Ind-Varieties of Generalized Flags: A Survey

Abstract

This paper is a review of results on the structure of homogeneous ind-varieties G/P of the ind-groups G = GL∞(ℂ), SL∞(ℂ), SO∞(ℂ), and Sp∞(ℂ), subject to the condition that G/P is the inductive limit of compact homogeneous spaces Gn/Pn. In this case, the subgroup P ⊂ G is a splitting parabolic subgroup of G and the ind-variety G/P admits a “flag realization.” Instead of ordinary flags, one considers generalized flags that are, in general, infinite chains \( \mathcal{C} \) of subspaces in the natural representation V of G satisfying a certain condition; roughly speaking, for each nonzero vector 𝜐 of V , there exist the largest space in \( \mathcal{C} \), which does not contain 𝜐, and the smallest space in \( \mathcal{C} \), which contains v. We start with a review of the construction of ind-varieties of generalized flags and then show that these ind-varieties are homogeneous ind-spaces of the form G/P for splitting parabolic ind-subgroups P ⊂ G. Also, we briefly review the characterization of more general, i.e., nonsplitting, parabolic ind-subgroups in terms of generalized flags. In the special case of the ind-grassmannian X, we give a purely algebraic-geometric construction of X. Further topics discussed are the Bott–Borel–Weil theorem for ind-varieties of generalized flags, finite-rank vector bundles on ind-varieties of generalized flags, the theory of Schubert decomposition of G/P for arbitrary splitting parabolic ind-subgroups P ⊂ G, as well as the orbits of real forms on G/P for G = SL∞(ℂ).