# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 129, 2013, pp. 277–297**

**DOI: 10.4171/RSMUP/129-16**

Berezin Quantization and Holomorphic Representations

Benjamin Cahen^{[1]}(1) Dép. de Mathématiques, LMMAS, ISGMP, Université de Metz, Ile de Saulcy, 57045, Metz CEDEX, France

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 ${\pi} (\kern 1pt g)$ (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.

*Keywords: *Berezin quantization; Berezin transform; quasi-Hermitian Lie group; coadjoint orbit; unitary representation; holomorphic representation; reproducing kernel Hilbert space; coherent states; Weyl correspondence; Stratonovich-Weyl correspondence

Cahen Benjamin: Berezin Quantization and Holomorphic Representations. *Rend. Sem. Mat. Univ. Padova* 129 (2013), 277-297. doi: 10.4171/RSMUP/129-16