Stratonovich-Weyl correspondence via Berezin quantization

Abstract

Let G be a quasi-Hermitian Lie group and let K be a maximal compactly embedded subgroup of G. Let π be aunitary representation of G which is holomorphically induced from a unitary representation ρ of K. We introduce and study a notion of complex-valued Berezin symbol for an operator acting on the space of π and the corresponding notion of Stratonovich-Weyl correspondence. This generalizes some results already obtained in the case when ρ is an unitary character, see [19]. As an example, we treat in detail the case of the Heisenberg motion groups.