isometric actions of lie groups and invariants [jnl article] - p. michor

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1、ISOMETRICACTIONSOFLIEGROUPSANDINVARIANTSPeterW.MichorErwinSchrodingerInstituteforMathematicalPhysics,Boltzmanngasse9,A-1090Wien,AustriaandInstitutfurMathematik,UniversitatWien,Strudlhofgasse4,A-1090Wien,Austria.July31,1997ThisarethenotesofalecturecourseheldbyP.Mi

2、chorattheUniversityofViennaintheacademicyear1992/1993,writtenbyKonstanzeRietsch.Thelecturecoursewasheldagainintheacademicyear1996/97andthenoteswerecorrectedandenlargedbyP.Michor.Tableofcontents1.Introduction...........................12.PolynomialInvariantTheory.

3、...................513.C-InvariantTheoryofCompactLieGroups.............114.TransformationGroups.......................345.ProperActions..........................416.RiemannianG-manifolds......................487.RiemannianSubmersions......................578.Sect

4、ions.............................709.TheGeneralizedWeylGroupofaSection...............7710.BasicDierentialForms......................8411.Basicversusequivariantcohomology................911991MathematicsSubjectClassication.53-02,53C40,22E46,57S15.Keywordsandphra

5、ses.invariants,representations,sectionofisometricLiegroupactions,slices.Supportedby`FondszurForderungderwissenschaftlichenForschung,ProjektP10037PHY'.TypesetbyAS-TXME11.IntroductionLetS(n)denotethespaceofsymmetricnnmatriceswithentriesinRandO(n)theorthogonalgroup

6、.Considertheaction:`:O(n)S(n);!S(n);1t(AB)7!ABA=ABAIfisthespaceofalrealdiagonalmatricesandSthesymmetricgrouponnnletters,thenwehavethefollowing1.1.Theorem.(1)meetseveryO(n)-orbit.(2)IfB2,then`(O(n)B),theintersectionoftheO(n)-orbitthroughBwith,equalstheS-o

7、rbitthroughB,whereSactsonB2bynnpermutingtheeigenvalues.(3)intersectseachorbitorthogonallyintermsoftheinnerproducthABi=ttr(AB)=tr(AB)onS(n).O(n)(4)R[S(n)],thespaceofalO(n)-invariantpolynomialsinS(n)isisomor-SnphictoR[],thesymmetricpolynomialsin(byrestriction)

8、.O(n)11(5)ThespaceC(S(n))ofO(n)-invariantC-functionsisisomorphictoSn11C(),thespaceofallsymmetricC-functionsin(againbyrestric-1tion),andtheseagainareisomorphi