On certain domains in cycle spaces of flag manifolds (Q1849481)

Let \(G^\mathbb{C}/K^\mathbb{C}\) be the affine homogeneous space associated to a real semisimple Lie group \(G\) with maximal compact subgroup \(K\). The author studies some naturally defined \(G\)-invariant neighborhoods of its real points \(M_\mathbb{R}=G/K\). For example, the universal Iwasawa domain \(\Omega_I\) is introduced from the point of view of incidence geometry and some of its properties are obtained: it is Stein, Kobayashi hyperbolic and contains the domain \(\Omega_{AG}\) introduced by Akhiezer and Gindikin. The paper develops some methods to study the Wolf domain \(\Omega_W(D)\) of cycles in an open \(G\)-orbit \(D\) in a flag manifold \(G^\mathbb{C}/P\). The key method is the study of the Schubert domain \(\Omega_S(D)\) which is defined by Schubert cycles of complementary dimension to the cycles.