# Berezin Quantization and Holomorphic Representations

### Benjamin Cahen

Université de Metz, France

## Abstract

Let $G$ be a quasi-Hermitian Lie group and let $π$ 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 $π$ , 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 $π(g)$ (for $g∈G$) and $dπ(X)$ (for $X$ in the Lie algebra of $G$) and we show that $S$ provides an adapted Weyl correspondence in the sense of [B. Cahen, *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 $dπ(X)$ and we give the expression of $W(dπ(X))$. As an example, we study the case when $π$ is a generic unitary representation of the diamond group.

## Cite this article

Benjamin Cahen, Berezin Quantization and Holomorphic Representations. Rend. Sem. Mat. Univ. Padova 129 (2013), pp. 277–297

DOI 10.4171/RSMUP/129-16