Berezin Quantization and Holomorphic Representations

Abstract

Let $G$ be a quasi-Hermitian Lie group and let $\pi$ be a unitary highest weight representation of $G$ realized in a reproducing kernel Hilbert space of holomorphic functions. We study the Berezin symbol map $S$ and the corresponding Stratonovich-Weyl map $W$ which is defined on the space of Hilbert-Schmidt operators acting on the space of $\pi$ , generalizing some results that we have already obtained for the holomorphic discrete series representations of a semi-simple Lie group. In particular, we give explicit formulas for the Berezin symbols of the representation operators $\pi (g)$ (for $g\in G$) and $d\pi (X)$ (for $X$ in the Lie algebra of $G$) and we show that $S$ provides an adapted Weyl correspondence in the sense of [B. Cahen, {\it Weyl quantization for semidirect products,} Differential Geom. Appl. 25 (2007), 177-190]. Moreover, in the case when $G$ is reductive, we prove that $W$ can be extended to the operators $d\pi (X)$ and we give the expression of $W(d\pi (X))$. As an example, we study the case when $\pi$ is a generic unitary representation of the diamond group.