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Invariant domains in the complexification of a noncompact Riemannian symmetric space. (English) Zbl 1018.32030

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 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 subspace, where \({\mathfrak g}={\mathfrak f}\oplus{\mathfrak p}\) is the Cartan decomposition).
For some \(G\) of Hermitian type there are other known Stein invaraint domains \(S_{\pm W}\), studied by K.-H. Neeb [Ann. Inst. Fourier 49, 177-225 (1999; Zbl 0921.22003)]. Here it is shown that a Stein \(\Omega\) is contained in \(S_{\pm W}\) or in \(D_i\), unless the boundary contains no generic orbit. But this is the case for \(\Omega=D_i\), and it is not known whether the \(D_i\) are Stein.

MSC:

32V15 CR manifolds as boundaries of domains
22E46 Semisimple Lie groups and their representations
32E10 Stein spaces
32V25 Extension of functions and other analytic objects from CR manifolds
32Q28 Stein manifolds
53C35 Differential geometry of symmetric spaces
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