On-the-affine-analogue-of-Jacks-and-Macdonalds-pol

ONTHEAFFINEANALOGUEOFJACK’SANDMACDONALD’SPOLYNOMIALSPavelI.Etingof,AlexanderA.Kirillov,Jr.DepartmentofMathematicsYaleUniversityNewHaven,CT06520,USAe-mail:[e-mail],[e-mail]’sandMacdonald’spolynomialsareanimportantclassofsymmetricfunc-tionsassociatedtorootsystems.InthispaperwedeneandstudyananalogueofJack’sandMacdonald’spolynomialsforanerootsystems.OurapproachisbasedonrepresentationtheoryofaneLiealgebrasandquantumanealgebras,andfollowstheideasofourrecentpapers[EK1,EK2,EK3].WestartwithareviewofthetheoryofJack(Jacobi)polynomialsassociatedwiththerootsystemofasimpleLiealgebrag.ThistheorywasdescribedinthepapersofHeckmanandOpdam[HO,H1,O1,O2].Inthesepapers,Jack’spolynomialsaredenedasabasisinthespaceofWeylgroupinvarianttrigonometricpolynomialswhich1)diersfromthebasisoforbitsumsbyatriangularmatrix(withrespecttothestandardpartialorderingondominantintegralweights)withonesonthediagonal,and2)isaneigenbasisforacertainsecondorderdierentialoperator(theSutherland-Olshanetsky-Perelomovoperator,[Su,OP]).ItturnsoutthattheseconditionsdetermineJack’spolynomialsuniquely.OrbitsumsandcharactersforgturnouttobespecialcasesofJack’spolynomials.Thesepolynomialshaveaq-deformation,whichiscalledMacdonald’spolynomials;theyhavebeenintroducedbyI.Macdonaldinhispapers[M1,M2]andhavebeenintensivelystudiedsincethattime.WegeneralizethedenitionofJack’spolynomialstothecaseofanerootsystems.Weassignsuchapolynomialtoeverydominantintegralweightoftheanerootsystem.Itisdoneinthesamewayasfortheusualrootsystems:theonlythingonehastodoisreplacetheSutherlandoperatorbyitsaneanalogue.Thisanalogueisconstructedinthesamewayasforusualrootsystems,anditturnsouttobe(afterspecializationoflevel)aparabolicdierentialoperatorwhosecoecientsareellipticfunctions.Thisoperatorwasintroducedin[EK3](fortherootsystemAn1)andiscloselyrelatedtotheSutherlandoperatorwithellipticcoecientsconsideredin[OP],butismoregeneral.Analogouslytothenite-dimensionalcase,orbitsumsandcharacters(ofintegrablemodules)fortheaneLiealgebra^garespecialcasesofaneJack’spolynomials.TypesetbyAMS-TEX1FororbitsumsandcharactersofaneLiealgebras,thereisabeautifultheoryofmodularinvariancedescribedin[K].WegeneralizethistheorytogeneralaneJack’spolynomials.Itturnsoutthatthenite-dimensionalspacespannedbytheJack’spolynomialsofagivenlevelismodularinvariantwithacertainweight.Moreover,asinthecharactercase,therepresentationofthemodulargroupinthisspaceis(conjecturedly)unitary,withrespecttoaquitenontrivialinnerproductwhichgeneralizestheMacdonaldinnerproduct.ThisinnerproductcoincideswiththeinnerproductonconformalblocksoftheWess-Zumino-Wittenconformaleldtheory,anditsexistencestillremainsaconjecture.However,weshowthatunlikethecharactercase,theimageofthecorrespondingprojectiverepresentationofthemodulargroupmaybeinnite.FortherootsystemAn1,itispossibletogiveaninterpretationofJack’sandMacdonald’spolynomialsintermsofrepresentationtheoryoftheLiealgebraslnandquantumgroupUq(sln),respectively[EK1,EK2].Morespecically,Macdon-ald’spolynomialsareinterpretedascertain(renormalized)vector-valuedcharacters(tracesofintertwiners)forquantumgroups{anotiongeneralizingtheusualchar-acters.Analogously,inthispaperweshowthatfortherootsystem^An1theaneJack’spolynomialsdenedaseigenfunctionsofacertainsecondorderdierentialoperatorcanberepresentedasrenormalizedtracesofintertwinersbetweencertainrepresentationsoftheaneLiealgebra^g.Thisproofisanalogoustotheonegivenin[EK2]forthenite-dimensionalcase.Finally,wedenetheaneMacdonald’spolynomials(i.e.q-deformedJack’spolynomials)fortherootsystem^An1toberenormalizedtracesofintertwinersforquantumanealgebras,andformulate(asaconjecture)theaneanalogueoftheMacdonaldspecialvalueidentitiesfrom[M2].Thepaperisorganizedasfollows.InSection1,wegivebasicdenitionscon-cerningrootsystems.InSection2,wedenetheSutherlandoperatoranditseigenfunctions(Jack’spolynomials)andquotesomeknownresultsaboutthem.InSection3,weconstructJack’spolynomialsfortherootsystemAn1viarepresen-tationtheoryofsln.InSection4,wemakebasicdenitionsconcerninganerootsystems.InSection5,wedenetheaneanalogueofthegroupalgebraoftheweightlattice.InSection6,wedeneandstudytheaneCalogero-SutherlandoperatorandintroducetheaneJack’spolynomials.InSection7,weconstructtheaneJack’spolynomialsviatracesofintertwinersforcsln.InSection8,wegiveacomplex-analyticdescriptionoftheaneJack’spolynomials.InSection9,westudymodularpropertiesoftheaneJack’spolynomials.InSection10,wegiveabriefintroductiontotheWess-Zumino-WittenmodelandformulateaconjectureontheunitarityoftheactionofthemodulargrouponaneJack’spolynomials.InSection11,wedenetheaneMacdonald’spolynomials,andconjecturethatananeanalogueoftheMacdonaldspecialvalueformulaistrue.Also,inthissectionwediscusstheextensionoftheresultsoftheprevioussectionstonon-integervaluesofthecentralchargeofthe(quantum)anealgebra.Finally,Section12isdevotedtothediscussionofsomeinterestingproblemswhichstillremainopen.Acknowledgements.WewouldliketothankouradvisorI.Frenkelforusefulsuggestion