  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 $\Omega_{AG}$. In the exceptional
Hermitian cases, the cycle spaces are equivalent to the associated bounded
domain.

Complex Variables;Representation Theory