On the Cycle Spaces Associated to Orbits of Semi-simple Lie Groups
Let G be a semi-simple Lie group and Q a parabilic subgroup of its complexification G^\mathbb C, then Z:=G^\mathbb C/Q is a compact complex homogeneous manifold. Moreover, G as well as K^\mathbb C, the complexification of the maximal compact subgroup of G, acts naturally on Z with finitely many orbits. For any G-orbit, there exist a K^\mathbb C-orbit so that their intersection is non-empty and compact. This duality relation with consideration of cycle intersection at the boundary of a G-orbit lead to the definition of the cycle space associated to any G-orbit. Methods involving Schubert varieties, transversal Schubert slices together with geometric properties of a certain complementary incidence hypersurface and results about the open orbits yield a complete characterisation of the cycle space associated to an arbitrary G-orbit. In particular, it is shown that all the cycle spaces except in a few Hermitian cases are equivalent to the domain $Ω_{AG}$. In the exceptional Hermitian cases, the cycle spaces are equivalent to the associated bounded domain.