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, {\it 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