Cycles in \(G\)-orbits in \(G^{\mathbb{C}}\)-flag manifolds (Q1434240)

There is a natural duality between orbits \(\gamma \) of a real form \(G\) of a complex semisimple group \(G^{\mathbb{C}}\) on a homogeneous rational manifold \(Z=G^{\mathbb{C}}/P\) and those \(\kappa\) of the complexification \(K^{\mathbb{C}}\) of any of its maximal compact subgroups \(K\): \((\gamma , \kappa)\) is a dual pair if \(\gamma\cap \kappa\) is a \(K\)-orbit. The cycle space \(C(\gamma)\) is defined to be the connected component containing the identity of the interior of \(\{g:g(\kappa)\cap \gamma\) is non-empty and compact\}. Using methods which were recently developed for the case of open \(G\)-orbits, geometric properties of cycles are proved. It is shown that \(C(\gamma )\) is contained in a domain defined by incidence geometry. In the non-Hermitian case, this is a key ingredient for proving that \(C(\gamma )\) is a certain explicitly computable universal domain.