Connection matrices on the Siegel-Jacobi upper half space and extended Siegel-Jacobi upper half space

TL;DR

This paper derives the inverse metric matrices on the Siegel-Jacobi upper half space and its extension, invariant under specific group actions, with implications for Berezin quantization and explicit calculations for the case n=2.

Contribution

It provides explicit formulas for inverse metric matrices on these spaces, invariant under the Jacobi group, advancing geometric quantization methods.

Findings

Explicit inverse metric matrices for ${ ext{Siegel-Jacobi}}$ spaces.

Invariance under the real Jacobi group actions.

Calculations demonstrated for the case n=2.

Abstract

The inverse of the metric matrices on the Siegel-Jacobi upper half space ${\mathcal{X}}^J_n$, invariant to the restricted real Jacobi group $G^J_n(\mathbb{R})_0$ and extended Siegel-Jacobi $\tilde{{\mathcal{X}}}^J_n$ upper half space, invariant to the action of the real Jacobi $G^J_n(\mathbb{R})$, are presented. The results are relevant for Berezin quantization of the manifolds ${\mathcal{X}}^J_ n$ and $\tilde{\mathcal{X}}^J_n$. Explicit calculations in the case $n=2$ are given.