Ind-varieties of generalized flags: a survey of results

Abstract

This is a review of results on the structure of the homogeneous ind-varieties $G/P$ of the ind-groups $G={\mathrm{GL}}_{\infty }(\mathbb{C})$, ${\mathrm{SL}}_{\infty }(\mathbb{C})$, ${\mathrm{SO}}_{\infty }(\mathbb{C})$, ${\mathrm{Sp}}_{\infty }(\mathbb{C})$, subject to the condition that $G/P$ is a inductive limit of compact homogeneous spaces $G_n/P_n$. In this case the subgroup $P\subset G$ 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 $\mathcal{C}$ 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 $\mathcal{C}$ which does not contain $v$, and a smallest space in $\mathcal{C}$ 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 $P\subset G$. 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 $P\subset G$, as well as the orbits of real forms on $G/P$ for $G={\mathrm{SL}}_{\infty }(\mathbb{C})$.

This is a corrected version. The most important correction is in Corollary 7.10. Besides that we have corrected some typos

References in corpus (2)

• Cartan subalgebras of root-reductive Lie algebras
• On the Barth-Van de Ven-Tyurin-Sato theorem

Cited by in corpus (1)

• The Cohomology of the Grassmannian is a $g{l}_n$-module