; TeX output 2001.10.10:1205header=pstricks.proheader=pst-dots.proheader=pst-node.proiҍe.iҍ cN cmbx12POLARIZATIONٚANDQUANTIZATIONrK`y cmr10KENROUUFURUT*ANI &A29- cmcsc10Abstract.jInthisarticleweexplainamethoGdofageometricquantization,pair- 29ingOofpGolarizations,andexhibitsomeofknownresultsfortheexamplesofsuch29symplectictmanifoldstowhichthismethoGdwasappliedtoconstructaquantization29opGeratorUUofgeodesic ows,anddiscussafurtherproblem.kƋ- cmcsc10Contents69XQ cmr121. InrtroSductionC1692. ProlarizationsandtheirpairingV393. Compactrankonesymmetricspacesw694. QuaternionprojectivrespacesO1495. RemarksQ6*209References\"217q卍U1.Y_Introduction9In\themostprimitivresenseof@ cmti12quantization,y1wemeanacorrespSondencefromfunc-9tionsonaphasespacetoopSeratorsactingonfunctionsonacon gurationspace.9Ofcourseacon gurationspaceisthoughrtasasetofallstatesofacertainphysical9system4andtheywillconstituteamanifoldinthemathematicalsenseandaphase9space~gisitscotangenrtbundlewiththenaturalsymplecticstructure.So,Wfunctions9onthephasespacearecalledclassicalobservXablesandfunctionsonthemanifold9areHunderstoSodasrepresenrtingquantumstates.RSuchaninterpretationofthecor-9respSondencesycanbeextendedbetrweenyhigherdegreequanrtitieslike(co)homology9classesMofthemanifoldorits(co)tangenrtbundleandopSeratorinvXariants([Ar `],[AS15],9[AS25],[APST],[DG1a],[WVeP],[Go1 ],[Fl ],[YVo ;],these,arenotcomplete).lInthispapSer9wreconsideraproblemofageometricquantizationunderthisunderstandingofthe9phrysicalsenseofquantization.9Norw^let g cmmi12MBbSeacompactmanifold.2rWVedenoteby#!", cmsy10F12|{Ycmr80(M@)thegroupofinvertible9FVourierOinrtegralopSeratorsoforderzeroactingonfunctions(ormorepreciselyhalf9 ff<  1991': cmti10MathematicsSubje}'ctClassi cation.53D12,UU58J30,58B15,53D50. 늍 Key;wor}'dsandphrases.symplecticPAmanifold,QEgeometricquantization,pGolarization,Lagrangian foliation,F*ourierEintegralopGerator,Kahlerstructure,quaternionpro8jectivespace,pairingofpGo-larizaton,UUpuncturedcotangentbundle. O?/o cmr91*iҍe.&e924~KENR9OTFURUT:ANIiҍ 9densitiesonM@,ɵandbryDSifGf21RA!2cmmi8syI{mp (T2$K cmsy8RA0aM)thegroupofhomogeneoussymplectomor-M9phisms lofthepuncturedcotangenrtbundleT2RA0aM@.+AlsobyP20(M@)wedenotethe69group;ofinrvertible;pseudo-di erentialopSeratorsoforderzeroonM@.+WVeonlycon-9sider7theopSeratorsinF120(M@)harvingclassicaltotalsymbSols,Kthatis,totalsymrbSols9harveasymptoticexpansionsbryhomogeneousfunctionsofintegerdegreesoneach9loScalcoordinateneighrborhood.8Thenwrehavetheexactsequence9Ս9(1.1)咋5[F0UR !URP20(M@)!F120(M@)ߒ!?DSifGf21RAsyI{mp (T2RA0aM@)UR !UR0:9ԍ9HerethemapϹisthecorrespSondencefromaFVourierinrtegraloperatortoits9canonical.relation.>rInourcasecanonicalrelationsarethegraphsofhomogeneous9symplectomorphisms.9ThereTwillbSenonaturalwraysTtoconstructasectionofthismaponthe\whole"9group|DSifGf21RAsyI{mp (T2RA0aM@),Ehorweverforexamples,Eifweconsiderasubgroupconsisting9ofgVthehomogeneoussymplectomorphismscomingfromdi eomorphismsoftheun-9derlyingPmanifoldM@,jjthenwrehavethenaturalrealizationofeachdi eomorphism9as>aFVourierinrtegralopSeratorandthisgivesusasectionofthemapTwrestrictedto9this$subgroup._OrifwretakeacompactsubgroupinDSifGf21RAsyI{mp (T2RA0aM@),>then$thereisa9sectionofthemap,realizingeacrhsymplectomorphismasanHermitetypSeFVourier9inrtegralopSerator(see[BGE]).8InthesecasesthesectionofthemapXisnotunique.9TVo ndasectionforthemapXwreconsiderthefollowingway:9(1)%Let0%n eufm10hbSeaHilbertspaceconsistingofcertainclassof\nice"functionsde ned9on T2RA0aM@.^ IfwrehaveanopSeratorT 0:jjh!L2(M@) whichgivesanisomorphism,%then9brytakingasubgroupGUR?DSifGfsyI{mp (T2RA0aM@)whichpreservesh,then9(2)theproblemisreducedtodeterminewhethertheopSeratorTn(CFTƟ21isa9FVourierݐinrtegralopSeratorforC122Gsuchthatitscanonicalrelationisthegraphof9Cܞ.9TVoconstructsucrhasubspacehandanopSeratorTѹ: h!L2(M@)weemploya9methoSdX devrelopedbryJ.H.Rawnsley([Ra1e],sf[Ra2],[Wh]).WVeX explainthisinthe9nextoRsectionx2.TherewreputseveralstrongassumptionsA(1)A(5)onsymplec-9ticmanifolds,&thentheygivreusquiteexplicitresults.uAlthougheachassumption9isEsatis edbrymanysymplecticmanifolds,forthemomentitisnoteasyto nd9symplecticemanifoldswhicrhsatisfyalltheassumptions([FT'm],=[Ii1 `],[Ii2],[Sz1],[Sz2],9[Sz3]).In;x3wregatheruptheconcreteresultsforseveralsuchsymplecticmanifolds9(= puncturedcotangenrtbundlesofcompactrankonesymmetricspaces,,[Ra2e],[FYy],9[F1 ],[F2]).9InEx4wredescribSepreciselythespacehconsistingof\nice"functionsandan9opSeratornJTforthepuncturedcotangenrtbundleofthequaternionprojectivespace9([F2 ]). iҍe.&ewPOLARIZA:TION!ANDQUANTIZATIONy#3iҍ zx2.|qPolariza32tionsandtheirpairing69In thissectionwreexplainbrie yamethoSd,?socalled,pairingofpSolarizationsfor69symplecticmanifoldssatisfyingsomewhatstrongconditions,Hsinceinx4wreonly9exhibitexplicitcalculationsforamorespSecialcase.9Let:(XJg;8!n9)bSeasymplecticmanifoldwiththesymplecticform!.^WVeassume9that,thecohomologyclass[!n9]isaninrtegerclass,}Xanddenoteby(L,}Xr)theline9bundle|withtheconnectionrcorrespSondingtothesymplecticform!n9.WVesarythat9a&EcomplexsubbundleF inthecomplexi edtangenrtbundleTX , msbm10Cisapffolarization,9ifb dimT-ppmsbm8C >mFP= 1=2dimPXSandb satis esinteffgrabilityb conditions. AmongpSolarizations9wreonlytreatwithrealandcomplexpSolarizations,thatis,&h$(a)2(Fnisreal,ifF=UR\-z < ӍF H,}(b)2(G꨹iscomplex,ifG\\-z D ӍGD=URf0g.9WVemruststillassumethatacomplexpSolarizationsispositivre,thatis,Tp Ήz5S 215U!n9(s;ꦟ: z7  ݹ)UR0;ꦹforanry.2?G:9WVeknorwthatifasymplecticmanifoldhasapSositivecomplexpSolarization,then69itRisaK ahlermanifold,andthesymplecticformcanbSeaKahlerform.-AlsoifFkis9arealpSolarization,wremaythinkFnisacomplexi cationofaLagrangianfoliation./9Prop`osition62.1.$qLffet+F͂bearealpolarizationandGapositivecomplexpolarization,9then35wehavealwaysT9(2.1):FLn\\-z D ӍGD=URf0g:9Note/thatvrectorsinapSositivecomplexpSolarizationGaretheanti-holomorphic9tangenrtvectors.9Let*|FBbSearealpolarizationandGapositivrecomplexpolarizationonXasaborve,9thenbrytheprecedingpropSosition2.1,wehave9Prop`osition2.2.p9(2.2):Kܞ F rzN _KܞGPH԰0=$max&d&u cmex10^7wiTƟ aX+ C;kj9andTƟ2aX+ CistrivializedbrytheLiouville35volumeform{1 X r۹=ō(1)2n(n1)=2[z="w ΍mn!B/!n9 n;ꦹ2nUR=dimXJg:29Hencecwrecande neapffairing<UR ;c >,~for(smoSoth)sections h2(Kܞ2F;X)and9 2UR(Kܞ2G6;X),brye,9(2.3)pΉz5S 21\1nY<UR ; > X r۹= 7^: zLm :T9Ingeneral,forapSolarizationFnwredenotetheannihilatorXSFƟ  =URfm2TƟ aM Cj?<s;x>=0;foranry.x2Fg;09andVbryKܞ2F Թ=max fk )V_FƟ2aʹ,socalled,thecanonicallinebundleofthepSolarizationF69(\max"lmeansthehighestdegreeexteriorproSduct).Onthecanonicallinebundleܠiҍe.&e944~KENR9OTFURUT:ANIiҍ 9wrehaveaconnectionr2F alongthepSolarizationFƹ. cThisisde nedbytheLie69derivXativre:΍r FڍO(s)UR=iRLods;s]9where%i istheinrteriorproSductbya(complex)vector elds2(Fƹ)takingvXalues9inFnandsUR2(Kܞ2F;X).9ThepairingbSetrweenKܞ2F andKܞ2G չ(Fereal,PGpositivrecomplex)isde nedina9natural`wrayasabSove,butwhatwewanttode neisapairingoftheirsquareroSot.獑9In'general,NtheexistenceofthesquareroSoty-p 'y-z ӍKܞF":andy-p 'y-zN ӍKܞG"J`ofcanonicallinebundles9is6guaranrteedbythepropSertythattheirChernclassesaredivisibleby2(inwhich9case,wrecallXbSeingmetaplectic).mInourcasesbelorw,bNthatisXFY=TT2RA0aPƟ2nJHthe9punctured`Hcotangenrtbundleofthequaternionprojectivespace,}therearenowhere9vXanishingsmoSothglobalsectionF ofKܞ2F andanorwherevanishingholomorphic9global sectionG -^ofpSositivrecomplexpolarizationKܞ2G6.;So,Vwrede nesimplythe9pairingofAp ꪟAz KFMandAp ꪟAz DKGas׍9(2.4)l:<URp UTz TDFO;ꦟpꨟz DTDG! >>=UR'spUT'szJu ؍jUR<FO;G t>jZT:썑9InwSection4wrepresentsomeresultsofexplicitcalculationsofpairings,explicit9constructionsofaHilbSertspacehandoperatorTforthecaseXqbeingthepunctured9cotangenrt=bundleofthequaternionprojectivespace.2xSoFVromnowonweworkon9onlysucrhsymplecticmanifolds(XJg;!n9)likeapuncturedcotangentbundle,thatsatisfy9furthersevreralstrongconditionsA(1)A(5):UA(1)2(The35sympleffcticformisexact,i.e.,!Ë=URdwithaspeci crealone-formS,UA(2)2(therffe35existsarealpolarizationFsuchthat<URS;33F>=0,UA(3)2((i)35therffeexistsapositivecomplexpolarizationG.2(So.wecffanseefromthebeginningthatXisacomplex(Kahler)manifoldand2(G35isasubbundleof(0;1)-veffctors,andmoreoverassumethat2((ii)35therffeexistsa(1;0)-formnsuchthatdË=UR!(notethat<URn9;33G>=0).UA(4)2(Therffe35exist2((i)35anowherffevanishingsmoothglobalsectionF BofKܞ2F2(and,2((ii)X2anowherffevanishingholomorphicglobalsectionG wofthepositivecomplex2(pffolarization35G.UA(5)2(FinallyK/weassumethattheleffafspaceX=Fisasmoothmetalinear(ormore2(strffongly35orientable)manifoldfortherealpolarizationF.9UnderMtheseassumptionswredonotdistinguishhalfformsandhalfdensities.Infact9from^theassumptionA(5)wrecantakeanowherevXanishingsmoSothglobalsection9F as;thepullbacrkn92.=(dvn9ol)ofavolumeformdvn9ol3;ontheleafspaceX=FO(soA(4)9(i)issupSer uous).8ByA(1)oursymplecticmanifoldcannotbecompact.9Sincetheconnectionr2F C(resp.r2G)ofKܞ2F (resp.Kܞ2G6)isde nedasLiederivXativre,|9wre?hcanintroSducetheconnectionony-p?jy-z ӍKܞF$i׹(resp.7!y-p7#y-zN ӍKܞG"2q)alongF.(resp.7!G).Hence9wregalsohaveaconnectiononL y-p ϟy-z ӍKܞF#R(resp. L y-p ϟy-zN ӍKܞG)ginanobviouswayV,and9canddde nethespaceofsmoSothsectionsFO(Li y-p ky-z ӍKܞFp;dbX)dd(resp. G(L y-p ky-zN ӍKܞG;X))+iҍe.&ewPOLARIZA:TION!ANDQUANTIZATIONy#5iҍ9whicrhVconsistofparallelsectionswithrespSecttotheconnectiononL y-p y-z ӍKܞF!b(resp.|9L y-p y-zN ӍKܞG).69The*sF:9Also#tG:9LetsF d=URtG,thenmrustsatis es9(2.5)Ls()+2n9p n;Ήz5S 21UR<6n9;>=0; Nforanryvector eldn:9Thishasauniquesolutionexceptconstanrtmultiples.獑9UnderZthesetrivializationsoflinebundlesthesections'UR2G(L y-p y-zN ӍKܞGT;XX)Zare9writtenwithholomorphicfunctionsf2onX+asaform'UR=ftG @ p z DTDGw;^9andthesections Ë2URFO(L y-p y-z ӍKܞF;X)bry Ë=URn9 .=(gn9)sF 0p 0z dvol9withgË2URCܞ21 ܦ(X=Fƹ),:URXF!X=Fƹ.|9NorwweintroSduceinnerproductsonG(L i y-p ky-zN ӍKܞG;X)andFO(L y-p ky-z ӍKܞFp;X)as9follorws:8let'i,=URfitG @ Ap Az DKG@2G(L y-p y-zN ӍKܞG;X)(i=1;2),thens"9(2.6)]('1;'2)G tUR甆Z _Xf1j: z f2(tG;tG)'sp'szG% ؍jUR<G;G t>jXzh ;9where &9(2.7)pΉz5S 21@1n>J<URG;G t> =G @^dz DKGD9isde nedbrythesamewayas(2.3),andfor i,=URn92.=(gidڹ)@.sF O} lp@0lz, Bn9(dvn9ol)=2URFO(L %9y-p9y-z ӍKܞF*$;X)(iUR=1;2),N9(2.8)ߤ( 1; 2)F dUR甆Z _X|=F7g1jdz `[Kg2dvn9ol:,O9SothecompletionisjustisomorphictoL2(XJg;dvn9ol).9Finally,^wrede nethepairingbSetweenG(Le y-p gy-zN ӍKܞGҵ;,\X),^andFO(Le y-p gy-z ӍKܞFl;X)9bry9(2.9)glhgChghmd n9;'ii31i 3.UR甆Z _Xgf(sFO;tG)'sp'szG͟ ؍jUR<FO;G t>jXj ;⍑9where F=~ n92.=(gn9)!YsF 0 lp![lz, B.=(dvolC)?y2~ FO(L y-p ![y-z ӍKܞF `;X),ĒgF2Cܞ21 ܦ(X=Fƹ)and'=%9ftG @ Ap Az DKG@2URG(L y-p y-zN ӍKܞG;X)(f2:8holomorphiconX).>ݠiҍe.&e964~KENR9OTFURUT:ANIiҍ 9AlthoughatherearemanrysymplecticmanifoldswhichsatisfyonlyoneofA(1),69A(2)orA(3)likrecotangentbundlesorK ahlermanifolds,itisnoteasyto nd9symplecticmanifoldswhicrhsatisfyallourassumptionsandmoreoveratthisstagewe9doUnotguaranrteetheexistenceofgloballyde nednon-zeroholomorphicfunctionson9X.,Horweverforsucrhsymplecticmanifolds,5our^ rstprffoblem,#toconstructaHilbert9spffaceݡhcffonsistingof\nice"functionsonX$andanoperatorT:URh!L2(X=Fƹ),was9rffeducedtocffalculateexplicitlythequantities(2.4),(2.7),tosolvetheequation(2.5),9and35toshowthenon-deffgeneracy35ofthepffairing(2.9).#(lG3.{Comp32actrankonesymmetricspaces69In"thissectionwreexhibitapSositivecomplexpSolarizationforthepuncturedcotan-9genrtabundlesofthesphere,*quaternionprojectivespacesandtheCayleyprojective9plane,U/and?(1:0)-formssatisfyingA(3)(ii)foreacrhcase.8WVealsogiveanowhere9vXanishingVholomorphicglobalsectionforthecasesofthesphereandthequaternion9projectivre,space.Ofcoursecomplexprojectivreplanesatisfyallourassumptions9A(1)A(5),xbutwreomitit([FT'm]).RThespherecaseplaysanimpSortantrolefor9describing2thequaternioncase.8Inallourcases,therealpSolarizationisthevrertical9foliationoftheprojectionmaptothebasespace.9[@I]LetSן2n EϹbSeaunitsphereinR2n+1̹,=Z$Sן n y=URfx=(x0;:::ʜ;xnP)2R n+1otjꦟX*x 2ڍiV=k USxk*2=1g;yߍ9withthestandardRiemannianmetricinducedfromtheEuclideanspaceR2n+1̹.WVe9idenrtifytangentandcotangentbundlesbythestandardmetric:LXT ڍ0aSן nP y԰ a=["T0Sן n y=URf(x;yn9)2R n+1/tR n+1otjk xkL=1;yË6=0;(x;yn9)=Xxidyi,=0g>9It̓iswrellknown([Ra1e], [Ra2],[So2G])̓oftheK ahlerstructureonthepunctured9cotangenrtbundleandisrealizedbythemapr9(3.1)?S s4:URT2RA0aSן2n y=T0Sן2n!?C2n+1򍍑}(x;yn9)27!URS(x;yn9)=z5=(z0;:::ʜ;znP);zi,=k USyn9k~1xi+p ljz5S m91yid:r9Theimageiscrharacterizedbyanequation:9(3.2)xS(T0Sן n[')UR=fz52C n+1nf0gjꦟX*z 2ڍi9=0g:>9The(1,0)-formn9,dË=UR!,isgivrenbyŇË=URp UTΉz5S 22@kzkO;k zk>/=URqUT؉z'2r(XUVjzidj26;_9andwrehaveamorepreciserelation:4'9(3.3)RSr}";cmmi6n|=?thecanonicaloneform=ōs-ps/ljz5S m91s-[z5U Spljz 92"۵(@kzkO: z @ kzkݹ):R9iҍe.&ewPOLARIZA:TION!ANDQUANTIZATIONy#7iҍ 9Anorwhere!vXanishingholomorphicglobalsectionG 3ofthecanonicallinebundleof69thiscomplexstructureisgivrenby|ݍ9(3.4)G t=ō 2[z; mȍkzk7*2$unyX ㇍Zjv=0/I(1) jf dz \KzjAfdz0j^UN^dznP:"|ҍ9We35willdenotethisbyS Qherffeafter.M9[I`I],WVeexplainaK ahlerstructureonthepuncturedcotangenrtbundleofthe9quaternionprojectivrespacefollowing[F1 ]preciselyV.9LetHbSethequaternionnrumber eldorvertherealnrumber eldR,0whicrhis9generatedbrythebasisf5DF cmmib10e0 d;e1 d;e2;e3g꨹withtherelationse Wi 1ej钹givenbythetable69(3.5){&bffw J<͟h5ffN5ff5ffe!b0+3h5ff4e;1I>h5ffSeZR˟2hHh5ffrh&ey ՟3DRh5ffhw͟h5ff͟e q|c0M5ff5ffe!b0+3h5ff4e;1I>h5ffSeZR˟2hHh5ffrh&ey ՟3DRh5ff_ffw͟h5ff͟e q|c1M5ff5ffe!b1+3h5ff0Ife01h5ffSeZR˟3hHh5ffmze21h5ffffw͟h5ff͟e q|c2M5ff5ffe!b2+3h5ff0Ife31h5ffOpe01h5ffrh&ey ՟1DRh5ffffw͟h5ff͟e q|c3M5ff5ffe!b3+3h5ff4e;2I>h5ffOpe11h5ffmze01h5ffffw5⍑9WVeregardH2n asarighrtH-vectorspacewiththeH-innerproSduct9(3.6)shyh;kgiSßHa= knURX ㇍Si=1S(hidڹ)ki!E9wherehUR=(h1;:::ʜ;hnP);ko=UR(k1;:::ʜ;knP)UR2H2n QTandS(x)=x0e d0IM$x1e d1$x2e d2$x3e d3q9forxr =23 fk`P I퍓i=03_xide in2H.kThentheRbilinearformh3h;kgi#M=ō>1>[z ΍2 mfhh;kgiH%N+%h`kg;hiHgonH2nt9de nes"aEuclideaninnerproSductonH2n asarealvrectorspace.?WVewilldenoteits9extensionvtothecomplexi cationH2n+ fCasacomplexbilinearformwiththesame9notationh Q; UPi!.,.9Let M@(n;H)bSethespaceofnon H-matrices,T5andforX3'=A(xijJ)2M(n;H)9de nerespSectivrely5S(X)ؑ=UR(S(xijJ));9(3.7)u܍( t85X)ijؑ=URxjviJ;9(3.8)/dftrʊoXؑ=URXxiiI;9(3.9)JB TʊoXؑ=URS( t85X):9(3.10)k9Eacrh]matrixX 27M@(n;H)de nesarightH-linearmapX :2nH2n !7H2nP,zandwe9harve9(3.11)hRXh;kgi9HG=URG h; TGXk|G 0.8H8:5Í9The;groupSp(n)isthende nedasagroupconsistingofthosematricesX29M@(n;H)whicrhpreservetheH-innerproSduct.bJiҍe.&e984~KENR9OTFURUT:ANIiҍ 9ThespaceH(n;H)=fX2DM@(n;H)jX=D2TXgiscalledaJordanalgebrawith69theJordanproSduct9(3.12)JX+Y¹=ō1[z ΍2 (XYG+YpX):9NotethatforX2&H(n;H),?trczX2Re0P԰=*RandH(n;H)isequippSedwitha9EuclideaninnerproSductgivrenbyu 9(3.13) (hXJg;YpiRع=URtr x(X+Yp):9TheinnerproSducthasthepropertry:9(3.14)-hDX+YM.;Zܞi"՟RܿB=URhXJg;YGZki9rRA:9Asiswrell-known,'thequaternionprojectivrespacePƟ2nJHisthesetofallH-one-69dimensionalѢsubspacesinH2n+1̹.3HerewreidentifyPƟ2nJHwiththesubsetinH(nl+1;H):!׍9(3.15)6@PƟ nJ(H)UR=fP2H(n+1;H)URjP=(pidS(pjf ));pUR=(p0;:::ʜ;pnP)2H n+1;hp;pi=1g:9WVeUvconsidertheRiemannianmetriconPƟ2nJHde nedthroughtheHopf bration9: рSן24n+3g!hPƟ2nJH,pwhere"Sן24n+3=fh2H2n+1jhh ih;hi(r=h kh;hi!H-V=1g"hasthe9standardmetric.9LetusdenotetheisomorphismHUR!M@(2;C)ƍ9(3.16)WHUR3h7UY!qTx0j+p ljz5S m91x1`!x2j+p ljz5S m91x3򍍑*x2j+p ljz5S m91x3`x0jp ljz5S m91x1~q(2M@(2;C)9bryandwedenotewiththesamenotationitscomplexi cationq9(3.17)}H C(* O2pUR!M@(2;C): 9NorwweintroSducethefollowingspaces:O썑2aES s4=URf(p;qn9)2H n+1/tH n+1jUPh p;pi#ܹ=1;hp;qn9iwù=0;UPq+p(qn9;p)H 46=0g:{9ThespaceES !isSp(nj+1)-inrvXariantandalsoinrvariantundertherighrtactionof9Sp(1).6ִE 0ڍSI&Z=URf(p;qn9)2H n+1/tH n+1jUPh p;pi#ܹ=1;qË6=0;(qn9;p)H 4=0g;56;EHI&Z=URq US(PS;Q)URjP;Q2H(n+1;H);tr #P=UR1;PLnQUR=ō1[z ΍2 Q;Q6=0q;uX;8Sf6ִESI&Z=UR n US(B0;:::ʜ;BnP)UR2M@(2;C) n+1jz3^w6=0;XUTdet'Bi,=0 o&r|;Cݍ uiҍe.&ewPOLARIZA:TION!ANDQUANTIZATIONy#9iҍJ9whereb-Bi2M@(2;C)takrestheformBi= ŸqdXz2i4[w2i5 z2i+1.w2i+1ISqR,andz= (z0;:::ʜ;z2n+1й),59w=UR(w0;:::ʜ;w2n+1й).񍍟ඍR0mfJPDEߍ0bSc=UR n US(B0;:::ʜ;BnP)UR2{fESjUPXBidB ڍi\=JpXqƟ\-z ( ӍBi#^ȟt&BiJr o!:;'9where9J"=۲q[0%]15 1%]0+Yq8KandtheconditionPEBidB2RAi 6=۲JpP\-z ( ӍBOCit"EBiJV can9bSerewritten:9asꨟPSz2iޟdz5Kz 2i+1"+w2iޟdzݟKw]2i+1&4g=UR0.65Ez9fCEHV6=URG UTAUR2M@(2n+2;C)jJA?= tx/AJ;ꦹrank!AUR=2;A 2V=0G  :T9HerewredenotebyJ+퍍wJ=eUR0ǍURB38URBURBUR@sJRȈ@q cmti12O5)cqJ;.@A.Dߟ.ROV£JedW21ǍdW2C38dW2CdW2CdW2Ar,2URM@(2n+2;C):#Eō9WVewillde nethenormofamatrixAW=(aijJ)2M@(n;C)bryk Ak=WpWz'@ Pjaijj2:=9lp9lz* Btr #(AA)HɁandthenormofanelemenrtBX2n5eURES:Nbyk Bk$=UR%?qUT%?z+PkBi k&*2;.8$9Remarkx3.1.|5TheinnerproSducth ;i"!iR-FionH(nPD+1;H)istherestrictionoftheY9R-bilinear0Lformōc1c[z ΍2vtr4(X2T9Yv+Yp2TX)onM@(n+1;H)andcanbSeextendedtoM@(n+;>91;H)V; CP԰="M@(2n+2;C)asacomplexbilinearforminanaturalwrayV.,Wewill69denotethisbilinearformbry_4he޺k4; UPi|wC:URM@(n+1;H) CM(n+1;H) CUR!C:9ThentheHermitianinnerproSductonM@(n˹+1;H) CPUR԰n:=M(2n+2;C)isgivrenby9G A;\-z ( ӍB &G 1)8C:,and<+thenormofAUR2M@(2nFB+2;C)P<+԰U=ͭM(n+1;H) C,_whicrh<+weintroSducedxU9abSorve,equalstok Ak;=0q 0z0]ɟ ]2G A;\-z ӍA G $68C<]˹,thatis,G TA;\-z ӍA G &!\8C0S=1=2tr 0(AA2):܍9Nextwrede nethemaps ; O;H;S;S andH amongthesespaces:!B=lAM h:^ES!^fEH5f835ff[݄35ff[Ysd(p;qn9)}d 7UY!d(PS;Q);ǘP=UR(pidS(pjf ));Q=(piS(qjf )+qiS(pjf ))A)؍ ׍AM :sfqES!0fEH5xۿ35ff[ڛ35ff[Y3 (B0;:::ʜ;BnP)7UY!@6AUR=(AijJ);^,Aij 6=URBidJr2tTBjf J=lAMS s4:d)ESJ!Sן24n+35ks35ff[c35ff[^ʹ(p;qn9)o7UY!Zp iҍe.&e9104~KENR9OTFURUT:ANIiҍ卍=lAMH 4:gMEHC!:PƟ2nJH5n}35ff[ ߄35ff[_Iʹ(PS;Q)7UY!PA)؍ ׍AMS s4:bES!xfeOES5i35ff[F35ff[]ʹ(p;qn9)o7UY!dʹ(B0;:::ʜ;BnP);'\Bi,=UR(kqn9kpi 1+qi p Ήz5S 21); kqn9k"0=URqUT׉z 2)hqn9;qiଟH% ׍AMH 4:eEHC!9foBEH5mT}35ff[35ff[]ʹ(PS;Q)U7UY!AUR=(AijJ);+;]yAUR=k USQk*2c((PijJ))(RG=ō1[z ΍2tr8(Q1jQ2)UQ9for(PS;Q1),(P;Q2)2EH.9ES ?Dis!banopSensubspaceofthetangenrtbundleofSן24n+3w.Itconsistsofthosevectors9whicrharenotparalleltothe bSeroftheHopf bration.ff9The|mapS(resp.r[H)isanisomorphismbSetrween|thespacesES ^and%fESKL(resp.J9EH andQfEH).8AlsowrehaveS(E20bS)=ඍgfJEߍ0bSϤand9Prop`osition3.2.꨹(a)E20bS Qis35Sp(1)-invariant,andඍ^fJEߍ0bS`isSU@(2)-invariant,}(b)2(the35followingdiagrffamiscommutative:2M9(3.19)_dSן24n+3sX.SB UOcE20bSsX.SUO!ඍºfJEEߍ0bSKl3,r?38r?ry㍒s3,?38?y㍒ 3,»?38»?»y'¼ 6QPƟ2nJHB UO33(`.O+msbm6HFEH덒3q`HUO!˝fEH+H:;F˷(c)2(the35mapS Qcffommutes(onES)withtheactionofSp(1)  O2p!flSU@(2).O9Remark3.3.Themaps TS andH c 7donotcoincideonthewholespaceES.9BytheRiemannianmetricwreidentifythetangentbundleofthequaternionpro-69jectivrekspacePƟ2nJHwiththeircotangentbundleaswedidin[@I]kforkthesphere.Both9spaces4 ES RandEH canbSeseenascomplexmanifoldsthroughthemapsSandH.9Thenwrehave iҍe.&ewPOLARIZA:TION!ANDQUANTIZATIONt11iҍ 9Prop`osition~3.4.qPLffet+!H bethesymplecticformsonthecotangentbundleofthe69quaternion35prffojectivespace,thenkf!HFĹ=UR WڍH ㌹2O33Aacmr6133s^\)4 pΉz5S 21!`: z @(`@'sp 'sz ؍kAk!v o9:9(3.20);9Infactwrehaveamorepreciserelationsimilarto(3.3):!v9Prop`osition3.5.!9(3.21)y py Ήz5S 21_ WڍH(@'sp 'sz ؍kAk$ : z @ 'sp'sz ؍kAk*T)UR=2O33333s^\)4 P.:nLqHu9wherffe35P.:nLqHisthecanonicalone-formonthecotangentbundleTƟ2aPƟ2nJH.Ԧ9When:wreconsidertheactionofSLO(2;C)=(fr*2HD CjrSS(r)=1g):onJcfES9fromtherighrt,wehave9Prop`osition3.6.9(3.22) :'fESJ!fUREHu9is35aprincipffal berbundlewiththestructuregroupSL)J(2;C).9ThiseuprincipalbundlecanbSeseenasacomplexi cationoftheHopf brationwith9\the%complexi edbasespace"(=thepuncturedcotangenrtbundle)and\thecom-9plexi edstructuregroup".1WVeusethisstructureindescribinganorwherevXanishing9holomorphicglobalsectionofthecanonicallinebundleKܞ2G ޹ofT2RA0aPƟ2nJHinx4.E9[I`II]|Finally|6wredescribSeapositivrecomplexpolarizationforthepuncturedcotan-9genrt$bundleoftheCayleyprojectiveplaneandgivea(1;0)-forminAssumption9A(3)(ii)([F2 ]).9TheCaryleynumbSer eldO?isadivisionalgebraoverRgeneratedbythebasis9fei g27RAi=0includingċthequaternionnrumbSerċ eldH.ƊEspeciallyV,amongtherelations9eޢiC|e+j#N5,wrehaveu9(3.23)e=1dAe4F=URe 5;e 2de ^4=URe 6ande W3O[e 4 `=URe 7=a:9HenceO?isidenrti edwith9(3.24)POP԰Ð=UQHHe49andńthemrultiplicationbSetweenxUR=a^̹+be4*7andńyË=URh+kge ̟4T2HHe4*7isńgivrenby9(3.25)sxyË=URahS(kg)b+fka+bS(h)ge4 d;Fl9whereZ_hx=P*%3 U_%i=0 yhide i2H(hixR2R)andS(h)=h0e d0nh1e d1h2e d2h3e d3$,vLandZ_so69on.nWVeassumethatthebasisfei g27RAi=0=areorthonormalandwrewillsometimesomit9eޢ0N(=idenrtityelement)andidentifyRUR=Re0HOUV.-9FVorhUR=P*7 U_i=0 AShide iõ2OUV,wredenote$N9(3.26)S(h)UR=h0e d0_ 5R7X ㇍ti=1hide i nc;M9asforhUR2H. iҍe.&e9124~KENR9OTFURUT:ANIiҍ 9LetM@(3;OUV)bSethespaceof3w3matriceswithenrtriesinO.*Byidenrtifying69M@(3;OUV)PUR԰n:=M(3;R) ROUV,wredenoteforXF2URM@(3;OUV)H9(3.27)_S(X)UR=X0j e OW0 5R7X ㇍ti=1Xi e OWi 1;0P@9(3.28)6 tXFչ= 7URX ㇍Si=0 tand.theinnerproSduct(;)9tofthecomplexi cationJ R/CUR=J2C ȹinfanaturalwrayV. SoftheHermiteinnerproSduct9h;i꨹onJ2C 1ùisgivrenby@>9(3.38)hȖXJg;Ypi݄=UR(XJg;\-z m ӍY m);59wherev\-z ӍXϹ=ijP*7 U_ =0"(\-z ӍX- 6@_ e W` L,iX _2ijM@(3;C)vand\-z ӍXe risthecomplexconjugateofX .9ThetnormoftheseelemenrtsinJandJ2C Wisalwayswrittenask u eke̹,andwewritethe9normofthetangenrtvectorY2URTX(PƟ22aOUV)byk YpkXTP.Źiҍe.&e9144~KENR9OTFURUT:ANIiҍ 9NorwconsiderthemapO a:URT2RA0aPƟ22OUV(P԰= T0PƟ22O)UR !URJ2C 1ùde nedbryTmFXOx(XJg;Yp)9(3.39)op=UR(kYpkm*2-X+YGYp) 1+ō1۟[z Sp ljz 92 k Yk.^YG p Ήz5S 21䍍p=UR(2kYpk*m2 UUmP5X+YGYp) 1+kYkTP{YG p Ήz5S 21;9thenwrehave덑9Theoremj3.8.-ThemapO givesanisomorphismbffetweenT2RA0aPƟ22OFandE=fA269J2C mjURAA=0;A6=0g.fiMorffeover9(3.40)  WڍOx(p Ήz5S 215S: z @"S@kAka818s^\)2)UR=ō 1[z Sp ljz 92!O:}9In+Rthiscasewrehavealsoamorepreciserelationsimilarto(3.3)and(3.21):3Let9P.:2XO*bSethecanonicaloneformonTƟ2aPƟ22OUV,then9Prop`osition3.9.H9(3.41){O|pQ|Ήz 22MP.:2XO =pΉz5S 21  ڍO͹(@'sp 'sz ؍kAk$ : z @ 'sp'sz ؍kAk*T):`39Thetrwo-formpljz5S m92 Ci: z @'i@kAka818s^\)2#(isitselfaK ahlerformonJ2CGnf0g,Vsothatwecan9regard8J2CGnf0gisasymplecticmanifold.Onthissymplecticmanifoldthe orwftgt2R9de nedbry6Ѝ"t:URA7!tʹ(A)=e 2=p\=\) |4Í1t# AEō9is1aHamilton orw. |TheHamiltonianofthis owisgivenbythefunctionf$:MЍ9A7!Fu ž1;z U`ҡ`p\`\)@o2ákâAka*ſ1*ſs^\)20#.6SinceEisholomorphicandthe orwftgleavesEinvXariant,)thei͍9Hamiltonianwofthis orwonEisjusttherestrictionoffvtoE,thatis,theHamiltonian9is~thesquareroSotofthemetricfunction.cSothe orwftgisthebicharacteristic9 orwbofthesquareroSotoftheLaplacianonPƟ22aOUV.Especiallythe orwrestrictedto9thesunitsphere=f(XJg;Yp)=2TP22aO:=k =YkFP#Lm==1gscoincideswiththegeoSdesic orw.9Sowrehave9Corollary%3.10.4kThegeffodesics (t)onPƟ22aOthroughapointXwiththedirection9Yϥ(kYpkmP&=UR135andX+Y¹=Fu1z@2 Yp)isgivenby"n9(3.42)]} (t)UR=cos2t(X+ō۹1۟[z ΍2 YGYp)+ō1۟[z ΍2 sin^$2tY+ō1۟[z ΍2 YY:;:84.Qua32ternionprojectivespaces69InthissectionwredescribSeanowherevXanishingholomorphicglobalsectionG wof9thecanonicallinebundleofthecomplexstructuredescribSedintheprecedingsection9forthepuncturedcotangenrtbundleofthequaternionprojectivespaceandgivea9explicitformofthepairing(2.9),0togetherwiththeconcretevXaluesofpairings(2.4)9and-(2.7). pWVedenotebryH FinsteadofG,sincewehavealsothatforthesphere9case=S,andbrydvH^Xinsteadofdvn9olC,thevrolumeformoftheRiemannianmetric9onPƟ2nJH.dSo,wrechoSoseanowherevXanishingglobalsectionF aofthecanonicallineQiҍe.&ewPOLARIZA:TION!ANDQUANTIZATIONt15iҍ 9bundleoftherealpSolarizationasF d=UR2n9nXH(dvH),vandsimilarlywretakeforthesphere69case:8F d=UR2n9bS(dvS),dvS2istheRiemannvrolumeformonSן24n+3w.9LetZFbSeavrector eldonM@(2;C)UNM@(2;C)p|p4nv{zp4nv}Ս;#n+1a:nf0gde nedby/JA9(4.1)y&wZ1=ō1[zH. mȍkBk**2"z +2n+1,X ㇍.ţi=0AȴщzVm/ō33@D33[z ΍n@ziōZj@VRT[z, ΍@zii[+2n+1"dX ㇍ei=0vщz s/ō6@D33[z ΍@wiō5@//[z ΍@wiC\z!' ƍ9wherepwredenoteBX=UR(B0;:::ʜ;BnP)2M@(2;C) M(2;C)pandBi,=URqz2i3w2i5 *z2i+1.!Vw2i+1Iq:9asbSeforeandD>6=PSdet$ Bi9WVewritethe(4n+3)-formS onM@(2;C)UNM@(2;C)nf0g꨹(see3.4)bry͍9(4.2)E7S s4=ō \1[z< ә(2p ljz5S m915U)2n+2BaiZ6( ddz0j^UN^dz2n+1ox^dw0^UN^dw2n+1й):49Since<URdDS;Z1>?1,wrehave 9(4.3)@(2p Ήz5S 215U) 2n+2oxdD6^S s4=URdz0j^UN^dz2n+1^dw0j^UN^dw2n+19andtherestrictionofS %ƹtoPdet%IBi=0isholomorphicandnorwherevXanishingas9explainedininx2[@I].WVewilldenotetherestrictionofS tofES7withthesame69notation.m]9Letꨟq pljz5S m91>J0򍍑j0/Up ljz5S m91Rq]UW;q0$ 15 V1$ 0)꥟q6andꨟqj0/Up9Wljz5S m91򍍍 pljz5S m9190I*qUꪹbSeelemenrtsin~o9su(2)sl (2;C).6UWVedenotebryY1;Y2 andY3thevrector eldson`/fES`correspSonding9tothesethreeelemenrtsrespSectivelyde nedbytheactionofSLཹ(2;C).9Letusconsiderthe4n-formXongfESϤgivrenby 9(4.4)7Ë=URiYq3 B|iYq2iYq1 Թ(S):9Then޹isaholomorphic4n-form,rinrvXariantundertheactionofthegroupSp(n5+1)9onnf`ESfrom`theleftandSLu(2;C)-inrvXariant`fromtherighrt.MAlsonotethatdet(Adg)69= 1foranryg2~SL(2;C). SothereexistsauniquenowherevXanishingholomorphic94n-formH onQfEHjsucrhthat9(4.5)Ő O jS(H)UR=:9This4n-formH givresaholomorphictrivializationofthecanonicallinebundleKܞG5uH]I9ofQfEH,andisSp(n+1)-inrvXariant.69LetRV1;V2VandV3bSethreevrector eldsonthesphereSן24n+3\correspondingtothe9elemenrtsle z1~;e2|andle z3>Iinlsp(1)lHde nedthroughtheactionofSp(1)fromthe9righrt.9WVede nethreeone-forms1;2and3onSן24n+3bOinsucrhawaythat `idڹ(Vjf )Ih=URij9(4.6))idڹ(Vp)Ih=UR0 foranryV2TSן 4n+39(4.7)+Ѝ9whicrhisorthogonaltoVjf(j%=UR1;2;3).镠iҍe.&e9164~KENR9OTFURUT:ANIiҍ 9Norwfwehavethefollowingrelationsamongthesevector elds,one-formsand69vrolumeelements:5}(dvS)UR=2n9 2.=dvݟHv;9(4.8)}2n9 2.=n9 (dvHH)UR=iVq3 HiVq2iVq1 ߹(dvS);9(4.9)ƍ}1j^2^3^iVq3 HiVq2iVq1 ߹(dvS)UR=dvS:9(4.10)<9Heremeansthe bSerinrtegrationoftheHopfbundleË:URSן24n+3!PƟ2nJH.9Leti,=UR(SȊW18OS )2idڹ.8WVedecompSoseiOinrto9(4.11)(=i,=UR S0ڍi+ S00ڍi9withtheholomorphiccompSonenrt20RAi^ֹandtheanrti-holomorphiccomponenrt200RAi iinthe9complexi edcotangenrtbundleTƟ2aʹ(})fEST) C.9Then,wrehaveQ9Prop`osition4.1.9(4.12)f B S0ڍ1j^ S0ڍ2^ S0ڍ3^(iYq3 B|iYq2iYq1 Թ(S))UR=det( S0ڍidڹ(Yjf ))S:=9on^f35ES-.9Put9(4.13)GHSȊ^dzK ƍS9=URAS S9and9(4.14)HH c^dzK ƍH=URAH H9whereJ9(4.15)l S s4=ō(1)2(4n+3)(2n+1)[zNk ΍{(4n+3)!TW#!C4n+38OS]=ō1[z-m ΍(4n+3)!3_%!C4n+38OSa9and9(4.16) H 4=ō(1)22n(4n1)[z="w ΍(4n)!B/! C4nڍH7=ō=1[zJ ΍(4n)!Q! C4nڍHލ9aretheLiouvillevrolumeformsonTƟ2aSן24n+3bOandTƟ2PƟ2nJHrespSectivrelyV.69ThentheinrvXarianceofSand SunderthetransitivreactionofthegroupSOS(4no +94)ontheunitsphereinthetangenrtbundleTSן24n+3 jandtheinvXarianceofH Land9 H ̹underXthetransitivreactionofSp(n+1)Xontheunitsphereinthetangentbundle9TP2nJH꨹givreQ9Prop`osition4.2.ASܹ=URa }S ]k ^Bk(*4n+19(4.17)AHܹ=URa }H]k^Ak(uH*2n+29(4.18)9with35twocffonstantsa `Swanda `HB.Ġiҍe.&ewPOLARIZA:TION!ANDQUANTIZATIONt17iҍ 9AgainbrythesamereasonsasabSove,wecanputT,/(SȊW18OS ) (dvS)^dzK ƍS9=URBS S;9(4.19)lBS s4=URb OS/k0Bk';Y*19and(H cW1CH ) (dvHH)^dzK ƍH=URBH H;9(4.20)ƍBH 4=URb OHN/kN0Ak);9withtrwoconstantsb *S3/andb *Hㇹ.8Thenwehave%9Prop`osition4.3.ፍ9(4.21)W O jQqōAHrܟ[z ΍BH"Lq.v=URdet( S0ڍidڹ(Yjf ))ōBPAS33[z  ΍BS؍9onඍ^fJ35Eߍ0bS-.q9Byo(4.17)&to(4.20)%Xandk Bkrҟ*2"~=ꨟp ꪟljz 92ʦ O2jS(kAk)onඍgfJEߍ0bSϤwrehave9Corollary4.4. 9(4.22)2n9 2ōapa Sap[z ΍bSJdet'j( S0ڍidڹ(Yjf ))UR=qō]޹1 ]ݟ[z Sp ljz 92qq'Ff<2n+1ō?>gaF֒H?>g[zQ ΍bHN§:9Hencffe35det(2S0RAidڹ(Yjf ))35mustbeconstantonඍ^fJEߍ0bS-.9NorwwelisttheconcretevXaluesoftheseconstantsa ŸS5,b S,a -,H,b H wanddetS(2S0RAidڹ(Yjf )).9Prop`osition4.5.TQa9|S=URp Ήz5S 215U;9(4.23)b9|S=URp UTΉz5S 21;9(4.24)Qa|H=UR2 n2;9(4.25)Zdetjm( S0ڍidڹ(Yjf ))=UR2 3ι=det( S0ڍidڹ(Y p0ڍjj))=det( S00ڍip(Y p00ڍj))onඍfJEߍ0bS#;9(4.26)o^b|H=URō y133[z Sp ljz 92n92:9(4.27)9ލ9HeregkY2p0RAi(resp. !Y2p00RAi)istheholomorphic(resp.anrti-holomorphic)partofthevector9 eldYidڹ.8NotethatthevXaluedet=UR'spUT'sz ؍ja+HQ j&Q k,Q Ak;*n+1M;7Q9and}hereafterwredenotebyt7H'insteadoftG (globalsectionwhichtrivializesLwith9respSectQtothecomplexpolarizationG),thentheinnerproduct(2.6)isexpressedasy9(4.30)]4(; n9)G t=UR甆Z Z'a6cmex8f a 32@cmbx8E`HfG(A)`z pg(A)(tH ;tH)UR

UP H%nm_=UR甆Z Zf a E`HfG(A)`z pgn9(A)e 2 4b4ߟ`p ;`\)@o2?I{p I}׉z)k@Ak7sja+HQ jO33133s^\)2 k Ak_*n+11侹 H;9whereQ.=f΂t<HQ ΂Ap ΄Az fKH5,{ -=g<΂t<H ΂Ap ΄Az fKH c2QG(L΂ p΄zN iDKߍܞGnXH!ҹ). >Notethatthe9determination(of(tH ;tH)isdonebrysolvingtheequation(2.5)withthehelpofthe9relation(3.21),andisgivrenby[2tH=URe 2-33133J(w4 I{p I}׉z)k@Ak2)s HÊ:9WVedenotebryh2GnXH @thecompletionofthespace7&fft `H p z fTDH6jfGisapSolynomialoncM@(2n+2;C)restrictedtoDfB2EHRPg:čP;x԰;`=L1G!X'؍IlK=0[wP Hڍl ̍9withqrespSecttothenormde nedbrytheinnerproduct(4.30)!x,whereP2Hyl 'Bdenotesthe/9spaceğofpSolynomialsofdegreel.onM@(2n\+2;C)(restrictedğtoHfEHY).,2ThespacesP2Hyl9and5iP2H @ՍlK%q% cmsy60 areorthogonalforl6=ԐlCl0withrespSecttothisinnerproduct(4.30)"asCk9(4.31))<UR'spUT'sz2: ؍H x(dvHH)A;p z fTDH>=UR'spUT'sz ؍jb?H j&k,Aka<1<s^\)2DX;9then5 nallywrede nethepairingbSetweenthespacesFO(L+ p+ z iDKߍܞFnXH!)5andG(L+ O9p9zN iDKߍܞGnXH,5C)1asfollorws:Let'\=H2 x(fG)s ;H lplz2: BH(dvHH)Eand1 <=\gIKt̟Hj Ap Az fKHA,Cthen9wrede ney9(4.32)KhK㞪hL&hQ'; i]'iji gUR甆Z Zf a E`HH  x(fG)(A)gn9(A)(tH ;s_H)UP<UR'spUT'sz2: ؍H(dvHH)A;p z fTDH>UP H%ngi7=UR甆Z Zf a E`HH  x(fG)(A)gn9(A)e 2-33133J(w4 I{p I}׉z)k@Ak/'sp;'sz ؍jb?H jQ kW Akag*1g*s^\)2p H:iҍe.&ewPOLARIZA:TION!ANDQUANTIZATIONt19iҍ'9The|pairingabSorve|de nesanoperatorT:URgË7!Tƹ(gn9)2Cܞ21 ܦ(P2nJH)|forgË2URCܞ21 ܦ(ʩfEHT)69satisfyingasuitableinrtegrabilitycondition,thatiswrehaver鍍Gy甆ZN# _P.:nLqHbfGTƹ(gn9)dvHԹ=h hUPh 3. ڍH(fG)s Hn2 'sp'sz2: ؍H x(dvHH);gtbH p z fTDHE)iEm󞪫iE6iJ\:ꍑ9InfactwrehaveN9(4.33)lTƹ(gn9)dvHԹ=UR(H)(ge 2-33133J(w4 I{p I}׉z)k@Ak/'sp;'sz ؍jb?H jQ kW Akag*1g*s^\)2n H);{m9the bSerinrtegrationofthemapH.9Let2H bSetheLaplacianonPƟ2nJHwithrespecttotheRiemannianmetricde nedݍ9inex3andletfe`=p\=\) |4Í1t8p 8z1i0ȍH*+(2n+1)2Xgt2RibSetheoneparameterfamilyofunitaryand9ellipticOFVourierinrtegralopSeratorsgeneratedbythesquareroSotoftheoperator2H'0+9(2n+1)22.8T9The2Ybicrharacteristic owf`tgt2R5oftheopSeratorlp2[lzNC BH c+(2n+1)2bistheHamil-9tonͳ orwwhoseHamiltonianisthesquareroSotofthemetricfunctionandcanbe9expressedasfollorws:׸9Prop`osition4.7..9(4.34)`t:fUREH?!fUREH;`tʹ(A)UR=e 2=p\=\) |4Í1t"`A:9The/geoSdesic orwrestrictedtotheunittangentspherebundleofPƟ2nJHcoincides69withGtheHamilton orwabSoveundertheidenti cationofthetangentbundleandthe9cotangenrtbundlebytheRiemannianmetric.9FVromthetheoryofFourierinrtegralopSerators([Ho2]), weknowthatforeach9t2Rйthegraphofthesymplecticisomorphism`tDisthecanonicalrelationofthe9FVourierinrtegralopSeratore`=p\=\) |4Í1t8p 8z1i0ȍH*+(2n+1)2X.kThiscorrespondencecanbeseenas9aJquasi-classicalapprorximation.pAnoppSositecorrespondenceiswhatwrewantto9construct,andcarriedoutbrynotingthefollowings:'Since`2jRAt*(H)UR=e22=p\=\) |4Í1t(2n+1);AfH9and `2jRAt*(tH )/=tHd˹,-bwreshouldregardtheactionofthe owf`tgontheHilbSertspaceG㍑9hG5uH jsucrhthat(`tʹ)2(fz223t-H 23Ap 25Az fKHù)=e2=p\=\) |4Í1t(2lK+2n+1)Ff23t-H 23Ap 25Az fKHJforeacrhf.2/P2Hylx.9Thenwrehave׸9Theorem4.8.꨹(a)aThek?opfferatorT isanisomorphismbetweenthespacesh2GnXH and2(L2(PƟ2nJH).}(b)2(The35followingdiagrffamiscommutative8ƶ9(4.35)$Pih2GnXH[`-:!ft xxUO!&ih2GnXHKlT3,?38?y3,)?38)?)y1TL2(PƟ2nJH) dG!WPe"ttptɉO\ q71qNpqNz,HGH*+(2n+1)G2L2(PƟ2nJH):4=iҍe.&e9204~KENR9OTFURUT:ANIiҍ 5.Ц(Remarks6$(a)2(TVoshorwthattheopSeratorTisanisomorphism,Fweneedtodeterminecertain62( bSerinrtegrationsinaquiteexplicitformintermsofthe-function([F2 ]).}(b)2(TherearetrwoanotherwraystoconstructsimilaropSerators\鍒{Ti,:URh GڍH t!L2(PƟ nJH);i=1;2;2(to theopSeratorT=Ϲaborve.MThese areconstructedbrymakinguseofthecor-62(respSondingoperatorsconstructedforthesphere([Ra2e])andthecomplex2(projectivreGspace([FYy]),Hopf brationandthecomplexi edHopf bration2((3.6). !ItmighrtbSedesirabletoknowwhethertheopSeratorsTi,TƟ21are2(pseudo-di erenrtialornot.˷(c)2(LetgË2URSp(n+1),thenthe(dual)di erenrtialofgn9,\鍒7(dgn9) V:URT ڍ0aPƟ nJH!T ڍ0PƟ nJH2(israholomorphicandofcoursesymplectictransformationonT2RA0aPƟ2nJHand62(learvesthefunction[e 2-33133J(w4 I{p I}׉z)k@AkZ2(inrvXariant,swhichN~playsasaweightfunctioninthe bSerintegration for2(de ning+6theopSeratorTƹ.Thesefactsimplythatthenaturalaction(natural2(lifttoF120(PƟ2nJH))ofgn9,\鍒hgn9 :URL2(PƟ nJH;dvH)!L2(PƟ nJH;dvH);2(coincideswiththeopSeratorTLnd(gn921 ʵ)2jTƟ21 B.6}(d)2(EacrhofourassumptionsA(1)A(5)isnotstrongbyitself,howeveritis2(noteasyto ndmanifoldswhose(punctured)cotangenrtbundlesatis esall2(assumptions.simrultaneouslyV,Fandforthemomentweknowsuchexamplesas2(thexcotangenrtbundleoftheEuclideanspace,andthepuncturedcotangent2(bundlesM\ofcompactrankonesymmetricspaces.`Inalltheseexamplestheir2(geoSdesicx orwsareholomorphictransformations.rOInthepapers[Sz1],[Sz2],2([Sz3],k[Ii1 `]and[Ii2]theauthorsconstructedcomplexstructureson(punc-2(tured):cotangenrtbundlesofcompactrankonesymmetricspaces,Mwhichare2(notinrvXariantundertheactionofgeoSdesic owsandstudiedarelationbSe-2(trweentheircomplexstructuresandthatwreexplainedabSove.>(FVorthesesymplecticmanifoldsitwillbSeinrterestingtodeterminethegroup2(ofqholomorphicandsymplectic(homogeneous)transformations. ;WVeonly2(knorwccompletelyforthecaseofthepuncturedcotangentbundleofspheres2(([Ra1e],alsosee[FT'm]forlineartransformations).˷(e)2(HereEwredidnotexhibitanowherevXanishingholomorphicglobalsectionofthe2(canonicallinebundlefortheCaryleyprojectiveplanecase.˨WVewilldiscussthis2(problem]inaseparatepapSertogetherwithaconstructionofaquanrtization2(opSerator.GРiҍe.&ewPOLARIZA:TION!ANDQUANTIZATIONt21iҍ References9[Ar]4kV.fI.Arnold,iAchar}'acteristicenteringinquantizationconditions,iF*unct.fAnal.Appl. 4kB"V cmbx10Bv9ol.1(1)UU(1967),1-14.9[AS1]4kM.1F.Atiyah1andI.M.Singer, hThe,indexofellipticop}'erators,I,1Ann.Math.B87(1968),4k484-530.9[AS2]4kffT,UUTheindexofellipticop}'eratorsIII,Ann.Math.(1968),546-604. ㍍9[AS3]4kffT,)sIndexapthe}'oryforskew-adjointF;redholmoperators,)sInst.zHautesxEtudesSci.Publ.4kMath.UUB37(1969),5-26.9[APS]4kM.i\F.Atiyah,n^V.K.i\PatoGdiandI.M.Singer,n^Sp}'ectralTasymmetryandRiemanniange}'om-4ketry:]I,- Math.nPr}'oc.Camb.Phil.Soc.,v>B77(1975),543-69.- IGI,Math.Pr}'oc.Camb.Phil.Soc.,4kB78(1975),UU405-432.IGII,UUMath.Pr}'oc.Camb.Phil.Soc.,B79(1976),UU71-99.9[Ba]4kV.ABargmann,a;Irr}'educibleunitaryr}'epresentationsoftheL}'orentzgroup,Ann.AMath.4kB48(1947),UU568-640.9[Be]4kA.UUL.Besse,ManifoldsallofwhoseGe}'odesicsareClosed,UUSpringer-V*erlag.1978.9[BG]4kL.8%BoutetdeMonvel8%andV.Guillemin,=They sp}'ectralthoeryofT;oeplitzoperators,=Ann.8%of4kMath.UUStudies,PrincetonuniversityUUpress,No.99,1981.9[DG]4kJ.YFJ.DuistermaatandV.Guillemin,ZCThesp}'ectrumofpositiveellipticoperatorsandperi-4ko}'dicbicharacteristics,UUInventionesMath.,B29(1975),39-79.9[DH]4kJ.EXJ.DuistermaatandL.Hormander,{F;ourierinte}'graloperatorsII,EXActaMath.,{B128(1972),4k183-269.9[Fl]4kA.FloGer,.Amr}'elativeMorseindexforthesymplecticaction,.Comm.PureAppl.Math.4kB41(1988),UU393-407.9[FT]4kK.F*urutaniandR.Tanakqa,A$Kahlerkstructur}'eonthepuncturedcotangentbundleof4kc}'omplexandquaternionprojectivespacesanditsapplicationtogeometricquantizationI,4kJ.UUMath.KyotoUniv.B34(1994),719{737.9[FY]4kK.F*urutaniandS.Yoshizawa,AKahlerstructur}'eonthepuncturedcotangentbundleof4kc}'omplexLandquaternionprojectivespacesanditsapplicationtogeometricquantizationII,4kJapan.UUJ.Math.B21(1995),355{392.9[F1]4kK.5F*urutani,<Quantizationvofthege}'odesicv owonquaternionpr}'ojectivevspaces,to5appGear4kinUUAnnalsofGlobalAnalysisandgeometry*,KluwerUUAcademicPublishers.9[F2]4kffT,JAHFKahlerHtstructur}'eonthepuncturedcotangentbundleoftheCayleyprojective4kplane,toappGearintheProceedingsoftheConferenceintheHonorofJeanLeray*.Karl-4kskronaUU1999.Ed.M.deGosson(KluwerUUAcademicPublishers).9[Go1]4kM.deGosson,On;theL}'eray-Maslov;quantizationofL}'agrangian;manifold,J.Geom.Phys.4kB13(1994),UU155-168.9[Go2]4kffT,HMaslovclasses,eMetaple}'cticRepresentationandLagrangianQuantization,HMath.4kResearchUUB95(1997),AkqademieV*erlag.9[GS1]4kV.GuilleminandS.SternbGerg,Ge}'ometric'Asymtotics,Math.SurveysMonographs,Bv9ol.4k14,UU(1977),Amer.Math.SoGc..9[GS2]4kffT,pSymple}'cticT;echniquesinPhysics,pCambridgeUniv.Press,CambridgeMass.,4k1984.9[Ho1]4kL.XHormander,YPseudo-di er}'ential,operatorsandnon-ellipticboundaryproblems,YAnn.Xof4kMath.UUB83(1966),129{209.9[Ho2]4kffT,UUF;ourierinte}'graloperatorsI,UUActaMath,B127(1971),79-183.9[Ho3]4kffT,GTheLAnalysisofLine}'arPartialDi erentialOperatorsIII,Springer,GBerlin,1985.9[Ii1]4kK.SIi,qOniaBar}'gmann-typeitransformandaHilbertspaceofholomorphicfunctions,4kT^ohokuUUMath.J.(1)B38(1986),57{69.9[Ii2]4kK.IiandT.Morikqawa,KahlerBstructur}'esonthetangentbundleofRiemannianmanifolds4kofc}'onstantpositivecurvature,UUBull.Y*amagataUniv.Natur.Sci.B14(1999),141{154.Viiҍe.&e9224~KENR9OTFURUT:ANIiҍ 9[Le]4kJ.UULeray*,L}'agrangianAnalysis,UUMITpress,Cambridge,London1981. 9[Li]4kW. Lichtenstein,8@A:system:ofquadricsdescribingtheorbitofthehighestweightve}'ctor,4kProGc.UUAmer.Math.Soc.B84(1982),605{608.9[OF]4kN.OtsukiandK.F*urutani,daSp}'ectral owandMaslowindexarisingfr}'omLagrangian4kinterse}'ctions,UUT*okyoJ.Math.B14(1991),135{150.9[Ra1]4kJ.%H.Rawnsley*,Coher}'ent?2statesandKahlermanifolds,Quart.%J.Math.OxfordSer.4kB28(1977),UU403{415.9[Ra2]4kffT,Anon-unitaryp}'airingofpolarizationfortheKeplerproblem,T*rans.zAmer.Math.4kSoGc.UUB250(1979),167{180.9[So1]4kJ.UUM.Souriau,Structur}'edesSyst$emesDynamiques,UUDunoGd,Paris1970.9[So2]4kffT,UUSurlavari$etedeKepler,UUSympGosiaMath.B14(1974),343{360.9[Sz1]4kR.Sz}oke,Complexכstructur}'esonthetangentbundleofRiemannianmanifolds,Math.Ann.4kB291UU(1991),409{428.9[Sz2]4kffT,A}'daptedDcomplexstructuresandgeometricquantization,NagoyaYJ.Math.B1544k(1999),UU171{183.9[Sz3]4kffT,,Involutive1structur}'esontangentbundlesofsymmetricspaces,,Math.(Ann.B3194k(2001),UU319-348.9[W*e]4kA.W*eistein,~F;ourierInte}'gralOperators,|QuantizationandtheSpectraofRiemann-4kianManifolds,AInColloGqueInternationaldeGGeometrieSymplectiqueetPhysique4kMathGematiqueUUCNRSAix(June1974).9[Wh]4kN.!M.J.W*oGodhouse,[Ge}'ometricQuantization,2nd!ed.,OxfordMathematicalMono-4kgraphs.UUOxford:qClarendonPress.1997.9[Y*o]4kT.׈Y*oshida,Flo}'er homologyandsplittingsofmanifolds,Ann.׈ofMath.B134(1991),277{4k323.9KenroFurutUTani9DepUTartmentofMaUTthematics9FUaculUTtyofScienceandTechnology9ScienceUniversityofTokyo92641Noda,Chiba(278-8510)9JapUTan9E-mailaddr}'ess!:qC