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arXiv:math/0111033v1[math.RT]2Nov2001ComplexcrownsofRiemanniansymmetricspaces1ComplexcrownsofRiemanniansymmetricspacesandnon-compactlycausalsymmetricspacesSimonGindikin∗andBernhardKr¨otz†AbstractInthispaperwedefineadistinguishedboundaryforthecomplexcrownsΞ⊆GC/KCofnon-compactRiemanniansymmetricspacesG/K.ThebasicresultisthataffinesymmetricspacesofGcanappearasacomponentofthisboundaryifandonlyiftheyarenon-compactlycausalsymmetricspaces.IntroductionLetX=G/Kbeasemisimplenon-compactRiemanniansymmetricspace.WemayassumethatGissemisimplewithfinitecenter.WewriteG=NAKforanIwasawadecompositionofG.Byourassumption,GsitsinitsuniversalcomplexificationGCandsoX⊆XC:=GC/KC.NotethatGdoesnotactproperlyonXC.ThecomplexcrownΞ⊆XCofX(cf.[Gi98])wasfirstconsideredin[AkGi90].ItcanbedefinedbyΞ:=Gexp(iΩ)KC/KCwhereΩisapolyhedralconvexdomainina:=Lie(A)definedbyΩ:={X∈a:(∀α∈Σ)|α(X)|π2}.HereΣdenotestherestrictedrootsystemwithrespecttoa.NotethatGactsproperlyonΞ(cf.[AkGi90]).ThedomainΞisinterestinginseveralways.ItaccumulatesmanycrucialgeometricalandanalyticalpropertiesofX.Forexample,itwasshownin[GiMa01]thatΞparametrizesthecompactcyclesincomplexflagdomains.FromthepointofharmonicanalysisthedomainΞisuniversalinthesensethatalleigenfunctionsonXforthealgebraD(X)ofG-invariantdifferentialoperatorsextendholomorphicallytoΞ(cf.[KrSt01a]).Itwasconjecturedin[AkGi90]thatthedomainΞisStein;itwasprovedfordifferentcasesbydifferentauthorsduringthelastyear(cf.[GiKr01]forreferencesandamoredetailedaccountonthedomainΞ).∗SupportedinpartbytheNSF-grantDMS-0070816andtheMSRI†SupportedinpartbytheNSF-grantDMS-0097314andtheMSRI2SimonGindikin∗andBernhardKr¨otz†InthecaseofgroupsGofHermitiantype,XisisaHermitiansymmetricspaceandΞisG-equivariantlybiholomorphictoX×X,whereXreferstoXequippedwiththeoppositecomplexstructure(cf.[BHH01],[GiMa01]or[KrSt01b]).Thisexamplecanbeseeninamoregeneralframework.SupposethatXistherealformaHermitiansymmetricspace.ThismeansthatthereisaHermitiangroupScontainingGwithmaximalcompactsubgroupU⊇KsuchthattheinclusionG/K֒→S/UrealizesXasatotallyrealsubmanifoldofS/U.Forexample,ifGisHermitianwetakeS=G×GandU=K×K.TheexistenceofS/Uisguaranteedinmanycases.Inparticular,S/UexistsforallclassicalgroupsGinthesensethatwehavetoreplaceGsometimeswithG×R(e.g.Sl(n,R)withGl(n,R)).Bytheresultsof[BHH01]or[KrSt01b]thereexistsagenericG-invariantsubdomainΞ0ofΞwhichisG-biholomorphictoS/U.Moreover,equalityΞ0=ΞholdsifandonlyifΣisoftypeCnorBCnforn≥2orG=SO(1,n)withS=SO(2,n).ForthecaseswhereΞ0(Ξ,forexampleifG=SO(p,q)(p,q2)thegeometricstructureofΞbecomesverycomplicatedandespeciallycomplicatedcanbetheboundaryofΞ.InthispaperwestarttoinvestigatetheboundariesofΞ.Write∂ΞfortheboundaryofΞinXC.Apparently,theboundarywillbeaunionofG-orbits,butthestratificationoftheseorbitscanbeintricate.Firstly,notallorbitsontheboundaryintersectexp(ia)KC/KC–butinthispaperwefocusonlyonsuchorbits.Moreover,weareonlyinterestedinveryspecialorbitsofsuchtypewhichareinsomesenseminimal.Letusdefinethedistinguishedboundary∂dΞofΞastheunionofthefollowingG-orbitsin∂Ξ:∂dΞ:=Gexp(i∂eΩ)KC/KC,where∂eΩthesetofextremepointsinthepolyhedralcompactconvexsetΩ.Notethat∂eΩ=W(Y1)∐...∐W(Yn)isafiniteunionofWeylgrouporbits,whereWistheWeylgroupofΣ.Thedistinguishedboundaryisageometricallycomplicatedobject.Usuallyitisadiscon-nectedset.Nevertheless,weshowthatitisminimalfromsomeanalyticalpointsofviewandthatitfeaturespropertiesexpectedfromaShilovboundary.Inparticular,TheoremA.WriteA(Ξ)forthealgebraofboundedholomorphicfunctionswhichcontinuouslyextendtotheboundaryofΞ.Then(∀f∈A(Ξ))supz∈Ξ|f(z)|=supz∈∂dΞ|f(z)|.Further∂dΞ⊆∂ΞisminimalinacertainplurisubharmonicsenseasexplainedinSection1.InabovementionedcaseswhenΞ0=Ξthesituationissimpler.WecanrealizetheHermitiandomainD=S/UviatheCartanembeddinginthecompactHermitiansymmetricYspacedualtoS/U.HereithasacompactboundaryandacompactShilovboundary∂cD.Further,inthissituationtheSteinmanifoldXCcanberealizedasaZariskiopenpartofYwhichwillcontainΞbiholomorphicallyequivalenttoDand∂dΞwillbeonlyaZariskiopenpartofthecompactmanifold∂cD.LetusemphasizethattheShilovboundaryofDessentiallydependsontherealizationofD.Wedescribenowthedistinguishedboundary∂dΞinmoredetail.Writez′j:=exp(iYj)KC∈∂dΞforall1≤j≤n.DenotebyHjtheisotropysubgroupofGinz′j.ThenitfollowsfromthedefinitionofthedistinguishedboundarythatwehaveaG-isomorphism∂dΞ≃G/H1∐...∐G/Hn.ComplexcrownsofRiemanniansymmetricspaces3In[Gi98]itwasconjecturedthatnon-compactlycausalsymmetricspacesappearinthe“Shilovboundary”ofthecomplexcrownsΞ.Weestablishthisconjecture(inamoreexactform);namelyweprove:TheoremB.IfoneofthecomponentsG/Hjin∂dΞisasymmetricspace,thenitisanon-compactlycausalsymmetricspace.Moreover,everynon-compactlycausalsymmetricspaceoccursasacomponentofthedistinguishedboundaryofsomecomplexcrownΞ.Letussayafewwordsaboutthemotivationofthisconjecture.OnRiemanniansymmetricspaceswehaveanellipticanalysisandonnon-compactlycausalsymmetricspaceswehaveahyperbolicanalysis.Itisknowninmathematicalphysicsthatinmanyimportantcasesellipticandhyperbolictheoriescanbe“connected”throughcomplexdomains(Laplaciansandwaveequati