T. Nomura Berezin transforms related to multiplicity free actions (joint work with E. Fujita)

Let V be a finite-dimensional complex vector space on which a compact Lie group K acts linearly. The action is said to be multiplicity-free if the space P(V) of holomorphic polynomial functions on V has a multiplicity-free K-irreducible decomposition P(V) = ∑_{α∈A} P_{α}(V), where A is an index set. Fixing a Kinvariant hermitian inner product on V, we consider the normalized Gaussian measure μ and the corresponding L^{2}-space L^{2}(V, dμ). Then each P_{α}(V) is a subspace of L^{2}(V, dμ) with reproducing kernel κ_{α}. The Berezin transform B_{α} associated to P_{α}(V) is, by definition, an integral operator on L^{2}(V, κ_{α}(z, z) dμ) with integral kernel

\frac{|κ_{α}(z, w)|^{2}}{κ_{α}(z, z) κ_{α}(w, w)}.

Here κ_{α}(z, z) is strictly positive on a dense open subset O of V. My major interest consists in the spectral decomposition of the bounded positive selfadjoint K-invariant operator B_{α}. Some general facts and two case-studies are presented.