Invariant metric on the extended Siegel–Jacobi upper half space

The real Jacobi group $G^J_n(\mathbb{R})$, defined as the semidirect product of Heisenberg ${\rm H}_n(\R)$ with symplectic ${\mr {Sp}}(n,\mathbb{R})$, admits a matrix embedding in $\text{Sp}(n+1,\mathbb{R})$. modified pre-Iwasawa decomposition $\rm{Sp}(n,\mathbb{R})$ allows us to introduce convenient coordinatization $S_n$ which for $G^J_1(\mathbb{R})$ coincides $S$-coordinates. Invariant one-forms on $G^J_n(\mathbb{R})$ are determined. formula 4-parameter invariant metric $G^J_1(\R)$ obtained sum squares 6 is extended $G^J_n(\R)$, $n\in\mathbb{N}$. We obtain three parameter Siegel-Jacobi upper half space $\tilde{\mathcal{X}}^J_n\approx\mathcal{X}^J_n\times \mathbb{R}$ by adding square an one-form two-parameter balanced $ {\mathcal{X}}^J_n =\frac{G^J_n(\mathbb{R})}{\mr{U}(n)\times\mathbb{R}}$.