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