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.