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arXiv:0709.1286v1[math.RT]9Sep2007STUDYOFAZ-FORMOFTHECOORDINATERINGOFAREDUCTIVEGROUPG.LusztigIntroductionInhisfamouspaper[C1]Chevalleyassociatedtoanyrootdatumofadjointtypeandtoanyfieldkacertaingroup(nowknownasaChevalleygroup)whichinthecasewherek=CwastheusualadjointLiegroupoverCandwhichinthecasewherekisfiniteledtosomenewfamiliesoffinitesimplegroups.LetO′bethecoordinateringofaconnectedsemisimplegroupGoverCattachedtoafixed(semisimple)rootdatum.Inasequel[C2]to[C1],ChevalleydefinedaZ-formofO′.Later,anotherconstructionofsuchaZ-formwasproposedbyKostant[Ko].KostantnotesthatO′canbeviewedasa”restricted”dualoftheuniversalenvelopingalgebraUofLieG;hedefinesaZ-formUZofU(”theKostantZ-form”)andthendefinestheZ-formO′ZasthesetofallelementsinO′whichtakeintegralvaluesonUZ.ThenforanycommutativeringAwith1hedefinesO′AasA⊗O′Z;thisisnaturallyaHopfalgebraoverA.ItfollowsthatthesetGAofA-algebrahomomorphismsO′A−→Ahasanaturalgroupstructure.ThustherootdatumgivesrisetoafamilyofgroupsGA,oneforeachAasabove.UnlikeChevalley’sapproachwhichwasbasedonachoiceofafaithfulrepresen-tationofG,Kostant’sapproachisdirect(nochoicesinvolved)andgeneralizestothequantumcase.InthispaperwedevelopthetheoryofChevalleygroupsfollowingKostant’sapproach.Weshallprovethat:(I)ifAisanalgebraicallyclosedfieldthenO′AisthecoordinatealgebraofaconnectedsemisimplealgebraicgroupoverAcorrespondingtothegivenrootdatum.(Wetreatthereductivecaseatthesametime.)Notethat(I)wasstatedwithoutproofin[Ko].InthispaperwenotethatKostant’sdefinitioncanbereformulatedbyreplacingUbya”modifiedenvelopingalgebra”.Thetheoryisthendevelopedusingexten-sivelythetheoryofcanonicalbasesofsuchmodifiedenvelopingalgebras(presentedSupportedinpartbytheNationalScienceFoundationTypesetbyAMS-TEX12G.LUSZTIGin[L1]),comingfromquantumgroups.(SeetheNotesin[L1]forreferencestooriginalsourcesconcerningcanonicalbases.)Wenowpresentthecontentofthispaperinmoredetail.LetAbeafixedcommutativeringwith1withagiveninvertibleelementv∈A.In§1werecallthedefinitionandsomepropertiesofthemodified(quantized)envelopingalgebra˙UAoverAanditscanonicalbasis˙B.Wealsodefinea”comple-tion”ˆUAof˙UAwhichconsistsofformal(possiblyinfinite)A-linearcombinationsofelementsin˙B.Weshowthatthemultiplicationand”comultiplication”of˙UAextendnaturallytoˆUA.In§2weintroducesomeinvertibleelementss′i,eofˆUA(whereicorrespondstoasimplereflectionande=±1).Weshowthatconjugationbys′i,erestrictedto˙UAisessentiallytheactionofageneratorinthebraidgroupactionon˙UAstudiedin[L1].Thuss′i,eplaysthesameroleasanelementconsideredinasimilarcontext(withA=C(v))bySoibelman[So].ButwhileSoibelman’selementisnotexplicitanditsintegralitypropertiesarenotclear,ourelements′i,eisremarkablysimpleandhasobviousintegralityproperties.In§3wedefinefollowing[L1,29.5.2]theHopfalgebraOA(aquantumanalogueofO′Aabove.)WeprovethattheA-algebraOAisfinitelygenerated(see3.3)withanexplicitsetofgenerators.In3.7weshowthattheA-algebraOAcanbeimbeddedintothetensorproductoftwosimpleralgebras.(ForacloselyrelatedresultinthecasewhereA=Q(v),see[Jo,9.3.13].)Inthecasewherev=1inAthesesimpleralgebrascanbeexplicitlydescribedintermsofthebraidgroupaction,see3.13.Wededucethat,ifAisanintegraldomainandv=1inA,thenOAisanintegraldomain;seeTheorem3.15.(ForasimilarresultinthecasewhereA=Q(v),see[Jo,9.1.9].)In§4weassumethatv=1inAandweintroducethegroupGAinanalogywith[Ko].InTheorem4.11weshowthatOAhasapropertylike(I)above.In§5weidentify(assumingthatv=1inA)ourOAwithKostant’sO′A.Contents1.Thealgebras˙UA,ˆUA.2.Theelementss′i,e,s′′i,eofˆUA.3.TheHopfalgebraOA.4.ThegroupGA.5.Fromenvelopingalgebrastomodifiedenvelopingalgebras.1.Thealgebras˙UA,ˆUA1.1.Inthissectionwerecallthedefinitionofthemodifiedquantizedenvelopingalgebra˙UA(overA)attachedtoarootdatumandwerecallthedefinitionandsomeofthepropertiesofthecanonicalbasis˙Bof˙UA.WealsostudyacertaincompletionˆUAof˙UA.STUDYOFAZ-FORMOFTHECOORDINATERINGOFAREDUCTIVEGROUP3Wefixarootdatumasin[L1,2.2].ThisconsistsoftwofreeabeliangroupsoffinitetypeY,Xwithagivenperfectpairingh,i:Y×X→ZandafinitesetIwithgivenimbeddingsI−→Y(i7→i)andI−→X(i7→i′)suchthathi,i′i=2foralli∈Iandhi,j′i∈−Nforalli6=jinI;inaddition,wearegivenasymmetricbilinearformZ[I]×Z[I]→Z,ν,ν′7→ν·ν′suchthati·i∈2Z0foralli∈Iandhi,j′i=2i·j/i·iforalli6=jinI.Weassumethatthematrix(i·j)i,j∈Iispositivedefinite.LetX+={λ∈X;hi,λi≥0foralli∈I}.Forλ,λ′inXwewriteλ≥λ′orλ′≤λifλ−λ′∈Pi∈INi′.Theimageofν∈Z[I]underthehomomorphismZ[I]−→Xsuchthati7→i′fori∈I,isdenotedagainbyν.LetWbethe(finite)subgroupofAut(Y)generatedbytheinvolutionssi:y7→y−hy,i′ii(i∈I)orequivalentlythesubgroupofAut(X)generatedbytheinvolutionssi:x−hi,xii′(i∈I);thesetwosubgroupsmaybeidentifiedbytakingcontragredients.Fori6=jinIletnij=njibetheorderofsisjinW.Letl:W−→NbethestandardlengthfunctiononWwithrespectto{si;i∈I}.Letw0∈Wbetheuniqueelementsuchthatl(w0)ismaximal.1.2.Letvbeanindeterminate.LetA=Z[v,v−1].Fori∈Iwesetvi=vi·i/2.WefixacommutativeringAwith1withagivenringhomomorphismA−→Arespecting1.Forα∈AweshalloftendenotetheimageofαunderA−→Aagainbyα.WheneverwewriteA=Q(v)orA=AweshallunderstandthatAisregardedasanA-algebrainanobviousway.WheneverwewriteA=QorA=Zweshallu
本文标题:Study of a Z-form of the coordinate ring of a redu
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