Invariant domains in the complexification of a noncompact Riemannian symmetric space
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 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 \textit{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.
scientific article; zbMATH DE number 1846017
scientific article