Subrepresentations in the Polynomial Representatio

arXiv:math/0501272v1[math.QA]18Jan2005SUBREPRESENTATIONSINTHEPOLYNOMIALREPRESENTATIONOFTHEDOUBLEAFFINEHECKEALGEBRAOFTYPEGLnATtk+1qr−1=1MASAHIROKASATANIAbstract.WestudyaLaurentpolynomialrepresentationVofthedoubleaffineHeckealgebraoftypeGLnforspecializedparameterstk+1qr−1=1.WedefineaseriesofsubrepresentationsofVbyusingavanishingcondition.Forsomecases,wegiveanexplicitbasisofthesubrepresentationintermsofnonsymmetricMacdonaldpolynomials.Theseresultsarenonsymmetricversionsof[7]and[9].1.IntroductionIn1990’s,CherednikintroducedthedoubleaffineHeckealgebra[1].Intermsofthepolynomialrepresentationofthealgebra,Macdonald’sconjecturewassolvedin[2],[3]forreducedrootsystems(andin[12],[13]fornon-reduced(C∨,C)case).FortherootsystemoftypeA,aclassificationofirreduciblerepresentationsofacertainclassisgivenin[5],[14],[15].In[6],itisshownthatfinite-dimensionalquotientsofthepolynomialrepresentationbythekernelofsomedegeneratebilinearformareirreducible.ThedoubleaffineHeckealgebraHnoftypeGLnisanassociativealgebrawithtwoparameterst,qgeneratedbyXi,YiandTj(1≤i≤n,1≤j≤n−1)withsomerelations.ThealgebraHnhasabasicrepresentationUontheringofn-variableLaurentpolynomials.TherepresentationUisirreducibleandY-semisimple,namelytheoperatorsYiaresimultaneouslydiagonalizableonU.ThenonsymmetricMacdonaldpolynomialisdefinedtobeamonicsimultaneouseigen-vectorforYi.Inthispaper,wespecializetheparametersattk+1qr−1=1for1≤k≤n−1and2≤r.Tobeprecise,introduceanewparameteruandspecialize(t,q)=(u(r−1)/M,τu−(k+1)/M).(1)Here,Misthegreatestcommondivisorof(k+1,r−1)andτ=exp(2π√−1r−1).WedenotebyH(k,r)nandV,thecorrespondingalgebraanditspolynomialrepresen-tation.TherepresentationVcanhavesubrepresentationsandtheymaynotbeY-semisimple.In[7]and[9],aseriesofidealsintheringofsymmetricpolynomialswithn-variablesaredefinedbyvanishingconditions,andexplicitbasesoftheidealsare12MASAHIROKASATANIgivenintermsofsymmetricMacdonaldpolynomialsspecializedat(1).Thevan-ishingconditionsforsymmetricpolynomialfareasfollows.Fixm≥1.f(x1,···,xn)=0ifxil,a+1=xil,atqsl,a(1≤l≤m,1≤a≤k)whereil,aaredistinct,sj,i∈Z≥0,Pki=1sj,i≤r−1.Thisiscalledthewheelconditionforsymmetriccase.Inthecasem=1,thebasisoftheidealisgivenin[7]byMacdonaldpolynomialsPλspecializedat(1)withpartitionsλsatisfyingλi−λi+k≥rfor1≤i≤n−k.Inthecase(k,r,m)=(1,2,2),thebasisoftheidealisgivenin[9]bylinearcombinationsofMacdonaldpolynomialsat(1).Also,asimilaridealintheringofBC-symmetricLaurentpolynomialsisinvestigatedin[8].TheseidealsareinvariantunderthemultiplicationbysymmetricpolynomialsandMacdonald’sq-differenceoperators.Theformeractionsaresymmetricpolyno-mialsofXiandthelatteractionsaresymmetricpolynomialsofYi.Moreover,theactionofTionanysymmetricpolynomialisamultiplicationbyascalar.HencetheidealsarerepresentationsofthesubalgebraofH(k,r)ngeneratedby{C(u)[X1,···,Xn]Sn,C(u)[Y1,···,Yn]Sn,T1,···,Tn−1}.Inthispaper,weconsideranonsymmetricversionoftheseideals.Inotherwords,weconstructafiniteseriesofsubrepresentationsinVofthewholealgebraH(k,r)n.Inordertoobtainthem,wedefineavanishingconditionasfollows.Fixm≥1.f(x1,···,xn)=0ifxil,a+1=xil,atqsl,a(1≤l≤m,1≤a≤k)whereil,aaredistinct,sl,a∈Z≥0,Pki=1sl,a≤r−2,andil,ail,a+1ifsl,a=0.(2)Wecallthevanishingcondition(2)thewheelconditionfornonsymmetriccase.DenotebyI(k,r)mthespaceofLaurentpolynomialssatisfyingthewheelcondition(2).Letusstatethemaintheoreminthispaper.DefineB(k,r)={λ∈Zn;forany1≤a≤n−k,λia−λia+k≥r,orλia−λia+k=r−1andiaia+k}.Here,wedeterminetheindex(i1,···,in)=w·(1,···,n)usingtheshortestelementw∈W=Snsuchthatλ=w·λ+whereλ+isthedominantelementintheorbitWλ.TheresultisTheorem1.1.TheidealI(k,r)1isanirreduciblerepresentationofH(k,r)nanditisY-semisimple.Foranyλ∈B(k,r),thenonsymmetricMacdonaldpolynomialEλhasnopoleat(1).Moreover,abasisoftheidealI(k,r)1isgivenby{Eλ;λ∈B(k,r)}specializedat(1).FortheproofofTheorem1.1,wefirstshowthatthesepolynomialshavenopoleat(1)andtheysatisfythewheelconditioniftheyarespecializedat(1).WeusethedualityrelationfornonsymmetricMacdonaldpolynomials(see(3)inProposition2.2)andwecounttheorderofpolesandzerosinordertocheckthestatement.Thisgivesalowerestimateofthecharacteroftheideal.Next,wegiveanupperestimateofthecharacteroftheideal.WeintroducethefiltrationV(M)=span{xλ;λ∈Zn,|λi|≤M}anddefineanon-degeneratepairingSUBREPRESENTATIONSINTHEPOLYNOMIALREPRESENTATION3betweenV(M)andthen-thtensorspaceRM,n=(span{ed;−M≤d≤M})⊗n.WegiveaspanningsetofthequotientspaceofRM,nwhichhasthesamecharacterasI(k,r)1∩V(M).Finally,weshowirreducibilitybyusingintertwiningoperators.Theseoperatorssendoneeigenvectortoanothereigenvector.WeshowthatEλforanyλ∈B(k,r)isacyclicvectorofI(k,r)1.Forthecasen=k+1,wealsoshowthatV/I(n−1,r)1isirreducibleandwegiveanexplicitbasisofV/I(n−1,r)1.WeexpectthatallsubquotientsI(k,r)m/I(k,r)m−1oftheseriesareirreducible.Theplanofthepaperisasfollows.InSection2,wereviewthedoubleaffineHeckealgebra,thepolynomialrepresentationU,thenonsymmetricMacdonaldpolynomialsEλ,andintertwiners.InSection3,westatethewheelconditionandshowthatitdeterminesasubrepresentaionI(k,r)1.InSection4,wegivealowerestimateofthecharacterofI(k,r)1usingnonsymmetricMacdonaldpolyno