Ind-Varieties of Generalized Flags: A Survey of Results

This paper is a review of results on the structure of the homogeneous ind-varieties $G/P$ of the ind-groups $G=GL_{\infty}(\mathbb{C})$, $SL_{\infty}(\mathbb{C})$, $SO_{\infty}(\mathbb{C})$, and $Sp_{\infty}(\mathbb{C})$, subject to the condition that $G/P$ is the 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 that are, in general, infinite chains $\mathcal{C}$ 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 $\mathcal{C}$, which does not contain $v$, and the smallest space in $\mathcal{C}$ 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 $P\subset G$. 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 $P\subset G$, as well as the orbits of real forms on $G/P$ for $G=SL_{\infty}(\mathbb{C})$.