Berezin Quantization and Holomorphic Representations

  • Benjamin Cahen

    Université de Metz, France


Let GG be a quasi-Hermitian Lie group and let π{\pi} be a unitary highest weight representation of GG realized in a reproducing kernel Hilbert space of holomorphic functions. We study the Berezin symbol map SS and the corresponding Stratonovich-Weyl map WW which is defined on the space of Hilbert-Schmidt operators acting on the space of π{\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 π(g){\pi} (\kern 1pt g) (for gGg\in G) and dπ(X)d{\pi} (X) (for XX in the Lie algebra of GG) and we show that SS 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 GG is reductive, we prove that WW can be extended to the operators dπ(X)d{\pi} (X) and we give the expression of W(dπ(X))W(d{\pi} (X)). As an example, we study the case when π{\pi} 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