Lie algebra homology and the Macdonald-Kac formulas

《Lie algebra homology and the Macdonald-Kac formulas 》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、Inventionesmath.34,37-76(1976)Irlve~ltio~le$mathematicae9bySpringer-Verlag1976LieAlgebraHomologyandtheMacdonald-KacFormulasHowardGarland*(NewHaven)andJamesLepowsky**(Princeton)IntroductionI.G.Macdonald'snotedidentitiesconcerningDedekind'sq-functionwereorig-inallyobtainedin[

2、14]bymeansof"affinerootsystems".TheseformulashavesubsequentlybeeninterpretedbyV.G.KacIll(b)]andR.V.Moody[15(c)]asthepreciseanaloguesofWeyl'sdenominatorformulaforthe"EuclideanLiealgebras"(Moody'sterm)-certaininfinite-dimensionalanaloguesofcomplexsemisimpleLiealgebrasintroduce

3、dbyKacIll(a)]andMoody[15(a),(b)].In[ll(b)],KacalsosketchesanewproofoftheMacdonaldidentities,andinfactofamuchwiderclassofidentities-theanaloguesofbothWeyl'scharacteranddenominatorformulas,forafamilyofLiealgebrasconsiderablymoregeneralthantheEuclideanLiealgebras.Thesemoregener

4、alalgebras,alsointroducedbyKac[11(a)]andMoody[15(a)],aretheLiealgebrasdefinedbysymmetrizable(generalized)Cartanmatrices(seew2below).ThemainpurposeofthepresentpaperistogeneralizeB.Kostant'sfundamentalresult[12,Theorem5.14]onthehomology(orcohomology)ofnilradicalsofparabolicsub

5、algebrasincertainmodules,from(finite-dimensional)complexsemisimpleLiealgebrastotheKac-MoodyLiealgebrasdefinedbysymmetrizableCartanmatrices(seeTheorem8.6).Wethusobtaintheresultsin[11(b)],includingtheMacdonaldidentities,asimme-diateconsequencesoftheEuler-Poincar6principle(39),

6、justasKostantderivesWeyl'scharacteranddenominatorformulasin[12,w7]fromhishomologytheorem.Kac'smethodin[11(b)]istoadapttoLiealgebrasdefinedbysymmetrizableCaftanmatricesthesimpleproof,usingVermamodules,ofWeyl'scharacterformulagivenbyI.N.Bernstein,I.M.GelfandandS.I.Gelfandin[l(

7、a)].(Thisproofisalsopresentedin[6,w7.5]and[9,w24].)ButKacmustmakeacertainmodification:InplaceoftheHarish-Chandraisomorphismtheoremconcerningthecenteroftheuniversalenvelopingalgebrausedin[1(a)](see[-6,w7.4]or[9,w23]),heusesaCasimiroperator,whichineffectplaystheroleofasinglede

8、cisiveelementofthecenteroftheuniversalenvelopingalgebra.Whenappliedtocomple