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