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

In this paper we investigate invariant domains in $\, Ξ^+$, a distinguished $\,G$-invariant, Stein domain in the complexification of an irreducible Hermitian symmetric space $\,G/K$. The domain $\,Ξ^+$, recently introduced by Krötz and Opdam, contains the crown domain $\,Ξ\,$ 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 $\,Ξ$. We prove that the envelope of holomorphy of an invariant domain in $\,Ξ^+$, which is contained neither in $\,Ξ\,$ nor in $\,S^+$, is univalent and coincides with $\,Ξ^+$. This fact, together with known results concerning $\,Ξ\,$ and $\,S^+$, proves the univalence of the envelope of holomorphy of an arbitrary invariant domain in $\,Ξ^+\,$ and completes the classification of invariant Stein domains therein.