Cycle Spaces of Infinite Dimensional Flag Domains

Abstract

Let $G$ be a complex simple direct limit group, specifically $SL(\infty ;\mathbb{C})$, $SO(\infty ;\mathbb{C})$ or $Sp(\infty ;\mathbb{C})$. Let $\mathcal{F}$ be a (generalized) flag in ${\mathbb{C}}^{\infty }$. If $G$ is $SO(\infty ;\mathbb{C})$ or $Sp(\infty ;\mathbb{C})$ we suppose further that $\mathcal{F}$ is isotropic. Let $\mathcal{Z}$ denote the corresponding flag manifold; thus $\mathcal{Z}=G/Q$ where $Q$ is a parabolic subgroup of $G$. In a recent paper with Ignatyev and Penkov, we studied real forms $G_0$ of $G$ and properties of their orbits on $\mathcal{Z}$. Here we concentrate on open $G_0$--orbits $D\subset \mathcal{Z}$. When $G_0$ is of hermitian type we work out the complete $G_0$--orbit structure of flag manifolds dual to the bounded symmetric domain for $G_0$. Then we develop the structure of the corresponding cycle spaces ${\mathcal{M}}_D$. Finally we study the real and quaternionic analogs of these theories. All this extends an large body of results from the finite dimensional cases on the structure of hermitian symmetric spaces and related cycle spaces.

This revision improves the exposition and corrects a number of typos. Earlier revisions had clarified the ordering of subspaces in a flag relative to a given ordered basis of the ambient ${\mathbb{C}}^{\infty }$ as well as the product structure of the base cycles for flag domains of $Sp(\infty ;R)$ and $S{O}^{\ast }(\infty )$. These revisions had no effect on the results for the structure of the cycle spaces