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15 June 2003 The geometry of Grauert tubes and complexification of symmetric spaces
D. Burns, S. Halverscheid, R. Hind
Duke Math. J. 118(3): 465-491 (15 June 2003). DOI: 10.1215/S0012-7094-03-11833-5

Abstract

We consider complexifications of Riemannian symmetric spaces $X$ of nonpositive curvature. We show that the maximal Grauert domain of $X$ is biholomorphic to a maximal connected extension $\Omega\sb {{\rm AG}}$ of $X=G/K\subset G\sb {\mathbb {C}}/K\sb {\mathbb {C}}$ on which $G$ acts properly, a domain first studied by D. Akhiezer and S. Gindikin [1]. We determine when such domains are rigid, that is, when ${\rm Aut}\sb {\mathbb {C}}(\Omega\sb {{\rm AG}}=G$ and when it is not (when \Omega\sb {{\rm AG}}$ has "hidden symmetries"). We further compute the $G$-invariant plurisubharmonic functions on $\Omega\sb {{\rm AG}}$ and related domains in terms of Weyl group invariant strictly convex functions on a $W$-invariant convex neighborhood of $0\in \mathfrak {a}$. This generalizes previous results of M. Lassalle [25] and others. Similar results have also been proven recently by Gindikin and B. Krötz [8] and by Krötz and R. Stanton [24].

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D. Burns. S. Halverscheid. R. Hind. "The geometry of Grauert tubes and complexification of symmetric spaces." Duke Math. J. 118 (3) 465 - 491, 15 June 2003. https://doi.org/10.1215/S0012-7094-03-11833-5

Information

Published: 15 June 2003
First available in Project Euclid: 23 April 2004

zbMATH: 1044.53039
MathSciNet: MR1983038
Digital Object Identifier: 10.1215/S0012-7094-03-11833-5

Subjects:
Primary: 32Q28
Secondary: 32M15, 53C35

Rights: Copyright © 2003 Duke University Press

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Vol.118 • No. 3 • 15 June 2003
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