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