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

Geatti, Laura; Iannuzzi, Andrea

doi:10.5802/aif.3008

Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space
[Enveloppes d’holomorphie invariantes dans la complexification d’un espace symétrique hermitien]

Geatti, Laura ¹ ; Iannuzzi, Andrea ¹

¹ Università di Roma “Tor Vergata” Via della Ricerca Scientifica I-00133 Roma (Italy)

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

Résumé
Abstract

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

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

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.

Reçu le : 2013-11-27
Révisé le : 2014-10-20
Accepté le : 2014-10-23
Publié le : 2016-02-17

DOI : 10.5802/aif.3008

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

Affiliations des auteurs :

Geatti, Laura ¹ ; Iannuzzi, Andrea ¹

¹ Università di Roma “Tor Vergata” Via della Ricerca Scientifica I-00133 Roma (Italy)

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     title = {Invariant envelopes of holomorphy in the complexification of a {Hermitian} symmetric space},
     journal = {Annales de l'Institut Fourier},
     pages = {143--174},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Geatti, Laura; Iannuzzi, Andrea. Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space. Annales de l'Institut Fourier, Tome 66 (2016) no. 1, pp. 143-174. doi : 10.5802/aif.3008. https://aif.centre-mersenne.org/articles/10.5802/aif.3008/

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