The isotropy representation for homogeneous Siegel domains
The authors study relations between homogeneous Siegel domains (equivalently, homogeneous bounded domains) and different types of homogeneous spaces. As the main result they give new characterizations of symmetric domains among the class of homogeneous Siegel domains. Let D be a homogeneous Siegel domain, G the identity component of the automorphism group of D and K the isotropy subgroup at a point b of D. Then D is symmetric if and only if at least one of the following conditions is satisfied: a) The almost complex structure map on the tangent space \(T_ bD\) is in the image of the infinitesimal isotropy representation, b) There exist no nontrivial G-invariant vector fields, c) The algebra of G- invariant differential operators on D is commutative, d) The isotropy group acts transitively on the Shilov boundary, e) There exists a vector field X in the center of the isotropy algebra at b of \(D\subset {\mathbb{C}}^ n\) so that the differential of X is invertible as a linear transformation on \({\mathbb{C}}^ n\).
scientific article; zbMATH DE number 3950989