Complex Weyl correspondence for a generalized diamond group
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909779769688064 |
|---|---|
| author | Cahen, Benjamin |
| author_facet | Cahen, Benjamin |
| contents | 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 $π$ of $G$, proving in particular that these correspondences are covariant with respect to $π$. We give also some explicit formulas for the Berezin symbols and the complex Weyl symbols of the representation operators $π(g)$ for $g\in G$. These results are applied to recover various formulas involving the Moyal product. Moreover, we relate $π$ 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 $π$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08082 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complex Weyl correspondence for a generalized diamond group Cahen, Benjamin Representation Theory 22E45, 22E70, 81R05, 81S10, 81R30 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 $π$ of $G$, proving in particular that these correspondences are covariant with respect to $π$. We give also some explicit formulas for the Berezin symbols and the complex Weyl symbols of the representation operators $π(g)$ for $g\in G$. These results are applied to recover various formulas involving the Moyal product. Moreover, we relate $π$ 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 $π$. |
| title | Complex Weyl correspondence for a generalized diamond group |
| topic | Representation Theory 22E45, 22E70, 81R05, 81S10, 81R30 |
| url | https://arxiv.org/abs/2509.08082 |