Duke Mathematical Journal

The geometry of Grauert tubes and complexification of symmetric spaces

D. Burns, S. Halverscheid, and R. Hind

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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].

Article information

Duke Math. J., Volume 118, Number 3 (2003), 465-491.

First available in Project Euclid: 23 April 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32Q28: Stein manifolds
Secondary: 32M15: Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15] 53C35: Symmetric spaces [See also 32M15, 57T15]


Burns, D.; Halverscheid, S.; Hind, R. The geometry of Grauert tubes and complexification of symmetric spaces. Duke Math. J. 118 (2003), no. 3, 465--491. doi:10.1215/S0012-7094-03-11833-5. https://projecteuclid.org/euclid.dmj/1082744676

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