On closures of cycle spaces of flag domains

Abstract

Open orbits D of noncompact real forms G_0 acting on flag manifolds of their semisimple complexifications G are considered. The unique orbit C of a maximal compact subgroup K_0 of G_0 in D can be regarded as a point in the (full) cycle space of D. The group theoretical cycle space is defined to be the connected component containing C of the intersection of the G-orbit of C with the full cycle space of D. The main result of the present article is that the group theoretical cycle space is closed in the full cycle space. In particular, if they have the same dimension, then they are equal. This follows from an analysis of the closure of the Akhiezer-Gindikin domain in any G-equivariant compactification of the affine symmetric space G/K, where K is the complexification of K_0 in G.