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

Geatti, Laura; Iannuzzi, Andrea

Annales de l'Institut Fourier, Tome 66 (2016) no. 1, pp. 143-174.

Cet article est consacré à l’étude des domaines invariants dans ${\Xi }^{+}$, un domaine de Stein particulier dans la complexification d’un espace symétrique Hermitien irréductible $G/K$. Le domaine ${\Xi }^{+}$, introduit récemment par Krötz et Opdam, contient la couronne $\Xi$ et il est maximal en ce qui concerne la propreté de l’action de $G$. Dans le cas tubulaire, ${\Xi }^{+}$ contient aussi $S^{+}$, un domaine de Stein invariant lié à la structure causale d’une orbite symétrique dans le bord de $\Xi$.

On demontre que l’enveloppe d’holomorphie d’un domaine invariant dans ${\Xi }^{+}$, non contenu ni dans $\Xi$ ni dans $S^{+}$, est univalent et coincide avec ${\Xi }^{+}$. Ce fait, en combination avec des résultats connus pour $\Xi$ et $S^{+}$, démontre l’univalence de l’enveloppe d’holomorphie d’un domaine arbitraire dans ${\Xi }^{+}$ et complète la classification des domains de Stein invariants dans ${\Xi }^{+}.$

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 $\Xi$ 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.

Classification : 32D10, 32M15, 32Q28
Keywords: Hermitian symmetric space, Lie group complexification, envelope of holomorphy, invariant Stein domain
Mot clés : Espace symétrique hermitien, complexification, enveloppe d’holomorphie, domaine de Stein invariant

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