Ind--varieties of generalized flags as homogeneous spaces for classical ind--groups

Abstract

The purpose of the present paper is twofold: to introduce the notion of a generalized flag in an infinite dimensional vector space $V$ (extending the notion of a flag of subspaces in a vector space), and to give a geometric realization of homogeneous spaces of the ind--groups $SL(\infty )$, $SO(\infty )$ and $Sp(\infty )$ in terms of generalized flags. Generalized flags in $V$ are chains of subspaces which in general cannot be enumerated by integers. Given a basis $E$ of $V$, we define a notion of $E$--commensurability for generalized flags, and prove that the set $\cFl (\cF, E)$ of generalized flags E$--commensurablewithafixedgeneralizedflag$\cF$in$V$hasanaturalstructureofanind--variety.Inthecasewhen$V$isthestandardrepresentationof$G = SL(\infty)$,allhomogeneousind--spaces$G/P$forparabolicsubgroups$P$containingafixedsplittingCartansubgroupof$G$,areoftheform$\cFl (\cF, E)$.Wealsoconsiderisotropicgeneralizedflags.Thecorrespondingind--spacesarehomogeneousspacesfor$SO(\infty)$and$Sp(\infty)$.Asanapplicationoftheconstruction,wecomputethePicardgroupof$\cFl (\cF, E)$(andofitsisotropicanalogs)andshowthat$\cFl (\cF, E)$isaprojectiveind--varietyifandonlyif$\cF$isausual,possiblyinfinite,flagofsubspacesin$V$.