On certain domains in cycle spaces of flag manifolds

Abstract.

The affine homogeneous space $G^{\Bbb C}/K^{\Bbb C}$ associated to a real semi-simple Lie group G with maximal compact subgroup K contains a number of naturally defined{\it G}-invariant neighborhoods of its real points $M_{\Bbb R} = G/K$ which are of interest from various points of view. Here the universal Iwasawa domain $\Omega_I$ is introduced from the point of view of incidence geometry and certain of its properties are derived, e.g., it is Stein, Kobayashi hyperbolic and contains the domain $\Omega_{AG}$ introduced by Akhiezer and Gindikin which is now known to be equivalent to the maximal domain of definition $\Omega_{adp}$ of the adapted complex structure associated to the Killing metric in the tangent bundle $TM_{\Bbb R}$. One of the main goals of the paper is to develop methods which lead to a better understanding of the Wolf domain $\Omega_W(D)$ of cycles in an open G-orbit D in a flag manifold $G^\Bbb C/P$. The key is the Schubert domain $\Omega_S(D)$ which is defined by Schubert cycles of complementary dimension to the cycles. These are defined by a Borel subgroup containing an Iwasawa factor AN and consequently $\Omega_S(D)$ and $\Omega_I$ are closely related.