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

2013/10/28 by Laura Geatti, Geatti, Laura, Andrea Iannuzzi +1
Mathematics · #(2010): 32M05 #32Q28 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #math.CV #msc:32M05 #msc:32Q28

paper · pdf · doi:10.48550/arxiv.1310.7339

24 pages

arxiv created 2013/10/28 · openalex publication_date 2013/10/28 · arxiv updated 2013/10/29 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

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.

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