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

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 $SL(\mathrm{\infty })$, $SO(\mathrm{\infty })$ and $Sp(\mathrm{\infty })$ 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 $\text{cFl}(\text{cF},E)$ of generalized flags E$--commensurablewithafixedgeneralizedflag$\cF$in$V$ has a natural structure of an ind–variety. In the case when $V$ is the standard representation of $G=SL(\mathrm{\infty })$, all homogeneous ind–spaces $G\mathrm{/}P$ for parabolic subgroups $P$ containing a fixed splitting Cartan subgroup of $G$, are of the form $\text{cFl}(\text{cF},E)$. We also consider isotropic generalized flags. The corresponding ind–spaces are homogeneous spaces for $SO(\mathrm{\infty })$ and $Sp(\mathrm{\infty })$. As an application of the construction, we compute the Picard group of $\text{cFl}(\text{cF},E)$ (and of its isotropic analogs) and show that $\cFl (\cF, E)$isaprojectiveind--varietyifandonlyif$\cF$ is a usual, possibly infinite, flag of subspaces in $V$.