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$--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 $V$.