2004/03/26 by Ivan Dimitrov, Dimitrov, Ivan, Ivan Penkov +1
Mathematics · #14L35 #14M15 #14M17 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT #msc:14L35 #msc:14M15 #msc:14M17
paper · pdf · doi:10.48550/arxiv.math/0403471
arxiv created 2004/03/26 · openalex publication_date 2004/03/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \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(∞), 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(∞) and Sp(∞). 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.