The Centralizer of Invariant Functions and Divisio

arXiv:dg-ga/9506008v225Sep1996THECENTRALIZEROFINVARIANTFUNCTIONSANDDIVISIONPROPERTIESOFTHEMOMENTMAPYAELKARSHONANDEUGENELERMANAbstract.LetΦ:M−→g∗beapropermomentmapassociatedtoanactionofacompactconnectedLiegroup,G,onaconnectedsymplecticmanifold,(M,ω).AcollectivefunctionisapullbackviaΦofasmoothfunctionong∗.InthispaperwepresentfournewresultsabouttherelationshipbetweenthecollectivefunctionsandtheG-invariantfunctionsinthePoissonalgebraofsmoothfunctionsonM.Morespecifically,weshow:1.ThecentralizeroftheinvariantfunctionsconsistsofthealgebraofsmoothfunctionsonMthatareconstantonthelevelsetsofthemomentmap.ThisresolvesaconjectureofGuilleminandSternberg.2.Thequestionofwhetherthiscentralizerisequaltothealgebraofcollectivefunctionsorislargerisequivalenttoaformalalgebraicquestiononthelevelofpowerseries.3.IfthegroupGisatorus,thecentralizeroftheinvariantfunctionsconsistsofthecollectivefunctions.Wecloseagapinearlierproofsofthisfact.4.IfthegroupGisSU(2)andthecentralizeroftheinvariantfunctionsislargerthanthealgebraofofcollectivefunctions,theactionofSU(2)extendstoanactionofU(2)withthesameorbits,andthecentralizeroftheinvariantfunctionsconsistsoftheU(2)-collectivefunctions.Contents1.Introduction12.Thecentralizerofinvariantfunctions43.Divisionpropertycanbedetectedformally74.Divisionpropertyofatoralmomentmap125.ThecentralizerofSU(2)-invariantfunctions.15AppendixA.Localnormalformforthemomentmapandimplications19References211.IntroductionLetΦ:M−→g∗beamomentmapassociatedtoaHamiltonianactionofacompactconnectedLiegroupGonacompact1connectedsymplecticmanifoldDate:February7,2008.1991MathematicsSubjectClassification.Primary58F05;Secondary58C25,32B20.Keywordsandphrases.Momentmap,collectivefunctions,dualpairs.dg-ga/9506008.TheworkofE.LermanwaspartiallysupportedbyanNSFpostdoctoralfellowship.TheworkofY.KarshonwaspartiallysupportedbyNSFgrantDMS-9404404.1ThroughoutthisintroductionweassumethatthemanifoldMiscompactandthegroupGisconnected.Intherestofthepaperourassumptionsareoftenmoregeneral.12Y.KARSHONANDE.LERMAN(M,ω).PullbacksbyΦofsmoothfunctionsong∗arecalledcollectivefunctions.TheyformaPoissonsubalgebraofthealgebraofsmoothfunctionsonM.Itscentralizeristhealgebraofinvariantfunctions,i.e.,asmoothfunctionfonMisinvariantifandonlyif{f,h}=0foreverycollectivefunctionh,where{,}denotesthePoissonbracketcorrespondingtothesymplecticformω.Motivatedbyastudyofcompletelyintegrablesystemsin[GS1],GuilleminandSternbergconjecturedin[GS3]thatthecentralizerofthealgebraofinvariantfunc-tionsisthealgebraofcollectivefunctions.Theyprovedthisconjectureforneigh-borhoodsofgenericpointsinM.Acollectivefunctionisclearlyconstantonthelevelsetsofthemomentmap.Theconverseneednotbetrue.Forexample,thestandardlinearactionofthegroupG=SU(2)onC2hasamomentmapΦ(u,v)=(uv,12|u|2−12|v|2)whenweidentifythevectorspaceg∗withR×C.Thefunctionf(u,v)=|u|2+|v|2isconstantonthelevelsetsofΦbecauseitisequalto(|uv|2+(12|u|2−12|v|2)2)12=12||Φ||.Itisnotcollectivebecausethefunction||x||isnotsmoothonR×C.Insection2ofthispaperweshowthatthecentralizerofthealgebraofinvariantfunctionsisthealgebraoffunctionsthatareconstantonthelevelsetsofthemomentmap.Infact,thesetwoalgebrasaremutualcentralizersinthePoissonalgebraC∞(M).SeeTheorem1andCorollary2.12.Thiswasalreadyshowninthethesisofthefirstauthor[K],butourcurrentproofisshorter.Thisresultraisesthefollowingquestion:whatistheobstructionforafunctionthatisconstantonthelevelsetsofthemomentmaptobecollective?Insection3,Theorem2,weexpressthisobstructionasaconditionontheTaylorseriesofthefunction.TheproofusestheoremsofBierstoneandMilmanandofMarle,Guillemin,andSternberg.Theorem2essentiallyreducestheidentificationofthecentralizeroftheinvariantfunctionstoanalgebraicquestion.Basedonthis,F.KnoprecentlyannouncedacompletedescriptionofthecentralizeroftheinvariantfunctionsonaHamiltonianspaceintermsofthelittleWeylgroup(definedin[KN])ofthespace.IftheLiegroupGisabelian,everyfunctionwhichisconstantonthelevelsetsofthemomentmapisacollectivefunction.SotheconjectureofGuilleminandSternbergistruefortorusactions.Thehistoryoftheproofisasfollows.Alreadyintheirpaper[GS3],GuilleminandSternbergprovedthatforalineartorusactiononasymplecticvectorspacethecentralizeroftheinvariantsconsistsoffunctionsthatareconstantonthelevelsetsofthemomentmap.Theyclaimedthatthesefunctionsarecollective.Thisclaimisnotobvious;weproveitinsection4ofthispaper.In[L2],thesecondauthorshowedthatthisclaimimpliesthattheconjectureofGuilleminandSternbergistruefortorusactionsoncompactmanifolds,and,moregenerally,foractionsofcompactLiegroupsoncompactmanifolds,providedthattheimageofthemomentmapdoesnotintersectthewallsoftheWeylchambers.Werecall(slightlystrongerversionsof)theseresultsinsection4.ThefirstcounterexampletotheconjectureofGuilleminandSternbergwasgivenbythesecondauthorin[L2].ThisisthestandardactionofSU(2)onC2withmomentmapΦ(u,v)=(uv,12|u|2−12|v|2).ThisactionofSU(2)extendstothestandardactionofU(2)onC2withthesameorbits(spheres)andwithamomentmap˜Φ(u,v)=(uv,|u|2,|v|2).Thecentralizeroftheinvariants(foreitherSU(2)orU(2))consistsoftheU(2)-collectivefunctions.