Invariant metric on the extended Siegel-Jacobi upper half space

The real Jacobi group $G_n^J(\mathbb{R})$, defined as the semidirect product of the Heisenberg group ${\mathrm{H}}_n(\mathbb{R})$ with the symplectic group ${\mr {Sp}}(n,$\mathbb{R}$)$,admitsamatrixembeddingin$$\text{Sp}$(n+1,$\mathbb{R}$)$. The modified pre-Iwasawa decomposition of $\mathrm{S}\mathrm{p}(n,\mathbb{R})$ allows us to introduce a convenient coordinatization $S_n$ of $G_n^J(\mathbb{R})$, which for $G_1^J(\mathbb{R})$ coincides with the $S$-coordinates. Invariant one-forms on $G_n^J(\mathbb{R})$ are determined. The formula of the 4-parameter invariant metric on $G_1^J(\mathbb{R})$ obtained as sum of squares of 6 invariant one-forms is extended to $G_n^J(\mathbb{R})$, $n\in \mathbb{N}$. We obtain a three parameter invariant metric on the extended Siegel-Jacobi upper half space ${\tilde{\mathcal{X}}}_n^J\approx {\mathcal{X}}_n^J\times \mathbb{R}$ by adding the square of an invariant one-form to the two-parameter balanced metric on the Siegel-Jacobi upper half space $ {$\mathcal{X}$}^J_n =\frac{G^J_n($\mathbb{R}$)}{\mr{U}(n)\times$\mathbb{R}$}$.