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

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