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1、LettersinMathematicalPhysics30:119-123,1994.119©1994KluwerAcademicPublishers.PrintedintheNetherlands.TheStarExponentialandPathIntegralsonCompactGroupsII:TheWeylCharacterFormulaM.NAKASHIMADepartmentofMathematics,CaliforniaStatePolytechnicUniversity,Pomona,CA91768,U.S.A.(Received:16September1993)Abst
2、ract..ThisisacontinuationoftheworkbegunbyCadavidandNakashimainLett.Math.Phys.23,111-115(1991).AnexpressionfortheWeylCharacterFormulaisobtainedintermsofthestar-productpathintegral;andtherelationshipbetweenthestar-productpathintegralandthepathintegraldevelopedoncoadjointorbitsisestablished.Mathematic
3、sSubjectClassifications(1991).81S10,81S40,53C80,43A77.1.TheWeylCharacterFormulaLetGbeacompact,simplyconnectedLiegroupandHcGamaximaltorus.For2dominantintegral,theBorel-Weil-Bottconstructiongivesaholomorphiclinebundle7r:Lz~G/HandthespaceofholomorphicsectionsF(L~)withmetric(,)realizestheunitaryirreduc
4、iblerepresentation,p~,correspondingto2.ViathedequantizationprocedureofBerezin,aG-invariantstarproductcanbeobtained,andinl-l]apathintegralexpressionwasobtainedfortheassociatedstar-exponen-tial.Briefly,theconstructiongoesasfollows.LetLodenoteLzminusthezerosection.Foreachq~Lo,weleteqsF(L~)denotethecor
5、respondingcoherentstate.ThenwithnAt=tandIIeqII2=(eq,eq),Exp(*tH(x))=(eam*...*e~tn)(x)fore""f~lueAam~'")++mr.-,,~)]x1(eao,%)...(eq~_,,eq.)"-1x(eqo,~eqn11%112:l~e~._~k=l~(edyk);G/H""";G/HeAt[H(x'Yl)+"'"+H(yn-1,x)]Xxexp(n-~Lln(eq~,e~+,))1"fll(~dYk),k(eqo,%)~=1where120M.NAKASHIMAqo,q,~Trl(x)--{O},andqk
6、e~-l(Yk)--{O}.Let7:[0,7]-~G/Hwith~/(0)=~(t)=x,~(kAt)=y~,beaclosedpath.Takingtheqktolieontheliftof7withrespecttothemetricconnectionandlettingn~~gives~expEfoH(7(z))dz+;oy*~?ln(eq,eq)1,-~limk=l~lleq~ll'(1.1)where~(?)istheholonomyofthepath.TheintegraloftheconnectiongivesT(?),andIT(7)I=1asthebundlesatis
7、fiesintegrality.Sowritingthemeasureinpathspaceas1limI~edyk,thestar-exponentialisUsingthisresult,itisstraightforwardtoobtainapathintegralrepresentationoftheWeylCharacterFormula.ForgeG,g=eX,defineX(x)=(dpa(X)
