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.
Categories: math.DG