-simplefromgtogandcall2gnilpotent(orsemi-simple)if 1()isso.Onecanalsodenethesenotionsintrinsicallysayingthatissemi-simpleifandonlyiftheorbitG
isclosedifandonlyifthereexistsg2Gwith(g
)(n+n)=0.Andisnilpotentifandonlyif02 G
ifandonlyifthereexistsg2Gwith(g
)(b+)=0.Ageneral2ghasthenaJordandecomposition=s+nwithssemi-simpleandnnilpotentsuchthat 1()= 1(s)+ 1(n)istheJordandecompositioning.(Again,thiscanbedeneddirectlying,see[KW],section3.)ConsideraJordandecomposition=s+nasaboveandsetl=gs.Ourassumption(H2)impliesthatthereexistsaLevisubgroupLofGwithl=Lie(L).(ReplacingbyanelementinG
,onemayassumethats(n+)=0=s(n).Thenlisequaltothesumofhandallgwiths([g;g])=0.ThesetofallwiththispropertyisconjugateundertheWeylgrouptoasetoftheformR\ZIwhereIisasubsetofthesetofsimpleroots;hereoneusesthatpisgood.)There Liealgebrasinpositivecharacteristic9existsthenaparabolicsubgroupPinGsuchthatPisthesemi-directproductofitsunipotentradicalUPandofL.Thenp=Lie(P)satisesp=luwhereu=Lie(UP).Wehavenow(u)=0.(Thisfollowsfromthefactthat 1(s)and 1(n)commute.)IfweextendaU(l){moduletoap{modulelettingtheidealuactvia0,thenwegetthereforeaU(p){module.Nowonecanshow:Theorem:ThefunctorsV7!U(g)\nU(p)VandM7!Muarein-verseequivalencesofcategoriesbetweenfU(l){modulesgandfU(g){modulesg.Theyinducebijectionsofisomorphismclassesofsimplemod-ules.ThisgoesbacktoVeisfeilerandKatswhoshowedin[VK],Thm.2thatthesimpleU(g){modulesaretheU(g)\nU(p)EwithEasimpleU(l){module.ThemoreprecisestatementhereisduetoFriedlanderandParshall,see[FP1],Thm.3.2andThm.8.5.(Seealso[J3],7.4foraproofofthisresultbasedonatheoremofPremet.)TheunipotentgroupUPsatisesU(u)u=K1.UsingthePBWtheoremonecannowcheckthattheequivalenceofcategoriesM7!MutakeU(g)toadirectsumofpdcopiesofU(l)whered=dim(g=p).ThisimpliesthatU(g)isisomorphictothematrixringMpd(U(l)).B.9.TheoremB.8reducestheproblemofndingthesimpleg{modulestotheinvestigationofthesimpleU(g){modulesfor2gwithgs=gandtotheanalogousproblemforLiealgebrasofsmallerreductivegroupsthatagainsatisfy(H1){(H3).Bydenitiongs=gmeansthats([g;g])=0.Underourassump-tions[g;g]istheLiealgebraofthederivedgroupofG.Sog=[g;g]isarestrictedLiealgebra.LetEbeasimpleU\r(g=[g;g]){modulewhere\r(Y+[g;g])=s(Y)forallY2g.ThenEhasdimension1becauseg=[g;g]iscommutative,andEisaUs(g){modulewhenconsideredasag{module.NowV7!V\nEandV07!V0\nEareinverseequivalencesofcategoriesbetweenfUn(g){modulesgandfU(g){modulesg.ThisshowsthatitsucestostudyUn(g){modules.Sowehaveareductiontothecasewhereisnilpotent.B.10.Beforeweinvestigatethenilpotentcase,letuslookataspecialcaseofTheoremB.8.Supposethatisregularsemi-simple,i.e.,that=sandthatdim(g)=dim(h).ReplacingbyaconjugateunderG,wemayassumethatg=h.Thismeansthat(n+)=0=(n)andthat([g;g])=0forall2R.Wehavel=hinthenotationofB.8andmaychooseP=B+.SoTheoremB.8saysinthiscasethatallU(g){modulesaresemi-simple(sinceU(h){modulesareso)andthat 10J.C.JantzenthesimpleU(g){modulesaretheZ()with2.TheseZ()arepairwisenon-isomorphic.CNilpotentFormsWekeepthroughouttheassumptionsandnotationsfromB.1andB.6.C.1.ByB.8andB.9wemayrestricttothecasewhereisnilpotent.Replacingbysomeg
withg2G,wemayassumethat(b+)=0.Wehavetheninparticular=0.OnecanidentifythemultiplicativegroupoverKwithGL1,henceitsLiealgebrawithM1(K).ThisLiealgebrahasbasisHequaltothe(11){matrix(1).ThisshowsthatH[p]=Handthateachcharacter'n:t7!tnwithn2Zofthemultiplicativegrouphastangentmapd'nmappingHton.AlinearformonM1(K)satises(H)p(H[p])=0ifandonlyif(H)2Fpifandonlyif=d'nforsomen.Furthermore,wehaved'n=d'mifandonlyifnm(modp).ThisimpliesnowforT,adirectproductofmultiplicativegroups,that0=fdj2Xgandthatd=difandonlyif(modpX).WeshalloftenwriteKinsteadofKdandZ()insteadofZ(d)for2Xand2gwith(b+)=0.WehavetokeepinmindthatthenZ()=Z(+p)forall;2X.Let2gwith(b+)=0.WeknowbyB.4thateachsimpleU(g){moduleisthehomomorphicimageofsomeZ()with2X.Theproblemnowisthatisnotnecessarilyuniquelydetermined;weshallseeanexampleofthisinamoment.Furthermore,someZ()mayhavemorethanonesimplehomomorphicimage.Infact,thereexistevenZ()thataredecomposable,see[J3],6.9.C.2.Fix2gwith(b+)=0.ForeachnitedimensionalU(g){moduleMdenoteby[M]theclassofMintheGrothendieckgroupofallnitedimensionalU(g){modules.IfE1,E2;:::;ErisasystemofrepresentativesfortheisomorphismclassesofsimpleU(g){modules,then[E1],[E2];:::;[Er]isabasisoverZforthisGrothendieckgroup.AnarbitraryMthensatises[M]=Pri=1[M:Ei][Ei]where[M:Ei]isthemultiplicityofEiasacompositionfactorofM.LetWdenotetheWeylgroupofGwithrespecttoT.Foreach2Rdenotebys2Wthecorrespondingre\rectiongivenbys()=h;_iwhere_isthecorootto.Weshalloftenusethe\dotaction"ofWonX,givenbyw=w(+)where2X\nZQishalfthesumofallpositiveroots. Liealgebrasinpositivecharacteristic11Proposition:Wehave[Z(w)]=[Z()]forallw2Wand2X.Thiswasrstshownin[H1],Thm.2.2incase=0;theproofinthatcasegeneralises.Itsucestotakew=swithasimpleroot.Letddenotetheintegerwithh+;_id(modp)and0dp.Thensd(modpX);sowehavetoshowthat[Z(d)]=[Z()].Thisistrivialifd=0.Ifd&#x-342;&#x.422;0,thenrank1calculationsshowthatthereexistsahomomorphismofg{modules':Z(d)!Z()givenby'(vd)=Xdv.(WeuseherenotationslikevasinB.4.)If(X)=0,thenXpX[p]=Xpactsasthenon-zeroscalar(X)ponZ().Itthenfollowsthatv=(X)pXpv=(X)pXpd'(vd)belongstotheimageof'.Therefore'issurjective,hencebijectivebydimensioncomparison.Sointhiscase'isanisomorphismZ(d)!Z().If(X)=0,thenonechecks|workingwithbasesasinB.4suchthatN=|thatthekernelof'isgeneratedbyXpdvd.Furthermorethereisahomomorphism fromZ(p)=Z()toZ(d)with (v)=Xpdvd.Onegetsthenker(')=im( )andker( )=im('),hence[Z()]=[ker( )]+[im( )]=[im(')]+[ker(')]=[Z(d)]asclaimed.C.3.If(asinC.2)satises(X)=0forallsimpleroots,thenthe(sketched)proofofPropositionC.2showsthatZ(w)'Z()forallw2Wand2X.Inthiscaseis\regularnilpotent",i.e.,satisesdim(G
)=2dim(g=b)=2N.(IfcorrespondstoY2gunderanisomorphismg!gasinB.8,thenY=P2R+aXwithsuitablea2Kandwitha=0forallsimpleroots.Suchelementsingareregularnilpotent,see[J6],6.7(1).)NowTheoremB.7saysinthiscasethatpNdividesthedimensionofeachU(g){module.SinceallZ()havethisdimension,theyhavetobesimple.Wehavethusshownmostofthefollowingresult(provedin[FP1],4.2/3forcertaintypes,in[FP2],2.2{4ingeneralunderslightlymorerestrictiveconditionsonp;forG=SLnseealso[P1],Thm.5):Proposition:Let2gwith(b+)=0and(X)=0forallsimpleroots.TheneachZ()with2Xissimpleandeachsimple 12J.C.JantzenU(g){moduleisisomorphictosomeZ()with2X.Given;2XwehaveZ()'Z()ifandonlyif2W+pX.C.4.Theonlyargumentmissingaboveistheproofoftheclaim:IfZ()'Z(),then2W+pX.Forthisonelooksatthesubalge-braU(g)GofallAd(G){invariantelementsinU(g).ThissubalgebraiscontainedinthecentreZ(g)ofU(g).Ifwewereworkingincharacteris-tic0,thenU(g)GwouldbeallofZ(g).Inourpresentset-up,however,therearemanyelementsoftheformYpY[p]withY2gthatbelongtoZ(g),butnottoU(g)G.OnecanshowthatZ(g)isgeneratedbyU(g)GandZ0(g),cf.[BG],Thm.3.5.(However,thisdoesnotimplythatthecanonicalmapU(g)!U(g)mapsU(g)GontothecentreofU(g);seethecounter-examplebyPremetin[BG],3.17.)Onechecksnowthatthereexistsforeachu2U(g)Gandeach2Xascalarcen(u)2Ksuchthatuv=cen(u)vinZ()forall2gwith(b)=0.Itthenfollowsuactsasmultiplicationwithcen(u)onallofZ(),hencealsoonallcompositionfactorsofZ().EachcenisanalgebrahomomorphismfromU(g)GtoK.NowonehasanalogouslytotheHarish-Chandratheoremincharacteristic0:Proposition:If;2X,thencen=cenifandonlyif2W+pX.Infact,onehasasincharacteristic0aHarish-ChandraisomorphismU(g)G!U(h)W.Thiswasrstprovedin[H1],Thm.3.1forlargep(largerthantheCoxeternumber)andin[KW]foralmostsimpleG.(Theargumentsthereextendtothepresentset-up.See[J3],9.6foraproofthatusesreductionmoduloptechniques,asin[H1].)C.5.PropositionC.4hasanobviouscorollaryforthedescriptionoftheblocksofU(g):IftwosimpleU(g){modulesEandE0belongtothesameblock,thenU(g)GhastoactviathesamecharacteronbothEandE0.So,ifEisacompositionfactorofZ()andifE0isoneofZ(0),andifE,E0belongtothesameblock,then02W+pX.ButthenPropositionC.2impliesthatE0isalsoacompositionfactorofZ().Thisprovestheimplications\(i))(ii))(iii)"and\(iii))(ii)"in:Proposition:Let2gwith(b+)=0.LetEandE0besimpleU(g){modules.Thenthefollowingareequivalent:(i)EandE0belongtothesameblockofU(g).(ii)U(g)GactsviathesamecharacteronEandE0. Liealgebrasinpositivecharacteristic13(iii)Thereexists2XsuchthatbothEandE0arecompositionfactorsofZ().Remark:If(i)or(ii)holdsforone,thenitholdsforallg
withg2G.Thereforetheequivalenceof(i)and(ii)holdsforallnilpotent2g,notonlyforthosewith(b+)=0.Notethattheimplication\(iii))(i)"[thatwedidnotprovehere]isobviousinthecaseswhereallZ()areindecomposable.Thisholdsfor=0by[H1],Prop.1.5,andmoregenerallyforinstandardLeviform,seeD.1below.(Itactuallysucestondforeach2Xone02W+pXsuchthatZ(0)isindecomposable;thatisinmanyadditionalcasespossible,forexamplealwayswhenRhasnocomponentofexceptionaltype,see[J5],C.3andH.1.)Therstgeneralproofof\(ii))(i)"wasgivenin[BG],Thm.3.18(assumingp�2).Amoredirectproof(thatworksalsoforp=2)isduetoGordon,see[Go],Thm.3.6.Letm0denotethenumberoforbitsofWonX=pXwithrespecttothedotaction.Theimplication\(i))(iii)"showsthatthenumberofblocksofU(g)isatleastequaltom0forall2(b+)?=f2gj(b+)=0g.Sotheimplication\(iii))(i)"isequivalenttotheclaimthateachU(g)hasatmostm0blocks.NowonechecksforeachmthatthesetDm=f2(b+)?jU(g)hasatmostmblocksgisclosedin(b+)?.PropositionC.3impliesthatallregularnilpotentelementsin(b+)?belongtoDm0.Astheseelementsaredensein(b+)?,wegetDm0=(b+)?,hencetheclaim.C.6.Semi-continuityargumentsliketheoneusedinC.5canbeusedformanypurposesinthepresenttheory.Theyoftenrelyonthefollowingobservation:Fix2Xandchooseanumbering1,2;:::;Nofthepositiveroots.Forall2(b+)?andm=(m(1);m(2);:::;m(N))inZNwith0m(i)pforalliletzm;denotethebasiselementXm(1)1Xm(2)2:::Xm(N)NvofZ(),cf.B.4.TherearethenforallY2gandallm,nelementscn;m(Y;)2KsuchthatYzm;=Xncn;m(Y;)zn;cf.theproofofA.7(2)in[J5].Eachcn;m:g(b+)?!KisalinearfunctionofY2gandapolynomialfunctionof2(b+)?.Usingthisonecancheckthatbothf2(b+)?jZ()issimplegandf2(b+)?jZ()isprojectiveg 14J.C.Jantzenareopensubsetsof(b+)?.TheyarealsoAd(T){stable.Itiseasytoseethat02 Ad(T)
forall2(b+)?.SoifZ()isnotsimple(ornotprojective)forsome2(b+)?,thenalsoZ0()isnotsimple(ornotprojective).NowclassicalresultsontheSteinbergmoduleintherestrictedcase(cf.[J1],II.3.18andII.10.2)yield:Proposition:Let2Xwithh+;_i0(modp)forallroots.ThenZ()issimpleandprojectiveforall2(b+)?.KacseemstohavehadinmindaproofalongthelinesindicatedabovewhenheclaimedthesimplicityofZ()forthesein[Ka].Proofsoftheseresultsappearedin[FP2],Thms.4.1/2.Comparealso[BG],Cor.3.11.Remark:Letmeindicatehowonecanprovethatthetwosetsaboveareopenin(b+)?orratherthattheircomplementsareclosed.AllZ()havedimensionr=pN;wehavebijectionsf:Kr!Z()mappingafamily(am)mtoPmamzm;.SetNdequaltothesetofall2(b+)?forwhichZ()hasasubmoduleofdimensiond.WewanttoshowthateachNdwith0drisclosedin(b+)?.LetGd;rdenotetheGrassmannianofalld{dimensionalsubspacesofKr.ThedescriptionoftheactionofY2gonzm;impliesthatthesetMd(Y)ofall(V;)2Gd;r(b+)?withYf(V)f(V)isclosed.HencesoistheintersectionMdofallMd(Y)withY2g.NowthesecondprojectionmapsMdontoNd,andthisimageisclosedbecauseGd;risacompletevariety.InordertogettheclaimonprojectivityIshallusesupportvarieties.SetNp(g)equaltothesetofallx2gwithx[p]=0.Thisisaclosedandconicsubvarietyofg.SotheimagePNp(g)ofNp(g)nf0gintheprojectivespaceP(g)isclosed.ForanyU(g){moduleMsetg(M)equaltothesetofallKx2PNp(g)suchthattherankofx(x)actingonMisstrictlylessthan(p1)dim(M)=p.ThenMisaprojectiveU(g){moduleifandonlyofg(M)=;,see[FP1],Thm.6.4.NowKx2PNp(g)belongstog(Z())ifandonlyifall(mm){minorswithm=(p1)pN1ofthematrixofx(x)withrespecttothezm;are0.Thereforethesetofall(Kx;)2PNp(g)(b+)?withKx2g(Z())isclosed.HencesoisitsimageunderthesecondprojectionsincePNp(g)isacompletevariety.Thatimageisexactlythesetofall2(b+)?suchthatZ()isnotprojective.DStandardLeviFormWekeepthroughoutthesameassumptionsandnotationsasinthepre-cedingsection. Liealgebrasinpositivecharacteristic15D.1.Alinearform2gissaidtohavestandardLeviformif(b+)=0andifthereexistsasubsetIofthesetofsimplerootssuchthat(X)=0forall2Iwhile(X)=0forall2R+nI.(Thisdenitiongoesbackto[FP2],3.1.)IfhasstandardLeviform,then([n;n])=0and(n[p])=0.Thisimpliesthatdenesaone-dimensionaln{modulethatisthenaU(n){module.Sincenisunipotent,thisistheonlysimpleU(n){module(uptoisomorphism),cf.[J3],3.3.ItthenfollowsthatU(n)istheprojectivecoverofthissimplemodule,hencehasauniquemaximalsubmodule.EachZ()with2XisisomorphictoU(n)asann{module.Anyproperg{submoduleofZ()isthencontainedinthatuniquemaximaln{submodule.Takingthesumofalltheseg{submoduleswesee:Lemma:If2ghasstandardLeviform,theneachZ()with2Xhasauniquemaximalsubmodule.WethendenotebyL()theuniquesimplequotientofZ().LemmaB.4tellsusnowthateachsimpleU(g){moduleisisomorphictosomeL()with2X.However,willnotbeunique:WehaveatleastL()'L(+p)forall2X;buttheremaybeadditionalisomorphisms.D.2.BeforereturningtothequestionwhenL()'L(0)inD.1,letuslookatthespecialcase=0thatclearlyhasstandardLeviform.Thiswastherstcasetobeinvestigated.IfVisaG{module(i.e.,avectorspaceoverKwitharepresentationG!GL(V)thatisahomomorphismofalgebraicgroups),thenVbecomesag{moduletakingthetangentmapat1oftherepresentation.OnegetsthusU0(g){modules.ThesimpleG{modulesareclassiedbytheirhighestweight.Thereisforeach2Xwith0h;_iforall2R+asimpleG{moduleL()withhighestweight;eachsimpleG{moduleisisomorphictoexactlyoneoftheseL(),cf.[J1],II.2.7.NowCurtisprovedin[Cu]:Theorem:a)TheL()with2Xand0h;_ipforallsimplerootsaresimplealsoasg{modules.b)EachsimpleU0(g){moduleisisomorphictooneoftheseL().If;02Xbothsatisfytheconditionina),thenL()andL(0)areisomorphicasg{modulesifandonlyif02pX.Note:if;02Xbothsatisfytheconditionina)andif02pX,thenh;_i=h0;_iforall2R;incaseGissemi-simplethisimpliesthat=0. 16J.C.JantzenIfonecompareswiththesetupinD.1,specialisedtothecase=0,thenwegetthatL()withasina)isisomorphicasag{moduletoL0().Andwegetforall;2XthatL0()'L0()ifandonlyif(modpX).D.3.ReturntothemoregeneralsituationinD.1andconsider2ginstandardLeviformwithasetIofsimplerootsasinthedenition.The(sketched)proofofPropositionC.2showsthatZ()'Z(w)forall2Xandw2WIwhereWIisthesubgroupoftheWeylgroupWgeneratedbyallre\rectionsswith2I.Wegetthusonedirectionof:Proposition:SupposethathasstandardLeviformandthatI=f2Rj(X)=0g.Let;2X.ThenL()'L()ifandonlyif2WI+pX.The\onlyif"partisprovedin[FP2],3.2/4;forG=SLnseealso[P1],Thm.3.GeneralisationswerefoundbyShenGuangyuandbyNakano,cf.[J3],10.7.D.4.TheclassicationaboveofthesimpleU(g){modulesforinstandardLeviformleadsimmediatelytoaclassicationofthesimpleU(g){modulesforintheG{orbitofsome0instandardLeviform.Consideranisomorphismg!garisingfromabilinearformasin(H3).IfhasstandardLeviformandifI=f2Rj(X)=0g,thencorrespondsunderthisisomorphismtosomeY2gthatisregularnilpotentintheLevifactorgIofgspannedbyhandallgwith2R\ZI.IfG=GLnorG=SLn,thentheclassicationofnilpotentorbitsingbytheJordannormalformshowsthateachnilpotentelementingisconjugatetoaregularnilpotentoneinsomeLevifactor,henceeachnilpotentlinearformongconjugatetooneinstandardLeviform.ThereforePropositionD.3andtheearlierreductionsyieldacompleteclassicationofallsimpleg{modulesincaseGisisomorphictosomeGLn(oraproductofsuchgroups).AlsoforRoftypeB2orB3eachnilpotentlinearformongisconjugatetooneinstandardLeviform.Thisisnolongertruefortheothertypes.InthosecasesPropositionD.3doesnotyieldacompleteclassicationofallsimplemodules.D.5.LetusstaywithinstandardLeviform.Havingachievedaclassicationofthesimplemodulesonewouldliketoknowmoreabouttheirstructure;atleasttheirdimensionsoughttobedetermined.Inthecase=0Lusztig'sconjecturein[L1]onformalcharactersofsimpleG{modulesyieldsviaTheoremD.2aconjectureforsimpleg{modules Liealgebrasinpositivecharacteristic17thatdetermines(amongotherthings)theirdimensionsprovidedpisnottoosmall.OnemayhopethatpgreaterthantheCoxeternumberofRwilldo.ItisknownthatLusztig'sconjectureistrueforplargerthananunknownboundthatdependsontherootsystem,see[AJS].ThereisasimilarconjectureforanyinstandardLeviform.Inordertoformulateit,wehavetoreplacethecategoryofU(g){modulesbyacertaincategoryofgradedU(g){modules.Fix2ginstandardLeviformandsetI=f2Rj(X)=0g.TheenvelopingalgebraU(g)isinanaturalwayZR{gradedsuchthateachXhasdegreeandeachH2hdegree0.However,itwillnowbemoreusefultoregardU(g)asZR=ZI{gradedsuchthateachXhasdegree+ZIandeachH2hdegree0+ZI.ThishastheadvantagethatthekernelofthecanonicalmapU(g)!U(g)ishomogeneous:ItisgeneratedbyallHpH[p]withH2h|homogeneousofdegree0+ZI|andbyallXp(X)pwith2R|homogeneousofdegreep+ZIsincep2ZIwhenever(X)=0.WegetnowaZR=ZI{gradingonU(g).Itwillbeconvenienttore-gardthisasagradingbythelargergroupX=ZIandnowtostudyX=ZI{gradedU(g){modules.Forexample,wecangiveeachZ()with2XagradingsuchthateachbasiselementXm(1)1Xm(2)2:::Xm(N)NvasinB.4ishomogeneousofdegreePNi=1m(i)i+ZI.DenoteZ()withthisgradingbybZ().Generalresultsongradedmodules(cf.[J4],1.4/5)implythattheradicalofanyZ()isagradedsubmodule.ItfollowsthateachL()hasagradingsuchthattheimageofvinL()ishomogeneousofdegree+ZI.DenotethisgradedmodulebybL().NotethatanybZ(+p)isjustbZ()withthegradingshiftedbyp+ZI,similarlyforbL(+p)andbL().If2ZI,thenwestillhavebZ(+p)'bZ();butthisisnolongertruewhen=2ZI.Onecannowshow(see[J3],11.9):Proposition:Let;2X.ThenbL()'bL()()bZ()'bZ()()2WI+pZI:D.6.KeeptheassumptionsonandIuntiltheendofSectionD.WebeganinD.5withadiscussionofgradedU(g){modules.However,weshallnotconsiderallpossiblemodulesofthistype.IfM=L\r2X=ZIM\risanX=ZI{gradedU(g){module,theneachM\rish{stable,henceadirectsumofweightspacesforh.Wenowmaketheadditionalcondition: 18J.C.JantzenIf\r=+ZIwith2X,thenallweightsofhonM\rhavetheformd(+)with2ZI.WedenotebyCthecategoryofallX=ZI{gradedU(g){modulessatisfyingthiscondition.(Incase=0wehaveI=;anddealherewithX{gradedU0(g){modulesM=L2XM.InthiscasetheextraconditionmeansthathactsoneachMviad.ItfollowsthatwecanidentifyCfor=0withthecategoryofG1T{modulesasin[J1],II.9.)Nowoneeasilychecks(inthegeneralcase)thatallbZ()belongtoC.Sodoalltheircompositionfactors,inparticularallbL().Anargu-mentlikethatinB.4showsthateachsimplemoduleinCisisomorphictosomebL()with2X,cf.[J4],2.5.SetCI=f2Xj0h+;_ipforall2R+\ZIg:ThisisafundamentaldomainforthedotactiononXoftheanere\rectiongroupgeneratedbyWIandbythetranslationsbyallpwith2I.ThereforePropositionD.5implies:Corollary:EachsimplemoduleinCisisomorphictoexactlyonebL()with2CI.D.7.LetWpdenotetheaneWeylgroupofR,generatedbyWandthetranslationsbyelementsinpZR.Itisalsogeneratedbyallanere\rectionss;rpwith2Randr2Zwheres;rp()=(h;_irp).WeshallusuallyconsiderthedotactionofWponXwherew=w(+).LetdenotetheusualorderrelationonX:Wehaveifandonlyif2P2R+N.There\rectionss;mp2WpareusedtodeneanotherorderrelationonXthatwillbedenotedby".If2R+,m2Z,and2X,thenwesaythats;mp"ifandonlyifh+;_imp.Ingeneral"isdenedasthetransitiveclosureofthisrelation.Itisthenclearthat"impliesand2Wp.(Cf.[J1],II.6.4.)Onehasnowastronglinkageprinciple,see[J3],11.11and[J4],4.5:Proposition:Let;2CI.If[bZ():bL()]=0,then".ThisisanalogoustoclassicalresultsforG{modulesorG1T{modules(thecase=0here),cf.[J1],II.6.16andII.9.12.Wehavefurthermore(see[J4],2.8(1))[bZ():bL()]=1forall2CI. Liealgebrasinpositivecharacteristic19Itfollowsthatthe(innite)\decompositionmatrix"ofall[bZ():bL()]islowertriangularwithonesonthediagonal(withrespecttosometotalorderingonXthatrenes.Thiswillbecrucialforuswhenweshallwanttouseinformationonthisdecompositionmatrixtodeterminecharactersanddimensionsofthesimplemodules,cf.thediscussioninD.12below.WithoutintroducingthegradingthiswouldnotworkbecausePropositionC.2saysthatallZ()inablockhavethesamecompositionfactors.D.8.OnepossibleproofofPropositionD.7involvesasumformulaforacertainltration:EachbZ()hasaltrationbZ()bZ()1bZ()2


withbZ()=bZ()1=bL()andbZ()i=0fori0suchthatforallsimplemodulesEinCXi�0[bZ()i:E]=X(Xi0[bZ((ip+n)):E]Xi�0[bZ(ip):E])wherenistheintegerwith0npandh+;_in(modp)andwherewesumoverall2R+nZIwithnp.Thisformulaisprovedin[J4],3.10generalisingthecase=0treatedin[AJS],6.6.(TheinnitesumintheformulamakessensebecauseonecancheckforeachEthatthereareonlynitelymanynon-zerotermsinthesum.)D.9.ThesumformulainD.8isalsooneofthemaintoolsindetermin-ingallmultiplicities[bZ():bL()]in\easy"cases.OtherimportanttoolsusedtherearePremet'stheoremB.7,translationfunctors,andin-decomposableprojectivemodules.Projectivemoduleswillbediscussedlateron,seeG.1.ThetranslationfunctorsonCaredenedinawaysimilartotheoneusedforG{orG1T{modules(forthesecompare[J1],II.7andII.9.19)andtheyhavesimilarproperties:Translationfroma\regularweight"(i.e.,onewithtrivialstabiliserinWp)toanarbitraryweighttakesababyVermamoduletoababyVermamodule,andittakesasimplemoduletoasimplemoduleorto0,andoneknows,whenonegets0,see[J4],4.9and4.11.UsingsuchtechniquesIhavebeenabletodetermineall[bZ():bL()]inthefollowingcases(sometimesunderadditionalrestrictionsonp):G AnBnB2An+1G2Dn I An1Bn1A1A1An1eA1Dn1 20J.C.JantzenHereGisassumedtobesemi-simpleandsimplyconnectedofthetypementionedintherstrow.ThesecondrowthendescribesthetypeoftherootsystemR\ZI.InthecasesoftworootlengthsIoftypeA1(resp.eA1)meansthatIconsistsofalong(resp.short)simpleroot.Forn=2theBn1underBnhastobeinterpretedaseA1.Thersttwocasesinthetableweretreatedin[J2];theretheonlyrestrictionsonparethoseimposedby(H3)or(H2):Onehasp-n+1fortypeAn,andp=2fortypeBn.(IfoneworksintherstcasewithGLn+1insteadofSLn+1,thennorestrictiononpisneededaccordingto[GP],Rem.9.3.)Thecase(B2;A1)wasrstdealtwith(forp=2)bybruteforcecalculationsinanAarhuspreprint(1997:13);in[J4],5.2{10thiscaseistreatedforp�3usingthegeneralideasmentionedabove.IntheremainingcasesIassumethatpislargerthantheCoxeternumberofR.FileswiththelengthycalculationsintheAn+1andDncasesareavailablefrommeuponrequest.D.10.TheexplicitresultsreferredtoinD.9conrmineachcaseaconjecturebyLusztig.Asinthecase=0(seeD.5)onewillhavetoexpectsomerestrictionsonpforthisconjecturetobetrue;onemayalsoherehopethatpgreaterthantheCoxeternumberofRwillsuce.Theconjecturesaysthatanymultiplicity[bZ():bL()]with;2CIisthevalueofacertainpolynomialat1or1(dependingonsomenormalisationofthepolynomial).Thesepolynomialswererstconstructedin[L2]whereLusztigthen(in13.17)expresseshishopethattheywouldplayarolesimilartothatofsomepreviouslyconstructedpolynomialsinthecase=0.(Seealsotheexplicitformulationin[J3],11.24.)Asinothersituationsonemayspeculatewhetheralsothecoecientsofthesepolynomialshavearepresentationtheoreticinterpretation.Forexample,onemayaskwhethertheyyieldthemultiplicitiesinthefactorsofsubsequenttermsintheltrationmentionedinD.8orintheradicalltrationofbZ().[Theseltrationsmaywellcoincide.]SetA0=f2Xj0h+;_ipforall2R+g:Thisisthesetofintegralweightsintheinteriorofthe\rstdominantalcove"withrespecttoWp.ThisistheintersectionofXwiththe\rstdominantalcoveoverR"asin[J1],II.6.2(6).LetusassumethatA0=;,i.e.,thatpisatleastequaltotheCoxeternumber,cf.[J1],II.6.2(10).WecallsetsoftheformwA0withw2Wpalcoves(inX).WedenotethesetofallalcovesbyA;thenw7!wA0isabijectionfromWpontoA.SetAIequaltothesetofallalcovesA2AwithACI. Liealgebrasinpositivecharacteristic21Choose02A0.ForeachA2AletAdenotetheuniqueelementinA\Wp0;ifA=wA0withw2Wp,thenA=w0.WehavenowCI\Wp0=fAjA2AIg.Lusztig'sconjectureintheversionof[L5],17.3says:Conjecture([L5]):[bZ(A):bL(B)]=B;A(1)forallA;B2AI.HeretheB;AareLusztig's\periodicpolynomials"normalisedasin[L5],9.17.ThankstothelinkageprincipletheconjecturewoulddetermineallcompositionfactorsofallbZ()with2CI\Wp0.ThentranslationfunctorswouldyieldallcompositionfactorsofallbZ()with2CI.SincebabyVermamodulesarenitedimensional,thereareforeachA2AIonlynitelymanyB2AIwith[bZ(A):bL(B)]=0.Accord-ingtoConj.9.20(b)in[L5]theB;Aareexpectedtohavealternatingsigns.ThereforeB;A=0shouldimplyB;A(1)=0.SogivenAthereshouldbeonlynitelymanyB2AIwithB;A=0.Thatisknowntoholdinmanyexamples,butisonlyconjecturedbyLusztigingeneral,see[L5],12.7/8.Also,itisnotclearwhetherthereisagoodalgorithmforcomputingtheB;A,cf.[L2],13.19.D.11.TheformalcharacterofanitedimensionalX=ZI{gradedU(g){moduleM=L2X=ZIMisdenedaschM=X2X=ZIdim(M)e()2Z[X=ZI]wherethee()with2X=ZIformthecanonicalbasisofthegroupringZ[X=ZI].NotethatZR=ZIisafreeabeliangroupofniterank:ThecosetsofthesimplerootsnotinIareabasis.ThereforethegroupringZ[ZR=ZI]isalocalisedpolynomialring,henceanintegraldomain.ThelargergroupringZ[X=ZI]containsZ[ZR=ZI]asasubringandisafreemoduleoverthissubring.ThestandardbasisofbZ()showsthatchbZ()=pN(I)e(+ZI)Y2R+nZI1e(p) 1e()(1)forall2X,whereN(I)=jR+\ZIj.ItisthenclearthatchbZ()=e(+ZI)chbZ()(2) 22J.C.Jantzenforall;2X.Ontheotherhand,wehavebL(+p)=e(p+ZI)chbL()(3)forall;2Xbecauseaddingponlyamountstoashiftofthegrading.WegetfromtheresultsinD.7thatforall2CIchbZ()=chbL()+X;2CI[bZ():bL()]chbL():ItfollowsthatwecanwriteeachchbL()asa(usuallyinnite)linearcombinationoftheformPachbZ()withalla2Zanda=1.Suchinnitesumsmakesensebecauseeache()with2X=ZIoccursonlyinnitelymanychbZ(),2CIwithanon-zerocoecient.Hereisaneasyexamplethatshowsalsohowonemayreplaceaninnitesumbyaniteone.ConsiderG=SL2and=0.Denotetheonlypositiverootby.Consider2Xandanintegerdwithh+;_id(modp)and0dp.Itiseasytoshow(andaclassicalresult)foralli2Zthat[bZ(ip)]=[bL(ip)]+[bL((ip+d))]and[bZ((ip+d))]=[bL((ip+d))]+[bL((i+1)p)]intheGrothendieckgroup.ItfollowsthatchbL()=Xi0chbZ(ip)chbZ((ip+d)):ComparingwiththeanalogousformulaforchbL(p)onegetsusing(2)and(3)(1e(p))chbL()=chbL()chbL(p)=chbZ()chbZ(d)=(1e(d))chbZ():Nowplugin(1):(1e(p))chbL()=(1e(d))(1e(p)) 1e()e()andcancelthecommonfactor1e(p):chbL()=1e(d) 1e()e()=d1Xj=0e(j):Notethatsuchacancellationisquitegenerallypermittedbecauseweworkinafreemoduleoveranintegraldomain,seeabove. Liealgebrasinpositivecharacteristic23D.12.IwanttoconcludethissectionbyshowingthatitwillalwaysbepossibletogeteachchbL()asanitesummodulotwoconjecturesbyLusztig.ForthisIneedsomepropertiesofthepolynomialsB;AandthereforehavetolookcloseratLusztig'sconstructions.LusztigworkswiththelocalisedpolynomialringA=Z[v;v1]intheindeterminatevandheconsidersthefreeA{moduleMcwithbasisAIaswellastwocompletions,MandM,ofMc.Inordertodenethem,weneedanorderrelationonA:WesetABifandonlyifA"Bintheset-upofD.10.(Thisistheorderrelationonthealcovesdenotedby"in[J1],II.6.5.)Thesupportofafunctionf:AI!AisthesetfA2AIjf(A)=0g.WeidentifyMcwiththeA{moduleofallfthathavenitesupport.ThenM(resp.M)consistsofallfunctionsf:AI!Awhosesupportisboundedabove(resp.below)relativeto.Wewritesuchfunctionsasformalsumsf=PA2AIf(A)A.In[L5],9.17/19Lusztigintroduces(foreachB2AI)elementsB=PAB~A;BA2MandB=PABB;AA2MandB=PABA;BA2Mwherethecoecients(~A;Betc.)belongtoZ[v1].Moreprecisely,wehave~B;B=1and~A;B2v1Z[v1]ifA=B,simi-larlyfortheotherpolynomials.Lemma11.7(b)in[L5]says@(CkB)=C;Bwhichmeansbythedenitionin[L5],11.6thatXAB;AA;C=C;B(1)forallB;C2AI.ThefundamentaldomainpropertyofCIshowsforallA2AIand2ZRthatthereexistsanalcove\r(A)2AIsuchthat\r(A)=w(A+p)+pforsuitablew2WIand2ZI.If2ZI,thenobviously\r(A)=AforallA.ItfollowsthatwegetanactionofZR=ZIonAI.WedenotethisactionbyandwriteA=\r(A)if=+ZI.Thisactionpreserves,see[L2],2.12(c).TheelementB2MisdeterminedbytheformofthecoecientsstatedabovetogetherwithitsinvarianceunderacertaininvolutiononM.ThisinvolutioncommuteswiththeactionofZR=ZIdescribedabove.Usingthisonecancheckthat(B)=(B)forallB,hencethatB;A=B;AforallB;A.Similarresultsholdfortheotherpolynomials.WeshallusebelowthecorrespondingresultfortheB:A;B=A;B:(2)In[L5],8.3Lusztigconstructsahomomorphism0I:ZR!Zsuchthat0I()=2forall2Iandsuchthatthereareintegerscwith 24J.C.Jantzen0I()=P2Ich;_iforall2ZR.Then0Iinducesahomomor-phism7!"fromZR=ZItof1gsuchthat"+ZI=(1)0I()forall.LusztigextendstheA{modulestructureonMc,M,andMtothegroupalgebraA[ZR=ZI]bysettinge()A="(A)forallA2AIand2ZR=ZI.Thereisacertainelementz=Pze()2A[ZR=Z]withB=zB,see[L5],9.19.ThismeansthatB=XAXA;B"z(A)=XAX()1A;B"zA;hence~A;B=X"z()1A;B=X"zA;B(3)whereweused(2)forthelaststep.Combining(3)and(1)wegetnowXAC;A~A;B=X"zC;B:(4)IfnowLusztig'sconjectureasinD.10holds,thenwegetforallB2AI(intheGrothendieckgroup)XAB~A;B(1)[bZ(A)]=X"z(1)[bL(B)]:(5)ThismeansonthecharacterlevelforeachBXAB~A;B(1)ch(bZ(A))=ch(bL(B))X"z(1)e(p):(6)Thereisaringhomomorphism':A[ZR=ZI]!Z[ZR=ZI]with'(v)=1and'(e())="e(p)forall.Inthisnotationtherighthandsidein(6)isequalto'(z)ch(bL(B)).Theelementzisintroducedin[L5],Lemma8.15.Bythatlemma(comparealsotheproofofLemma3.2.ain[L5])zisaproductoffactorsoftheform1vme()withm2Zand2(ZR=ZI)n0,inparticularwithe()=1.Itfollowsthat'(z)=0.Thecharactersin(6)belongtoafreemoduleoverZ[ZR=ZI].Thereforech(bL(B))isuniquelyde-terminedby'(z)ch(bL(B))andwecanuse(6)tocomputeit.Finally,LusztigconjecturesthatforeachBoneshouldhaveonlynitelymanyAwith~A;B=0,see[L5],12.7/8.Sothelefthandsidein(6)shouldbeanitesum. Liealgebrasinpositivecharacteristic25ERepresentationsandSpringerFibresWekeepthroughoutthesameassumptionsandnotationsasinthepre-cedingsection.LethdenotetheCoxeternumberoftherootsystemR.E.1.SetB=G=B+;thisisthe\ragvarietyofG.ItcanbeidentiedviagB+7!gB+g1withthesetofallBorelsubgroupsofG,orviagB+7!Ad(g)(b+)withthesetofallBorelsubalgebrasofg.Foreach2gsetB=fgB+2Bj(Ad(g)(b+))=0g:ThisisaclosedsubvarietyofB.UsingtheidenticationabovewecanthinkofBasthesetofallBorelsubalgebrasofgcontainedinthekernelof.NotethatB=;ifandonlyif(g1
)(b+)=0forsomeg2Gifandonlyifisnilpotent,cf.B.8.Supposethatweidentifygandgusingabilinearformasin(H3)andthatY2gcorrespondstosomenilpotent2g.ThenwegetB=BYwhereBYisthesetofallgB+withY2Ad(g)(n+).SoBisequaltothebreoverYinSpringer'sresolutionofthenilpotentconeing,cf.[J6],6.6(3).WethereforecallBtheSpringerbreof.TheSpringerresolutionanditsbreshaveplayedanimportantroleinotherpartsofrepresentationtheory,e.g.,inSpringer'stheoryofWeylgrouprepresentations.Humphreyshassuggestedforsometime(cf.[H3],23)thattheremightbeconnectionsbetweenthetheoryofU(g){modulesandthegeometryofB.Thismotivatedgeometriccon-structionsofrepresentationsbyMirkovicandRumyninin[MR].LusztigmadethensomeexplicitconjecturesinvolvingB(in[L3],[L4],and[L5])partofwhichhasnowbeenprovedbyBezrukavnikov,Mirkovic,andRumyninin[BMR].Weshalldescribetheirmainresultsinthissection.E.2.RecallfromC.2thesubalgebraU(g)GofthecentreZ(g)ofU(g)andthealgebrahomomorphismscen:U(g)G!K.ForanynilpotentandanythesimpleU(g){modulesonwhichU(g)GactsviacenarethesimplemodulesinoneblockofU(g),seeC.5.Forany2gletU0denotethequotientalgebraofU(g)bytheidealgeneratedbytheimageofker(cen0)underthecanonicalmapU(g)!U(g).SothesimpleU0{modulesarefornilpotentthesimpleU(g){modulesinonespecicblockofU(g).Nowoneofthemainresultsin[BMR](Cor.5.2.2andSection6)is:Theorem:Supposethatp�2h2.Let2gbenilpotent.ThenthenumberofsimpleU0{modulesisequaltotherankoftheGrothendieck 26J.C.JantzengroupofthecategoryofcoherentsheavesontheSpringerbreB.Thisrankisalsoequaltothedimensionofthel{adiccohomologyofB.HerehistheCoxeternumberofR;oneshouldexpectthatthetheoremshouldextendtoallp�h.TakeforexampleinstandardLeviformandIasinthedenitioninD.1.ThenPropositionD.3showsthatthenumberofsimpleU0{modulesisequaltojW=WIjincasep�h.Thisiscompatiblewiththepresenttheorem.Forphthenumbergetssmallerindicatingthatthetheoremcannotextendtosuchp.Thetheoremproves(forp�2h2)aconjectureofLusztig.InhisformulationtheconjectureactuallydidnotinvolveB,butananalogueoverC:LetGCdenotetheconnectedreductivegroupoverCwiththesamerootdataasG,setgC=Lie(GC)andletBCdenotethe\ragvarietyofGC.UnderourassumptiononpwehaveabijectionbetweenthesetofnilpotentorbitsingandthesetofthoseingC,hencealsoabijectionbetweenthesetofnilpotentorbitsingandthesetofthoseingC.Chooseanilpotent(C)2gCintheorbitcorrespondingtothatof.ThenLusztigpredictsthatthenumberofsimpleU0{modulesshouldbeequaltothedimensionoftheordinarycohomologyofBC(C)withcoecientsinaeldofcharacteristic0.However,thatdimensionisequaltothedimensionofthel{adiccohomologyofB,thankstoworkbyLusztig,cf.[BMR],6.4.3.E.3.Let2gbenilpotent.TheoremE.2yieldsforp�2h2thenumberofsimpleU(g){modulesinaspecicblockofU(g).Onemayaskabouttheremainingblocks.Considerrst2Xsuchthat+pXhastrivialstabiliserforthedotactiononX=pX.ThenthenumberofsimpleU(g){modulesintheblockdeterminedbycenisequaltothenumberofsimpleU0{modules.Infacttherearetranslationfunctorsthatareequivalencesbetweentheblocksdeterminedbycenandbycen0andthatinduceinversebijectionsbetweenthetwosetsofsimplemodules,cf.[J5],Prop.B.5.Itisnotclearwhatwillhappenwhen+pXhasanon-trivialstabiliserforthedotactiononX=pX.Supposethatp�h.IfhasstandardLeviform(orifG
containsanelementinstandardLeviform),thenthetranslationfunctorfromtheblockof0totheblockoftakessimplemodulesto0ortosimplemodules;wegetthuseachsimplemoduleintheblockoffromexactlyonesimplemodule(uptoisomorphism)intheblockof0.(ThisfollowsfromtheanalogousresultinthegradedcasementionedinD.9.)WegetthesamebehaviourinthesubregularnilpotentcasesinSectionF.Onemayspeculatewhethersuchresultsgeneralise.Maybefurtherworkinthespiritof[BMR]willleadtosomeanswerstothesequestions. Liealgebrasinpositivecharacteristic27E.4.SetU0equaltothequotientofU(g)bytheidealgeneratedbyker(cen0).SowecandescribeanyU0alsoasthequotientofU0bytheidealgeneratedbytheimagesinU0ofallYpY[p](Y)pwithY2g.Theworkin[BMR]leadingtoTheoremE.2isinspiredbythepaper[BB1].ThereBeilinsonandBernsteinshowthattheanaloguetoU0overCisisomorphictothealgebraofglobalsectionsofthesheafDCofdierentialoperatorsonthe\ragvariety,andthattheglobalsectionfunctorinducesanequivalenceofcategoriesfrom(certain)sheavesofDC{modulestoU0{modules.(ThisresultisoneofthemainstepsinBeilinson'sandBernstein'sproofoftheKazhdan-LusztigconjectureoncharactersofsimplehighestweightmodulesovergC.)Thegeneralisationofthisresulttoprimecharacteristicinvolvestwomajorchanges:Onehastoreplacedierentialoperatorsby\crystalline"dierentialoperators(usingtheterminologyof[BMR]),andonegetsattheendnotanequivalenceofcategories,butanequivalenceofderivedcategories.E.5.IfAisthealgebraofregularfunctionsonasmoothanevarietyoverK,thenthealgebraof\crystalline"dierentialoperatorsonAisanalgebrageneratedbyAandtheLiealgebraDerK(A)ofallK{linearderivationsofA.Oneimposessomeobviousrelations,e.g.,DaaD=D(a)foralla2AandD2DerK(A).Forexample,ifAisapolynomialringA=K[X1;X2;:::;Xn],thenthealgebraof\crystalline"dierentialoperatorsonAisafreemoduleoverAwithbasisallmonomialsinthepartialderivatives@i=@=@Xi,1in.ThisalgebraactsonAvia\true"dierentialoperators,butthisactionisnotfaithfulbecause(e.g.)any(@i)pactsas0.Thisconstructionhasasheafversionthatleadstoasheafof\crys-talline"dierentialoperatorsDYonanysmoothvarietyYoverK.OnereplacesAbythesheafOYofregularfunctionsonY,andDerK(A)bythetangentsheafTY,cf.[BMR],1.2.Thisisaspecialcaseofaconstructionfrom[BB2],1.2.5.E.6.ConsidernowY=B,the\ragvariety,andsetD=DB.TheactionofGonB=G=B+denesaLiealgebrahomomorphismfromgtotheglobalsectionsofthetangentsheafTB.Thisinducesbyconstruc-tionahomomorphismfromtheenvelopingalgebraU(g)tothealgebraH0(B;D)ofglobalsectionsofD.ThishomomorphismfactorsoverU0([BMR],3.1.7)andinducesforgoodpanisomorphismU0!H0(B;D)(1) 28J.C.Jantzensee[BMR],Prop.3.3.1.Theproofusesatransitiontotheassociatedgradedalgebras;thistechniqueshowsalsothatHi(B;D)=0foralli�0.Denotebymodc(D)thecategoryofcoherentD{modules;herecoher-entmeanslocallynitelygenerated.Writemodfg(U0)forthecategoryofnitelygeneratedU0{modules.ForeachcoherentD{moduleFthespaceH0(B;F)ofglobalsectionsisvia(1)aU0{modulethatturnsouttobenitelygenerated.OnegetsthusafunctorthatinducesafunctorontheboundedderivedcategoriesDb(modc(D))!Db(modfg(U0))be-causeHi(B;?)=0fori�dim(B).Nowtherstmainresultin[BMR](Thm.3.2)says:Theorem:Supposethatp�2h2.ThenthefunctorDb(modc(D))!Db(modfg(U0))(2)isanequivalenceofderivedcategories.Aspointedoutin[BMR](Remarks1and2followingThm.3.2)thistheoremdoesnotholdwithoutgoingovertothederivedcategories;italsowillnotholdforsmallp.Theequivalenceintheoppositedirectionin(2)isinducedbythe\localisationfunctor"thattakesaU0{moduleMtotheD{moduleD\nU0M.ForthistoworkonersthastoshowthatthederivedfunctorDL\nU0takesDb(modfg(U0))totheboundedderivedcategoryDb(modc(D)),see[BMR],Prop.3.8.1.Onethenwantstoshowthatthecompositionsofthesefunctorsareisomorphictotheidentity.Thisischeckedonlyoncertainspecialobjects;theproofthatitsucestolookatthesespecialobjectsrequirestheboundonpinthetheorem.E.7.ThetransitionfromTheoremE.6toTheoremE.2involvestwomainsteps.DenotebyDtherestrictionofDtoBB.Ifwewerelucky,thentheequivalenceinE.6(2)wouldinduceanequivalencebe-tweenDb(modc(D))andDb(modfg(U0))foreachnilpotent2g,andthederivedcategoryDb(modc(D))wouldalsobeequivalenttothederivedcategoryofcoherentmodulesoverthestructuresheafofB.Thisisalmosttrue:ItholdswhenwereplaceBbyaformalneighbour-hood,see[BMR],Thm.5.2.1.ThathoweverdoesnotmatterintheendbecausethistransitiondoesnotchangetheGrothendieckgroupsofthecategoriesinvolved,cf.theproofofCor.4.1.4in[BMR].FinallyonehastoobservethattheGrothendieckgroupsofthesecategoriesandthoseoftheirboundedderivedcategoriescoincide. Liealgebrasinpositivecharacteristic29FTheSubregularNilpotentCaseInadditiontoourearlierconventions,weassumeinthissectionthatGisalmostsimple.F.1.SinceweassumeGtobealmostsimplethereisexactlyonenilpotentorbitingofdimension2(N1)whereN=jR+j.Thankstoourgeneralassumption(H3)thereisalsoonlyonenilpotentorbitingofthisdimension.Wecalltheseorbitsthesubregularnilpotentorbits(ingorg).IfRisoftypeAnorBnforsomen,thenwecanchooseinthesubregularnilpotentorbitingsuchthathasstandardLeviform.InthesecasesthesimpleU(g){modulesaswellasthestructureofthebabyVermamodulesweredeterminedin[J2].ForRnotoftypeAnorBn,thenthesubregularnilpotentorbitingdoesnotcontainanelementinstandardLeviform.InthesecasesmanyresultsonU(g){moduleswereprovedin[J5].Theresultsweremostdetailedinthesimplylacedcases(i.e.,forRoftypeDnorEn)andprovedinthesespecialcasesLusztig'sconjecturesfrom[L4],2.4/6and[L5],17.2.Forothertypestheresultsin[J5]arelesscompleteandcannowbeimprovedusingtheresultsfrom[BMR].Thoughoutthissectionweusuallysay\simplemodules"whenwemean\isomorphismclassesofsimplemodules".F.2.Choosesomeinthesubregularnilpotentorbitingsuchthat(b+)=0.Wepicksome2Xsuchthat0h+;_ipforall2R+.EachorbitofWonX=pXhasarepresentativeofthisform.SowedescribeanarbitraryblockofU(g)ifwedescribetheblock\of",i.e.,theblockwhereU(g)Gactsonallsimplemodulesviacen,cf.C.5.WeknowalsothatthesimplemodulesinthisblockareexactlythecompositionfactorsofZ().Wecallregularifpdoesnotdivideanyh+;_i,or,equivalently,if0h+;_ipforall2R+,i.e.,ifbelongstotheinteriorA0oftherstdominantalcoveasinD.10.Weshallassumethroughoutthatp&#x-366;&#x.499;h.Thisimpliesthatregulardoexist.Denotethesetofallsimplerootsbyanddenoteby0thelargestshortroot.Setm0=1anddenepositiveintegers(m)2by_0=P2m_.Let($)2denotethefundamentalweights.Ifisregular,thenweassociatetoeachU(g){moduleMintheblockofaninvariant(M)denedby(M)=f2[f0gjT$(M)=0g 30J.C.Jantzenwhereweset$0=0.HereeachTisatranslationfunctor,cf.[J5],B.2.F.3.AssumeinthissubsectionthatRissimplylaced,i.e.,oftypeAn,Dn,orEn.IntypeE8theresultsstatedbelowareprovedonlyforp�h+1;thisadditionalrestrictionshouldbeunnecessary.Supposerstthatisregular.Thentheblockofcontainsjj+1simplemodules.WecandenotethesesimplemodulesbyLwith2[f0gsuchthatdim(L)=(h+;_ipN1;if2;(ph+;_0i)pN1;if=0;(1)(recallthatN=jR+j)and(L)=(fg;if2;[f0g;if=0:(2)Wehave[Z():L]=m(3)forall2[f0g.DenotebyQtheprojectivecoverofLinthecategoryofallU(g){modules.Thenwehave[Q:L]=jWjmm(4)forall;2[f0g.LetuswriteExtforExt-groupsinthecategoryofallU(g){modules.OnegetsnowExt1(L;L)'(K;ifh;_i0;0;ifh;_i=0;(5)forall;2[f0gwith=|exceptthatwehavetoreplaceKbyK2ifRhastypeA1.ThesizeoftheExtgroupincase=isunknowntome;onecanshowthatitisnon-zeroinmostcases.Thestrategyfortheproofoftheseresultsin[J5]isasfollows:UsinghomomorphismsbetweenbabyVermamodulesoneconstructsachainofsubmodulesinZ()wherethedimensionsand{invariantsareknownforthequotientsofsubsequentterms.ThentranslationfunctorsandPremet'stheoremareusedtoshowthatthesequotientsaresimple.Nextadeformationargumentyieldsthatsimplemoduleswiththesame{invariantareisomorphictoeachother.Thusonegetstheclassicationofthesimplemodulesand(1){(3),cf.[J5],Thm.F.5.Quitegeneralargumentsimplyinourcasethatdim(Q)=pNjWj[Z():L],see Liealgebrasinpositivecharacteristic31G.3(1)below.Inductiveargumentsusingtranslationfunctorsshowinthepresentcasethat[Q]=(dim(Q)=pN)[Z()]intheGrothendieckgroupforallprojectivemodulesQ.Now(4)followsfrom(3),see[J5],Thm.G.6.Theproofof(5)usesanideaofVogan,relatingExtgroupstothestructureofsimplemodulestranslatedthroughawall,see[J5],Prop.H.12.Thestatementaboutthenumberofsimplemodulesaswellastheresult(4)ontheCartanmatrixoftheblockproveconjecturesbyLusztigfrom[L4],2.4/6and[L5],17.2.Dropnowtheassumptionthatshouldberegular,butcontinuetoassumethath+;_ip.Thenthenumberofsimplemodulesintheblockofis1+jf2jh+;_i&#x-282;&#x.199;0gj.OnecandenotethesesimplemodulesbyLwith=0orwith2withh+;_i&#x-282;&#x.199;0.Then(1)and(3)abovestillholdaslongasweonlyallowfromthisnewparameterset.Sodoes(4)ifwereplacejWjbyjW(+pX)j.Also(2)survivesifwemodifythedenitionofappropriately.If2A0,thenTtakesLtoLifthismoduleisdened,to0otherwise.IhavenoinformationaboutExtgroupsbetweensimplemoduleswhenisnotregular(exceptforRoftypeAnwhereonecanconsult[J2],Prop.2.19).Theseresultsareactuallyprovedalsoforph(exceptintypeE8)aslongas(H2)and(H3)hold.Supposeontheotherhandthath+;_i=pforsome2R+.Thenwehaveinparticularh+;_0i=p.Inthiscasethesimplemodulesshouldbeparametrisedbythesimplerootswithh+;_i&#x-318;&#x.311;0,and(1),(3),(4)shouldstillholdwiththeappropriatemodicationsasintheprecedingparagraph.ThisistrueforRoftypeAnandDn,butopenintypeEn.F.4.Asmentionedbefore,theclassicationofthesimplemodulesintheregularcaseinF.3wastherstevidencebesidesthecaseofstandardLeviformthatTheoremE.2shouldhold.Toseethis,letusdescribeBexplicitlyforsubregularnilpotent.ForRoftypeAn,Dn,orEnthevarietyBhasjjirreduciblecomponents(`)2.Eachcomponent`isisomorphictotheprojectivelineP1.Twodistinctcomponents`and`donotintersectifh;_i=0;theyintersecttransversallyinonepointifh;_i0.ThisdescriptionshowsthatH0(B;Ql)'QlbecauseBiscon-nected,andthatdimH2(B;Ql)=jjbecauseallirreduciblecompo-nentsofBhavedimension1andtherearejjofthem.ThisfactimpliesalsothatHi(B;Ql)=0fori&#x-5.1;䡣2.Finally,H1(B;Ql)=0follows(e.g.)fromthefactthatonecanpaveBbyanelinesandonepoint. 32J.C.JantzenFortheremainingrootsystemsonecanreducethedescriptionofBtothecasesalreadyconsidered:IfRhastypeBn(resp.Cn,F4,G2),thenBisisomorphicasvarietytotheanalogousobjectforagroupoftypeA2n1(resp.Dn+1,E6,D4).SoagaineachirreduciblecomponentofBisomorphictotheprojectivelineP1.Buttheirnumberisnowlargerthanjj.IntypesBn,Cn,andF4onehasnowonecomponentforeachshortsimpleroot,buttwocomponentsforeachlongsimpleroot.IntypeG2thereareonecomponentfortheshortsimplerootandthreecomponentsforthelongsimpleroot.Inanycasethecentraliser Gofintheadjointgroup GofGactsonB,hencepermutestheirreduciblecomponentsofB.Theconnectedcomponent G0of1in GhastostabiliseeachcomponentofB,sothecomponentgroupC()= G= G0actsaspermutationgroupontheirreduciblecomponentsofB.ForRoftypeAn,Dn,orEnthegroupC()istrivial.ForRoftypeBn,Cn(withn2inbothcases),orF4thegroupC()iscyclicoforder2;thenontrivialelementinterchangesforeachlongsimplerootthetwocomponentsbelongingtoanditxesthecomponentsbelongingtoshortsimpleroots.ForRoftypeG2thegroupC()isisomorphictothesymmetricgroupS3;itxesthecomponentbelongingtotheshortsimplerootandactsasfullpermutationgrouponthethreecomponentsbelongingtothelongsimpleroot.Theactionof GonBinducesanactionofC()onH(B;Ql).ThisactionistrivialonH0(B;Ql).Ontheotherhand,H2(B;Ql)hasabasisindexedbytheirreduciblecomponentsofBandC()permutesthesebasiselementsinthesamewayasitpermutestheirreduciblecomponents.F.5.AssumeinthissubsectionthatRisoftypeBn,Cn(withn2inbothcases),orF4.Assumethatp�2h2ifRhastypeCnwithn3orF4,i.e.,p�4n2intypeCnandp�22intypeF4.Weneedthisboundsothatwecanapplytheresultsfrom[BMR].Lets(resp.l)denotethesetofallshort(resp.long)simpleroots.Welookagainrstatthecasewhereisregular.Thentheblockofcontainsjsj+2jlj+1simplemodules.WecandenotethembyLwith2s[f0gandL;1,L;2with2lsuchthatF.3(1){(3)alsoholdhereforallshort,i.e.,for2s[f0g,andsuchthatforall2landalli2f1;2gdim(L;i)=h+;_ipN1(1) Liealgebrasinpositivecharacteristic33and(L;i)=fg(2)and[Z():L;i]=m 2:(3)In[J5]IhadtoleaveopenthequestionwhetherintypesCnandF4thetwosimplemodulesL;1andL;2correspondingtosome2lareisomorphictoeachotherornot;theresultsfrom[BMR]yieldingthetotalnumberofsimplemodulesgivetheanswer:theyarenot.DenotebyQtheprojectivecoverofLforshortandbyQ;itheprojectivecoverofL;iforlong.ThenF.3(4)holdsforshortand.Ifisshortandislong,thenonehasforalli2f1;2g[Q:L;i]=[Q;i:L]=jWjmm 2:(4)FortheremainingCartaninvariantsIcanonlystatetheobvious:Conjecture:[Q;i:L;j]=jWjmm 4foralllongandandalli;j2f1;2g.(ThisshouldbeaspecialcaseofLusztig'sconjecturein[L5],17.2.)AtleastintypeBnthisformulaholdsthankstotheresultsin[J2].Ifbothandareshortwith=,thenF.3(5)extendstothepresentsituation.Ifisshortandislong,thenonegetsforalli2f1;2gExt1(L;L;i)'Ext1(L;i;L)'(K;ifh;_i0;0;ifh;_i=0:(5)Finally,onecanchoosethenumberingoftheL;isuchthatforalllongandwith=andalli;j2f1;2gExt1(L;i;L;j)'(K;ifh;_i0andi=j;0;ifh;_i=0ori=j:(6)IdonotknowwhatdimExt1(L;1;L;2)isforlong.Furthermore,usuallythesimplemoduleshavenon-trivialself-extensions,butasinF.3IdonotknowhowbigtheExtgroupis.(Forallofthis,see[J5],Prop.H.12.)IfwetwistaU(g){modulewithAd(g)forsomeg2 G,thecen-traliserofintheadjointgroupofG,thenwegetagainaU(g){module.Iftheoriginalmodulewassimple,thensoisthetwistedone.IfU(g)G 34J.C.Jantzenactsviacenontheoriginalmodule,thenalsoonthenewone.WegetthusanactionofthecomponentgroupC()asinF.4onthesetofisomorphismclassesofsimpleU(g){modulesintheblockof.InourpresentsituationC()iscyclicoforder2andthenon-trivialelementinC()interchangesL;1andL;2foralllong,anditxesallLwithshort,see[J5],Prop.F.7.Fornon-regularthesituationshouldbesimilartotheonede-scribedinF.3.Ifwestillhaveh+;_ipforall2R+,thenonegetssimplemodulesLandL;imoreorlessasaboveexceptthatwedropallwithh+;_i=0.Eachsimplemoduleisisomorphictooneofthese.TheonlyproblemisthatwenolongercanbesurethatL;1andL;2forlongarenotisomorphictoeachotherbecause[BMR]doesnottellusthenumberofsimplemodulesinthenon-regularcase.However,inthecasewherethereisonlyone2R+suchthatpdividesh+;_i,thenonecanshowthatthesimplemodulesL;1andL;2forlongarenotisomorphic,see[J5],LemmaH.11.b.F.6.ConsidernowRoftypeG2andassumethatp&#x-294;&#x.236;2h2=10.Denotethesimplerootsby1(short)and2(long).Letberegular.Then[J5],D.11shows:TheblockofcontainsasimplemoduleL0with{invariant[f0ganddimension(ph+;_0i)pN1;thisistheonlysimplemoduleinthisblockwith{invariant[f0g.Thereareoneortwosimplemoduleinthisblockwith{invariantf1g;thismoduleorthesemoduleshavedimensionh+;_1ipN1.Thereisatleastonesimplemodulewith{invariantf2ganddimensionh+;_2ipN1.Allremainingsimplemoduleshave{invariantf2gor;.Now[BMR]tellsusthattherearevesimplemodules.WeknowalsothatthecomponentgroupC()'S3permutesthesemodules.Thisactionpreservesthe{invariantby[J5],D.5(3),andisindependentofthechoiceofifweusetranslationfunctorstoidentifythesimplemodulesindierentblocksforregularweights,see[J5],B.13(2).ThepossibleorbitsizesofC()permutingthevesimplemodulesare1,2,and3.Therehavetobeatleastthreeorbits.Ifoneoftheorbitshassize3,thenthereareexactlythreeorbits;theothertwoorbitshavecardinality1.Ifthereisnoorbitofsize3,thenthealternatingsubgroupinC()'S3hastoxallsimplemodules.Wehaveby[BMR]anisomorphismbetweentheGrothendieckgroupoftheblockof0andtheGrothendieckgroupK0(B)ofcoherentsheavesonB.Bythenaturalityoftheconstructionin[BMR]onemayhopethatthisisomorphismiscompatiblewiththeactionsof G.Letusassumethatthisistrue.Thentheactionof GonK0(B)factorsoverC()sincethisholdsfortheactionontheblockside. Liealgebrasinpositivecharacteristic35DenotebyYtheirreduciblecomponentofBcorrespondingtotheshortsimpleroot,andbyUthe(open)complementofY.Thegroup GstabilisesYandhencealsoU.WehavethenasurjectivemapK0(B)!K0(U)thatiscompatiblewiththeactionof G.NowUisisomorphictoadisjointunionofthreeanelines.ThisimpliesthatK0(U)'Z3.Furthermore,theactionof GonK0(U)hastofactoroverC(),andtheactionofC()hastopermutethethreesummandsisomorphictoZinK0(U)becauseC()permutesthethreeanelines.ThereforeC()actsfaithfullyonK0(U),henceonK0(B).ThisshouldnowimplythatalsotheactionofC()ontheGrothen-dieckgroupoftheblockisfaithful.Aspointedoutabovethismeansthattherearethreeorbits,andthattheyhavecardinality1,1,and3.Thisholdsatrstfor=0andthenforallusingtranslationfunctors.Itthenfollowsthatthereisonlyonesimplemodule,sayL1,with{invariantf1g,andthattherearethreesimplemodules,sayL2;1,L2;2,L2;3,with{invariantf2g,allofdimensionh+;_2ipN1.Onegetsthen(using[J5],D.11oncemore)that[Z():L1]=2and[Z():L0]=1=[Z():L2;i]foralli.GProjectiveModulesWekeepthepreviousassumptionsasinSectionsDandE.G.1.EachU(g)with2gisanitedimensionalalgebra.SothereisabijectionbetweenthesetofisomorphismclassesofsimpleU(g){modulesandthesetofisomorphismclassesofindecomposableprojectiveU(g){modules.AnysimpleU(g){moduleEcorrespondstoitsprojectivecoverQE,theunique(uptoisomorphism)indecomposableprojectiveU(g){modulewithE'QE=rad(QE).Thematrixofall[QE:E0]withE,E0runningoverrepresentativesfortheisomorphismclassesofsimpleU(g){modulesisthentheCartanmatrixofU(g).G.2.Supposethat2ghasstandardLeviform.Denotethepro-jectivecoverofanyL()asinD.1byQ().RecallnowthegradedcategoryCfromD.5/6.EachsimplemodulebL()inChasaprojectivecoverbQ()inC.Ifweforgetthegrading,thenbQ()isisomorphictoQ(),cf.[J4],1.4. 36J.C.JantzenSetI=f2Rj(X)=0g.RecallfromD.6thedenitionofCIandthefactthat7!bL()yieldsabijectionbetweenCIandthesetofisomorphismclassesofsimplemodulesinC.Onecanshow(see[J4],Prop.2.9)thateachbQ()with2CIhasaltrationwithfactorsbZ()with2CI,eachbZ()occurringjWI(+pX)j[bZ():bL()]times.(Notethatthereisamisprintin[J4],page154,line2:ReplacejWI(+pX)jbyjWI(+pX)j.)ThisyieldsintheGrothendieckgroupofCusingPropositionD.7[bQ()]=X2CI;"jWI(+pX)j[bZ():bL()][bZ()]:(1)Wehaveintheungradedcategory[Z()]=[Z()]forall2W+pX,henceforallwith",seePropositionC.2.So(1)impliesthatintheGrothendieckgroupoftheungradedcategory[Q()]=[Z()]X2CI;"jWI(+pX)j[bZ():bL()]:(2)ItfollowssincedimZ()=pNwithN=jR+j[Q()]=dim(Q()) pN[Z()]:(3)SoallrowsintheCartanmatrixbelongingtoaxedblockarepropor-tionaltoeachother.ThereforetheCartanmatrixhasdeterminant0unlesseachblockcontainsonlyonesimplemodule.ThelatteristhecaseifandonlyifIconsistsofallsimpleroots,i.e.,ifandonlyifisregularnilpotent.InthatspecialcaseallcompositionfactorsofQ()areisomorphictoL()=Z(),andtherearejWI(+pX)jofthem.G.3.Foranarbitrarynilpotent2gwith(b+)=0oneknowslessabouttheprojectiveindecomposablemodules.IfEisasimpleU(g){moduleintheblockofsome2X,thenonecanshowthatdim(QE)=pNjW(+pX)j[Z():E](1)see[J5],B.12(2).ForinstandardLeviformthiscanbededucedfromG.2(2).Ingeneral(fornotinstandardLeviform)itisnotknownwhetherQEhasaltrationwithfactorsoftheformZ()thatwould\explain"(1)andthatwouldallowustogeneraliseG.2(3).SowehaveatpresentonlyaHope:[QE]=dimQE pN[Z()] Liealgebrasinpositivecharacteristic37forallsimpleEintheblockofsome.Resultsinthesubregularnilpo-tentcase(seeF.3andF.5)supportthishope.G.4.Let2gbenilpotentandlet2Xberegular.(ThismeansasinF.2thatpdoesnotdivideanyh+;_iwith2R.Soweassumethatph.)LusztighasageneralconjectureontheCartanmatrixoftheblockofofU(g),see[L4],2.4and[L5],17.2.Actually,theconjectureisstatedforagradedversionofthatcategory.OnechoosesamaximaltorusT0inthecentraliserofinG.ThentheadjointactionofT0onU(g)inducesoneonU(g).ThisleadsthentoagradingofU(g)bythecharactergroupX(T0)ofT0.OnenowconsidersX(T0){gradedU(g){modulessatisfyingacompatibilityconditionanalogoustotheoneinD.6;nallyonetakesablockofthiscategorywhereU(g)Gactsonallsimplemodulesviacen.NowLusztig'sconjecturepredictsthatthesimplemodulesinthisblockshouldbeinbijectionwithacertainbasisofsomeequivariantK{groupofsomevariety(aSlodowyvariety)relatedtoB,andthattheentryintheCartanmatrixcorrespondingtotwosimplemodulesshouldbegivenbyacertainbilinearformevaluatedatthetwocorrespondingbasiselements.(See[L5]formoredetails.)InthecasewhereissubregularnilpotentandGalmostsimplenotoftypeAnorBn,thenT0=1,sowecanforgetaboutthegrading.IfGisalmostsimpleoftypeDnorEn,then[L4],2.6givesanexplicitdescriptionoftheconjecturedCartanmatrixthatsincehasturnedouttobecorrect,seeF.3(4).SupposethathasstandardLeviform(forarbitraryG).InthiscaseonecanchooseT0TwithX(T0)=ZR=ZI.InthiscasethecategorydescribedaboveisadirectsummandofthecategoryCfromD.6wheregradingsbyX=ZIarepermitted.However,thissmallercategorycontainstheblockcorrespondingto=0;allotherblocksforregularareequivalenttothisone(assumingph).ForsuchLusztig'sconjectureontheCartanmatrixwillfollowusingG.2(1)fromLusztig'sconjecture,mentionedinD.10,onthecompositionfactorsofthebZ(),see[L5],17.3.G.5.Considerthecase=0.SothegradedcategoryCfromD.6isthesameasthatofallG1T{modulesasin[J1],II.9.LetusnowdropinthiscasetheindexandwritebL(),bZ(),bQ()insteadofbL(),etc.Sotheseobjectsareparametrisedby2X.EachbQ()hasaltrationwithfactorsoftheformbZ()suchthateachbZ()occurs[bZ():bL()] 38J.C.Jantzentimes.Wehaveinparticular[bQ():bL()]=X2X[bZ():bL()][bZ():bL()](1)forall;2X.ThereisnowtheolderconjecturebyLusztigpredictingall[bZ():bL()]forp�handprovedin[AJS]forp0.Andthereistheneweronethatpredictsall[bQ():bL()]forp�h,andthatfollows,accordingtoLusztig,fromtheolderoneusing(1).NowIwanttopointoutthatinthiscase(=0)converselytheolderconjecturefollowsfromthenewerone.SinceLusztighasalreadyshowntheotherdirection,itsucestoshowthattheCartanmatrixdeterminesthedecompositionmatrix.Moreprecisely:Claim:Let2A0.Ifweknowall[bQ(x):bL(w)]withx;w2Wp,thenweknowall[bZ(x):bL(w)]withx;w2Wp.HereA0XisasinD.10theinterioroftherstdominantalcoveinsideX,andWpistheaneWeylgroupasinD.7.G.6.BeforewecanstartprovingtheclaiminG.5weneedsomefactsonthealcovegeometry.LetX+denotethesetofalldominantweightsinX.Foreach2Xthereexistsw2Wsuchthatwisantidominant(or,equivalently,suchthatw2X+)andwisuniquewiththisproperty.Movingtheorigintoanotherpoint,weget:Fix2X.Foreach2Xthereexistsw2Wpsuchthatw(p)=pandpw2X+.Theweightwisuniquelydeterminedbythiscondition;weshalldenoteitbya().Indeed:Thereexistsx2Wwithx(p)2X+.Set\r=x()2ZRandsetw2Wpequaltothecompositionofrstx,thentranslationwithp\r.Thenw(p)=x(p)+p\r=pandpw=pw(+)=pp\rx(+)=x(p)2X+:Thisyieldstheexistence;theuniquenessisleftasanexercise.Keepasabove.Atrivialremark:Ifz2Wp,then:z(p)=p=)a(z)=a()forall2X.(1)(Ifa()=wwithw2Wpandw(p)=pthena()=(wz1)(z)and(wz1)(p)=p.) Liealgebrasinpositivecharacteristic39Weclaimthatnexta()"forall2X.(2)Itiswellknownthatx0"0forallx2Wif0+isdominant.Onehassimilarly0"x0forallx2Wif0+isantidominant.Wecanapplythistoa()psincea()+pisantidominant.Therearex2Wand\r2ZRwitha()=p\r+xand\r=x().Nowwegeta()p"x1(a()p)=x1(p\r+x(+)p)=phence(2).Claim:Ifandareantidominant,thena(z)"a()forallz2W.(3)Thisfollowsbyinductiononthelengthofzfromthefollowingclaim,whereisassumedtobeantidominantandisarbitrary:Let2R.Then"s=)a(s)"a():(4)Proof:Wemayassumethat�0.Wehaves=+mwithm=h+;_i0.Thereexistsw2Wpwithw(p)=panda()=w.Set0=w(s).Then(1)saysthata(0)=a(s);sowewanttoshowthata(0)"a().Therearex2Wand\r2ZRsuchthatwisthecompositionofrstxandthentranslationbyp\r.Theassumptionw(p)=pimpliesthat\r=x().Set=x().Wehavethenwsw1=s;rpwithr=h\r;_i.Now0=ws=s;rp(w)=s;rpa()showsthata()"0or0"a().Ontheotherhand,wehavews=p\r+x(+m)=p\r+x+m=a()+m:If0,thenm0showsthat0"a(),hencea(s)=a(0)"0"a()using(2).Supposenowthat&#x-5.1;䝀0.Setn=h;_i2Z.Itisclearthats;np(p)=p.Sowehavea(0)=a(s;np0).Notethats;np0=s;nps;rpa()=a()+p(nr)andnr=h\r;_i=hx();_i=h;x1()_i=h;_i: 40J.C.JantzenSinceisantidominant,thelastequationyieldsnr0,hencea()+p(nr)"a().Nowwegeta(s)=a(s;np0)"s;np0=a()+p(nr)"a()asclaimed.G.7.(ProofoftheclaiminG.5)Ourmultiplicitiesareperiodic:Wehave[bZ(+p):bL(0+p)]=[bZ():bL(0)]forall;0;2Xsinceaddingpamountsjusttoashiftofthegrading.Each2Xcanbewrittenintheform=0+p1with12Xand02X1whereX1=f2Xj0h;_ipforallsimplerootsg:Actually,itwillbemoreconvenientforustoreplaceX1byX1p.SetW=fw2Wpjw2X1pg.Bytheobservationabovewegetall[bZ(x):bL(w)]ifweknowthemforallx2Wpandw2W.Wegetfrom[J1],II.9.13that[bZ(yx):bL(w)]=[bZ(x):bL(w)](1)forallw2W,x2Wp,andy2W.Itisthereforeenoughtoknowall[bZ(x):bL(w)]withw2Wandx2Wpsuchthatxisantidominant.SetW0equaltothesetofallx2Wpsuchthatxisantidominantandsuchthatthereexistsw2Wwithw"x.ThisisanitesetcontainingW.WehavetoshowthattheCartanmatrixdeterminesall[bZ(x):bL(w)]withw2Wandx2W0.Usethenotationd(C)foranyalcoveCasin[J1],II.6.6(1).ChooseanumberingW0=fw1;w2;:::;wrgsuchthatd(wiA0)d(wjA0)impliesji.Itfollowsthatwi"wjimpliesji,cf.[J1],II.6.6.Inparticular,if[bZ(wj):bL(wi)]=0,thenji.Weuseinductiononi.Fori=1observethattheonlyrelevantmultiplicityis[bZ(w1):bL(w1)]=1.Letnowi&#x-270;&#x.122;1andassumethattheCartanmatrixdeterminesall[bZ(w):bL(ws)]withsiandw2Wp.Thereexistsaweightiwithwi2piX1.Sincewiisantidominant,soisi.Nowwipi2X1showthatthereexistswm2WwithwiA0=pi+wmA0.Ifi=0,thend(wmA0)&#x-318;&#x.308;d(wiA0),hencemi.Thereexists02A0withwipi=wm0.Weknowbyinduction(andtranslation)all[bZ(w0):bL(wm0)]andgetbyperiodicityall[bZ(w):bL(wi)]. Liealgebrasinpositivecharacteristic41Sosupposethati=0,i.e.,thatwi2W.WehaveintheGrothen-dieckgroup[bQ(wi)]=Xji[bZ(wj):bL(wi)]Xx2W[bZ(xwj)]:(2)Herewehaveused(1).Wewanttoshowbyinductiononlthatall[bZ(wl):bL(wi)]aredetermined.As[bZ(wi):bL(wi)]=1,wemayassumeli.Supposerstthatwl2W.Then[bQ(wi):bL(wl)]isby(1)and(2)equaltoXjijWj[bZ(wj):bL(wi)][bZ(wj):bL(wl)]:(3)Byourinductiononiweknowall[bZ(wj):bL(wl)].Thismultiplicityis0unlessjl.Ifjl,thenweknow[bZ(wj):bL(wi)]byourinductiononl.Sotheonlyunknownsummandin(3)istheoneforj=lwherewegetjWj[bZ(wl):bL(wi)].Thisshowsthat[bZ(wl):bL(wi)]isdeterminedbytheCartanmatrix.Nowturntothecasewherewl=2W.Thereexistsanantidominantweight=0suchthatwl2pX1.Inthiscase[J1],II.9.13impliesforallthat[bZ():bL(wl)]=[bZ(a()):bL(wl)]:Sowegetfrom(2)that[bQ(wi):bL(wl)]isequaltoXji[bZ(wj):bL(wi)]Xx2W[bZ(a(xwj)):bL(wl)]:(4)Againweknowbyourinductiononiall[bZ(a(xwj)):bL(wl)].Ifthismultiplicityisnon-zero,thenwl"a(xwj).Sinceandwjareantidominant,wegetthenfromG.6(3)thatwl"a(wj),hencefromG.6(2)thatwl"wjandjl.Ifjl,thenweknow[bZ(wj):bL(wi)]byourinductiononl.Sotheonlyunknownsummandin(4)istheoneforj=lwhereweget[bZ(wl):bL(wi)]Xx2W[bZ(xwl):bL(wl)]:(Wedonotneedaanylonger.)Nowthesumoverxispositivesincealltermsarenon-negativeandtheoneforx=1isequaltoone.Weseeasabovethat[bZ(wl):bL(wi)]isdeterminedbytheCartanmatrix. 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