Homogeneous Siegel domains

Abstract

In 1935 E. Cartan classified all symmetric bounded domains [6]. At that time he proved that a bounded symmetric domain is homogeneous with respect to its group of holomorphic automorphisms. Thus the more general problem of investigating homogeneous bounded domains arose. It was known to E. Cartan that all homogeneous bounded domains of dimension ≤3 are symmetric [6]. For domains of higher dimension little was known. The first example of a 4-dimensional, homogeneous, non-symmetric bounded domain was provided by I. Piatetsky-Shapiro [41]. In several papers he investigated homogeneous bounded domains [20], [21], [41], [42], [43]. One of the main results is that all such domains have an unbounded realization of a certain type, as a so-called Siegel domain. But many questions still remained open. Amongst them the question for the structure and explicit form of the infinitesimal automorphisms of a homogeneous Siegel domain.