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1、PROCEEDINGSOFTHEAMERICANMATHEMATICALSOCIETYVolume48,Number1,March1975MACDONALDIDENTITIESANDEUCLIDEANLIEALGEBRASR.V.MOODY1ABSTRACT.TheMacdonaldidentitiesonaffinerootsystemsareplacedinthecontextofEuclideanLiealgebras.Thisyieldsasimpli-fiedformoftheidentities,whichisusedtoinferseveralr
2、esultsonthepartitionfunctionoftherootsystemofaEuclideanLiealgebra.Introduction.Inaremarkablepaper[7],Macdonaldshowedthatfromeachreducedaffinerootsystemoneisabletoproduceanidentityintheformalexponentialsofrootsofthissystem,andthattheseidentitiesaregeneralizationsofaclassicalidentityo
3、fWeylforfiniterootsystemsandcertainidentitiesconnectedwithDedekind's»/-function.AfeatureoftheMacdonaldidentities(seeequation(M)of§2)istheappearanceofafactorPwhosedescriptionisquiteawkwardandwhosemeaningisveryobscure.Ourintentionhereistoshowthatitispossibletoplacetheidentitiesintheco
4、ntextofEuclideanLiealgebras,whereuponthemeaningofPbecomesobviousandtheidentitiestakeonasimplerandevenmorebeautifulappearance.Intheirnewform,theidentitiesgiveamarvellousrelationshipbetweentheWeylgroup,therootsystem,andthedimensionsoftherootspaces.Itisnotunreasonabletoexpectthatsimila
5、ridentitiesmayholdforalltheLiealgebrasdeterminedbyarbitraryCartanmatrices.Extensivecomputercal-culationsbyDr.K.L.Teogivestrongevidencetosupportthis.Tobepre-cise,weconjecturethatProposition2remainstruewhenQisanyreducedheffalumpLiealgebra.In§3weusethenewformoftheidentitytogeneralizeso
6、mewell-knownresultsonKostant'spartitionfunctiontopartitionfunctionsontheEuclideanrootsystems.ReceivedbytheeditorsNovember13,1973.AMS(MOS)subjectclassifications(1970).Primary17B65,17B35,17B15,10A20.1TheauthorgratefullyacknowledgesthesupportofaNationalResearchCoun-cilofCanadagrantwhil
7、ethisworkwasbeingdone.Copyright©1975,AmericanMathematicalSociety4344R.V.MOODY1.HeffalumpLiealgebras.LetZ,N,Z+denotetheintegers,thenaturalnumbers,andthenonnegativeintegers,respectively.LetKbeafieldofcharacteristiczero,andlet(A..)bean/x/matrixofintegerssuchthat(i)A,.=2,i=I,-",/,(ii)A.
8、.
