On closures of cycle spaces of flag domains

Abstract

Open orbits D of noncompact real forms G ₀ acting on flag manifolds Z = G/Q of their semisimple complexifications G are considered. Given D and a maximal compact subgroup K ₀ of G ₀, there is a unique complex K ₀–orbit in D which is regarded as a point $C_{0}\in \mathcal {C}_q(D)$ in the space of q-dimensional cycles in D. The group theoretical cycle space $\mathcal M_{D}$ is defined to be the connected component containing C ₀ of the intersection of the G–orbit G(C ₀) with $\mathcal {C}_q(D)$. The main result of the present article is that $\mathcal {M}_D$ is closed in $\mathcal {C}_q(D)$. This follows from an analysis of the closure of the universal domain $\mathcal {U}$ in any G-equivariant compactification of the affine symmetric space G/K, where K is the complexification of K ₀ in G.