Berezin Transforms and Laplace-Beltrami Operators

on Homogeneous Siegel Domains

— commutativity, symmetry of the domain and a Cayley transform —

Takaaki NOMURA (Kyoto University)

In this talk, I first introduce a Cayley transform C which maps a homogeneous Siegel domain D biholomorphically onto a bounded domain. The image C(D) is a slight modification of the Harish-Chandra realization of homogeneous bounded domains defined by Penney (1996). If D is quasi-symmetric, then C can be naturally identified with the Cayley transform studied by Dorfmeister (1980). In particular, when D is symmetric, C(D) is the Harish-Chandra realization of a bounded symmetric domain. My first theorem concerns a characterization of symmetric Siegel domains through C, or a characterization of symmetric domains among the Harish-Chandra realizations of homogeneous bounded domains. The second theorem is related to the (non-)commutativity of the Berezin transforms with the Laplace-Beltrami operators on D (the Riemannian metric concerned being defined through the corresponding normal j-algebra). The Berezin transforms are group-invariant integral operators which are bounded and positive selfadjoint on L^{2}(D). They play important roles in the theory of Berezin quantization. Berezin himself already gave their spectral decompositions when D is symmetric and classical (1978). Their commutativity with the Laplace-Beltrami operator is immediately seen in the symmetric case. However, the commutativity always fails when D is no longer symmetric. This is a non-trivial consequence of the above characterization of symmetric Siegel domains. In fact application of the Laplace-Beltrami operator to the integral kernel of the Berezin transform gives, after somewhat tedious computations, a formula involving the Cayley transform C, which reveals a relationship between the symmetry condition of D and the commutativity of the operators. The spectral decomposition itself is yet to be established.