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 Krötz and Opdam (GAFA Geom Funct Anal 18:1326–1421, 2008) and Krötz (Invent Math 172:277–288, 2008) suggests that \Xi^+ also admits the structure of a twisted bundle G ×_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^+ \setminus \Xi which is relevant in the classification of envelopes of holomorphy of invariant subdomains of \Xi^+.