Orbit structure of a distinguished Stein invariant domain in the complexification of a Hermitian symmetric space

We carry out a detailed study of Xi(+), a distinguished G-invariant Stein domain in the complexification of an irreducible Hermitian symmetric space G/K. The domain Xi(+) contains the crown domain Xi and is naturally diffeomorphic to the anti-holomorphic tangent bundle of G/K. The unipotent parametrization of Xi(+) introduced in Krotz and Opdam (GAFA Geom Funct Anal 18:1326-1421, 2008) and Krotz (Invent Math 172:277-288, 2008) suggests that Xi(+) also admits the structure of a twisted bundle G x (K) N+, with fiber a nilpotent cone N+. Here we give a complete proof of this fact and use it to describe the G-orbit structure of Xi(+) via the K-orbit structure of N+. In the tube case, we also single out a Stein, G-invariant domain contained in Xi(+)\Xi which is relevant in the classification of envelopes of holomorphy of invariant subdomains of Xi(+).