Ind-Varieties of Generalized Flags: A Survey
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IND-VARIETIES OF GENERALIZED FLAGS: A SURVEY M. V. Ignatyev and I. Penkov
UDC 512.745.4, 512.815, 512.554.32, 514.765
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 C of subspaces in the natural representation V of G satisfying a certain condition; roughly speaking, for each nonzero vector v of V , there exist the largest space in C, which does not contain v, and the smallest space in 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). Keywords and phrases: ind-variety, ind-group, generalized flag, Schubert decomposition, real form. AMS Subject Classification: 22E65, 17B65, 14M15
CONTENTS 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Ind-varieties and ind-groups . . . . . . . . . . . . . . . . . . . . 2.2. Generalized flags . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Isotropic generalized flags . . . . . . . . . . . . . . . . . . . . . 3. Linear Ind-Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Definition of linear ind-grassmannians . . . . . . . . . . . . . . 3.2. Standard extensions . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Classification of linear ind-grassmannians . . . . . . . . . . . . 4. Ind-Varieties of Generalized Flags as Homogeneous Ind-Spaces . . . 4.1. Classical ind-groups and their Cartan subgroups . . . . . . . . 4.2. Splitting Borel and parabolic subgroups of classical ind-groups 4.3. Homogeneous ind-spaces . . . . . . . . . . . . . . . . . . . . . 4.4. Borel and parabolic subalgebras: general case . . . . . . . . . . 5. Schubert Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Analogs of the Weyl group . . . . . . . . . . . . . . . . . . . . 5.2. Schubert decomposition . . . . . . . . . . . . . . . . . . . . . . 5.3. Smoothness of Schubert subvarieties . . . . . . . . . . . . . .
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