2017/01/30 by Mikhail V. Ignatyev, Ignatyev, Mikhail V., Ivan Penkov +1
Mathematics · #14M15 #17B65 #22E65 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1701.08478
openalex publication_date 2017/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is a review of results on the structure of the homogeneous ind-varieties G/P of the ind-groups G=GL∞(ℂ), SL∞(ℂ), SO∞(ℂ), Sp∞(ℂ), subject to the condition that G/P is a 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 which are, generally infinite, chains C of subspaces in the natural representation V of G which satisfy a certain condition: roughly speaking, for each nonzero vector v of V there must be a largest space in C which does not contain v, and a smallest space in C which contains v. We start with a review of the construction of the 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. We also briefly review the characterization of more general, i.e. non-splitting, parabolic ind-subgroups in terms of generalized flags. In the special case of an 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).