Berezin Quantization and Holomorphic Representations

  • Benjamin Cahen

    Université de Metz, France


Let be a quasi-Hermitian Lie group and let be a unitary highest weight representation of realized in a reproducing kernel Hilbert space of holomorphic functions. We study the Berezin symbol map and the corresponding Stratonovich–Weyl map which is defined on the space of Hilbert–Schmidt operators acting on the space of , 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 (for ) and (for in the Lie algebra of ) and we show that 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 is reductive, we prove that can be extended to the operators and we give the expression of . As an example, we study the case when 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