The Bergman kernels on super-Cartan domains of the first type (Q1974201)

The classical Cartan domains are defined as the unit balls in certain matrix spaces \(V\); in all cases one can introduce a polynomial \(p(Z, \overline Z)\) in \(Z\) and \(\overline Z\) so that a matrix in the domain satisfies \(p(Z, \overline Z)>0\) and the Bergman kernel of the domain is an integer power of \(p(Z, \overline Z)\). In the present paper the author introduces a class of domains in a complex vector space \({\mathbb C}^N\times V\), defined as the set of elements \((W, Z)\) in \({\mathbb C}^N\times V\) so that \(|W|^{2K}<p(Z, \overline Z)\), called super-Cartan domains of different type depending on the type of the classical domain in \(V\). For type I super Cartan domains the author finds the Bergman reproducing kernels and describes their groups of automorphisms.