Berezin Quantization and Holomorphic Representations

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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, 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.