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

TL;DR

This paper studies the maximal invariant Stein domains in the complexification of Hermitian symmetric spaces, proving the univalence of their envelopes of holomorphy and completing their classification.

Contribution

It establishes that the envelope of holomorphy of any invariant domain in the domain i+ is univalent and coincides with i+, advancing the classification of invariant Stein domains.

Findings

Envelope of holomorphy of certain invariant domains is univalent.

The envelope of holomorphy coincides with i+ for these domains.

Complete classification of invariant Stein domains in i+.

Abstract

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.