Equivalence of domains arising from duality of orbits on flag manifolds III

In Gindikin and Matsuki 2003, we defined a $G_{\mathbb{R}}$- $K_{\mathbb{C}}$ invariant subset $C(S)$ of $G_{\mathbb{C}}$ for each $K_{\mathbb{C}}$-orbit $S$ on every flag manifold $G_{\mathbb{C}}/P$ and conjectured that the connected component $C(S)_0$ of the identity would be equal to the Akhiezer-Gindikin domain $D$ if $S$ is of nonholomorphic type. This conjecture was proved for closed $S$ in Wolf and Zierau 2000 and 2003, Fels and Huckleberry 2005, and Matsuki 2006 and for open $S$ in Matsuki 2006. It was proved for the other orbits in Matsuki 2006, when $G_{\mathbb{R}}$ is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed $K_{\mathbb{C}}$-orbit when $G_{\mathbb{R}}$ is of Hermitian type. Thus the conjecture is completely solved affirmatively.