Siegel上半空间上的Bergman空间.docx

Siegel上半空间上的Bergman空间
Introduction:
TheSiegelupperhalf-spaceisacomplexmanifoldoftheseconddegreeanditisdenotedbyHg.ItsstructureisverysimilartothePoincaréhalf-planemodelofthehyperbolicspace,anditisanimportantspaceinthestudyofmodularforms.TheBergmanspaceisaveryimportantobjectinthetheoryofseveralcomplexvariables,whichwasintroducedbyBergmaninthe1950s.Itisaspaceofholomorphicfunctionsonacomplexmanifold,whichisequippedwithanaturalinnerproduct.Inthisarticle,wewilldiscusstheBergmanspaceontheSiegelupperhalf-space.
Definitions:
TheSiegelupperhalf-spaceHgisdefinedasthesetofg×gcomplexsymmetricmatricesZwithpositivedefiniteimaginarypart.TheBergmanspaceonHgisthespaceofholomorphicfunctionsonHg,denotedbyA2(Hg),whicharesquare-integrablewithrespecttotheBergmankernel.TheBergmankernelonHgisdefinedasfollows:
K(Z,W)=det((i/2π)(Z+W))^-g,
whereZ,W∈Hg.Afunctionf∈A2(Hg)ifitsatisfiesthefollowingconditions:
1.fisholomorphiconHg,
2.fissquare-integrablewithrespecttotheBergmankernel,i.e.,∫Hg|f(Z)|^2K(Z,Z)dV(Z)<∞,
3.Thesetoffunctions{K(Z,W)^-1/2f(W)}isanorthonormalbasisforA2(Hg),whereV(Z)isthevolumeformofHg.
Properties:
1.A2(Hg)isafinite-dimensionalHilbertspace.Itsdimensionisgivenbytheformula:
dim(A2(Hg))=(g+1)(g+2)/2.
2.TheBergmanspacehasareproducingproperty,i.e.,foranyfixedZ∈Hgandf∈A2(Hg),theevaluationfunctionalΦZ(f)definedbyΦZ(f)=f(Z)iscontinuousonA2(Hg).
3.TheBergmanspaceisinvariantundertheactionofthediscretegroupΓgofsymplecticmatriceswithintegercoefficientsanddeterminant1.ThisgroupactsonHgbyfractionallineartransformations,andtheactionpreservestheBergmankernelandthevolumeform.Thus,thebasisfunctions{K(Z,W)^-1/2f(W)}areautomorphicformsofweight0forΓg.
Applications:
TheBergmanspaceonHghasmanyimportantapplicationsincomplexanalysis,numbertheory,andphysics.Severalexamplesarelistedbelow:
1.TheBergmanspacesonthecomplexupperhalf-planeandtheSiegelupperhalf-spaceareusedtostudythemodularformsandautomorphicforms.Infact,theFouriercoefficientsofamodularformoranautomorphicformcanbeexpressedintermsofthenormsofcert