Invariant envelopes of holomorphy in the complexification of a hermitian symmetric space

In this paper we investigate invariant domains in Xi+, a distinguished G-invariant, Stein domain in the complexification of an irreducible Hermitian symmetric space G/K. The domain Xi+, recently introduced by Krötz and Opdam, contains the crown domain and it is maximal with respect to properness of the G-action. In the tube case, it also contains S+, an invariant Stein domain arising from the compactly causal structure of a symmetric orbit in the boundary of Xi. We prove that the envelope of holomorphy of an invariant domain in Xi+, which is contained neither in Xi nor in S+, is univalent and coincides with Xi+. This fact, together with known results concerning and S+, proves the univalence of the envelope of holomorphy of an arbitrary invariant domain in Xi+ and completes the classification of invariant Stein domains therein