Complex Weyl correspondence for a generalized diamond group

Abstract

The generalized diamond group is the semi-direct product $G$ of the abelian group ${\mathbb R}^m$ by the $(2n+1)$-dimensional Heisenberg group $H_n$. We construct the generic representations of $G$ on the Fock space by extending those of $H_n$. Then we study the Berezin correspondence and the complex Weyl correspondence in connection with a generic representation $\pi$ of $G$, proving in particular that these correspondences are covariant with respect to $\pi$. We give also some explicit formulas for the Berezin symbols and the complex Weyl symbols of the representation operators $\pi(g)$ for $g\in G$. These results are applied to recover various formulas involving the Moyal product. Moreover, we relate $\pi$ to a coadjoint orbit of $G$ in the spirit of the Kirillov-Kostant method of orbits. This allows us to establish that the complex Weyl correspondence is a Stratonovich-Weyl correspondence for $\pi$.