Symmetry Conditions for Homogeneous Siegel Domains

and

Their Applications

Takaaki NOMURA

In this talk I present two norm equalities which characterize symmetric domains among homogeneous Siegel domains. Each of the norm equalities involves a Cayley transform that is a visible generalization of the linear fractional Cayley transform in ℂ mapping the right half plane onto the unit disk. Our two Cayley transforms as well as Penney’s one differ slightly from each other in general, but coincide up to positive scalar multiples if the domain is symmetric. As applications I show:

1. the first norm equality leads us to the equivalence between the commutativity of the Berezin transform with the Laplace-Beltrami operator and the symmetry of the domain,

2. the second norm equality accounts for the equivalence between the vanishing of the Poisson-Szegö kernel under the Laplace–Beltrami operator and the symmetry of the domain.