A characterization of products of balls by their isotropy groups

In this paper we will characterize products of balls – especially the ball and the polydisc – in by properties of the isotropy group of a single point. It will be shown that such a characterization is possible in the class of Siegel domains of the second kind, a class that extends the class of bounded homogeneous domains, and that such a characterization is no longer possible in the class of bounded domains with noncompact automorphism groups. The main result is that a Siegel domain of the second kind is biholomorphically equivalent to a product of balls, iff there is a point such that the isotropy group of p contains a torus of dimension n. As an application it will be proved that the only domains biholomorphically equivalent to a Siegel domain of the second kind and to a Reinhardt domain are exactly the domains biholomorphically equivalent to a product of b alls. Received: 27 February 1998 / In final form: 6 August 1998