Linear cycle spaces in flag domains

Abstract

Let Z = G/Q, a complex flag manifold, where G is a complex semisimple Lie group and Q is a parabolic subgroup. Fix a real form \(G_0 \subset G\) and consider the linear cycle spaces \(M_D\), spaces of maximal compact linear subvarieties of open orbits \(D = G_0(z) \subset Z\). In general \(M_D\) is a Stein manifold. Here the exact structure of \(M_D\) is worked out when \(G_0\) is a classical group that corresponds to a bounded symmetric domain B. In that case \(M_D\) is biholomorphic to B if a certain double fibration is holomorphic, is biholomorphic to \(B \times \overline{B}\) otherwise. There are also a number of structural results that do not require \(G_0\) to be classical.