Characterization of cycle domains via Kobayashi hyperbolicity

A real form G of a complex semi-simple Lie group GC has only finitely many orbits in any given G(C)-flag manifold Z = G(C)/Q. The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits D generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of D which, with very few well-understood exceptions, are parameterized by the Wolf cycle domains Omega(W)(D) in G(C)/K-C, where K is a maximal compact subgroup of G. Thus, for the various domains D in the various ambient spaces Z, it is possible to compare the cycle spaces Omega(W) (D). The main result here is that, with the few exceptions mentioned above, for a fixed real form G all of the cycle spaces Omega(W) (D) are the same. They are equal to a universal domain Omega(AG) which is natural from the the point of view of group actions and which, in essence, can be explicitly computed. The essential technical result is that if Omega is a G-invariant Stein domain which contains Omega(AG) and which is Kobayashi hyperbolic, then Omega = Omega(AG). The equality of the cycle domains follows from the fact that every Omega(W) (D) is itself Stein, is hyperbolic, and contains Omega(AG).