Envelope of holomorphy of a tube domain over a complex Lie group (Q2434065)

According to \textit{A. Hirschowitz} [Invent. Math. 26, 303--322 (1974; Zbl 0275.32009)] and to \textit{L. Hörmander} [An Introduction to Complex Analysis in Several Variables, New York: American Elsevier Publishing Company, Inc. (1973; Zbl 0271.32001)] if \(D=B+i\mathbb R^n\) is a tube domain in \(\mathbb C^n\), where \(B\) is a domain in \(\mathbb R^n\), and if \(\widetilde B\) is the convex envelope of \(B\), then any holomorphic function on \(D\) extends to the tube domain \(\widetilde D=\widetilde B+i\mathbb R^n\), which is a univalent envelope of holomorphy of \(D\). The author gives a generalization of this result to (nonunivalent) tube domains over a complex Lie group, which admits a closed subgroup as a real form. As an application, he proves that if \((V,\phi)\) is a tube domain over \(\mathbb C^n\) and if \(B\) is the convex envelope of \(\phi (V)\cap\mathbb R^n\) in \(\mathbb R^n\), then \(\widetilde V=B+i\mathbb R^n\) is an envelope of holomorphy of \((V,\phi)\).