Invariant domains in the complexification of a noncompact Riemannian symmetric space
type: Item
instance of: scholarly article
title: Invariant domains in the complexification of a noncompact Riemannian symmetric space
zbMATH Open document ID: 1018.32030
author: Laura Geatti
published in: Journal of Algebra
publication date: 2002-12-16T00:00:00Z
review text: This paper studies \(G\)-invariant domains \(\Omega\) in \(G^{\mathbb C}/K^{\mathbb C}\), where \(G/K\) is a noncompact symmetric space. The aim is to classify those \(\Omega\) which are Stein, or which support a nonconstant \(G\)-invariant plurisubharmonic function. The results are very different from the known results in the compact case: it appears that there are few such \(\Omega\). The methods of the paper only apply when the boundary of \(\Omega\) contains a generic orbit (a closed orbit of maximal dimension). In that case the generic orbits in the boundary are of a very special type. The author proves this by computing the Levi form and Levi cone associated to the CR structure inherited by a generic orbit from the complex structure of the \(G^{\mathbb C}/K^{\mathbb C}\). These control the extension of CR functions to holomorphic functions near the generic orbit and hence determine whether that orbit can arise in the boundary of a Stein domain. Both describing the CR structures and the Levi form calculations require much time and effort, with numerous special cases. A generic orbit depends on a Cartan subset \((\exp{J{\mathfrak c}})p\) for \({\mathfrak c}\) a maximal-dimensional abelian subset of \({\mathfrak g}\) and a base point \(p\). This follows from results of \textit{T. Matsuki} [J. Algebra 197, 49-91 (1997; Zbl 0887.22009)], but the author chooses the base point to satisfy strong algebraic conditions (in most cases \(Gp\) is a semisimple symmetric subspace of \(G^{\mathbb C}/K^{\mathbb C}\) of minimal dimension). This forms the basis of her Levi form calculations. The region \({\overline{\mathbf X}}_0\) consisting of all \(G\)-orbits intersecting the compact dual contains several copies of \(G/K\), each with a maximal invariant neighbourhood \(D_i\). In many cases any \(G\)-invariant connected Stein domain must be contained in one of the \(D_i\) (it corresponds to the case \({\mathfrak c}={\mathfrak a}\subset {\mathfrak p}\), a fundamental Cartan subsp…
Mathematics Subject Classification ID: 32V15
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 32V25
Mathematics Subject Classification ID: 32Q28
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1846017
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: symmetric space
zbMATH Keywords: homogeneous CR-structure
reviewed by: G. K. Sankaran
MaRDI profile type: Publication
full work available at URL: jabr.2001.9150
OpenAlex ID: W2164200134
Wikidata QID: Q115395653
cites work: On Stein extensions of real symmetric spaces
cites work: Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces
cites work: Bi-invariant domains in semisimple groups: nonprincipal orbits and boundaries
cites work: Q4000245
cites work: Quotients for algebraic group actions over non-algebraically closed fields.
cites work: Q4867152
cites work: Q3134936
cites work: Q4225588
cites work: On Levi's problem and the imbedding of real-analytic manifolds
cites work: Grauert tubes and the homogeneous Monge-Ampère equation
cites work: Grauert tubes and the homogeneous Monge-Ampère equation. II
cites work: Q3897665
cites work: Q4348456
cites work: Q4893911
cites work: Q3741813
cites work: A quick proof of the classification of simple real Lie algebras
cites work: Séries de Laurent des fonctions holomorphes dans la complexification d'un espace symétrique compact
cites work: Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundle of Riemannian manifolds
cites work: Action d'une forme réelle d'un groupe de Lie complexe sur les fonctions plurisousharmoniques
cites work: Sur Certaines Operations Differentiables des Groupes de Lie
cites work: Double coset decompositions of reductive Lie groups arising from two involutions
cites work: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups
cites work: On the complex geometry of invariant domains in complexified symmetric spaces
cites work: On the removal of singularities of analytic sets
cites work: Endomorphisms of linear algebraic groups
cites work: Q3984388
cites work: Conjugate classes of Cartan subalgebras in real semisimple Lie algebras
cites work: Complex structures on tangent bundles of Riemannian manifolds
cites work: Q3807534
cites work: Q5657042
cites work: Linear cycle spaces in flag domains
DOI: 10.1006/JABR.2001.9150
description: scientific article; zbMATH DE number 1846017