Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space
Annales de l'Institut Fourier, Volume 66 (2016) no. 1, p. 143-174

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.

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 + .

Received : 2013-11-27
Revised : 2014-10-20
Accepted : 2014-10-23
Published online : 2016-02-17
Classification:  32D10,  32M15,  32Q28
Keywords: Hermitian symmetric space, Lie group complexification, envelope of holomorphy, invariant Stein domain
     author = {Geatti, Laura and Iannuzzi, Andrea},
     title = {Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {1},
     year = {2016},
     pages = {143-174},
     doi = {10.5802/aif.3008},
     language = {en},
     url = {}
Geatti, Laura; Iannuzzi, Andrea. Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space. Annales de l'Institut Fourier, Volume 66 (2016) no. 1, pp. 143-174. doi : 10.5802/aif.3008.

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