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

In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_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 [WZ2,WZ3,FH,M4] and for open S in [M4]. It was proved for the other orbits in [M5] when G_R is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed K_C-orbit when G_R is of Hermitian type. Thus the conjecture is completely solved affirmatively.