Ind-varieties of generalized flags: a survey of results

This is a review of results on the structure of the homogeneous ind-varieties $G\mathrm{/}P$ of the ind-groups $G={\mathrm{G}\mathrm{L}}_{\mathrm{\infty }}(\mathbb{C})$, ${\mathrm{S}\mathrm{L}}_{\mathrm{\infty }}(\mathbb{C})$, ${\mathrm{S}\mathrm{O}}_{\mathrm{\infty }}(\mathbb{C})$, ${\mathrm{S}\mathrm{p}}_{\mathrm{\infty }}(\mathbb{C})$, subject to the condition that $G\mathrm{/}P$ is a inductive limit of compact homogeneous spaces $G_n\mathrm{/}{P}_n$. In this case the subgroup $P\subset G$ is a splitting parabolic subgroup of $G$, and the ind-variety $G\mathrm{/}P$ 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 $G\mathrm{/}P$ 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 $G\mathrm{/}P$ for arbitrary splitting parabolic ind-subgroups $P\subset G$, as well as the orbits of real forms on $G\mathrm{/}P$ for $G={\mathrm{S}\mathrm{L}}_{\mathrm{\infty }}(\mathbb{C})$.