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 Kr\"otz and Opdam, contains the crown domain $\,\Xi\,$ 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 $\,\Xi\,$ 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.