Ind – varieties of generalized flags as homogeneous spaces for classical ind – groups Ivan Dimitrov and Ivan Penkov

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 .