Cycle Spaces of Infinite Dimensional Flag Domains

TL;DR

This paper investigates the structure of cycle spaces associated with open orbits of real forms of infinite-dimensional complex groups on flag manifolds, extending finite-dimensional theories to the infinite-dimensional setting.

Contribution

It develops the theory of cycle spaces for infinite-dimensional flag domains, including real and quaternionic cases, generalizing finite-dimensional results.

Findings

Complete orbit structure for hermitian type real forms

Development of cycle space theory in infinite dimensions

Extension of finite-dimensional symmetric space results

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.