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}}$.

中文翻译：

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扩展 Siegel-Jacobi 上半空间的不变度量

实际雅可比群 $G^J_n(\mathbb{R})$，定义为海森堡群 ${\rm H}_n(\R)$ 与辛群 ${\mr {Sp}} 的半直积(n,\mathbb{R})$, 允许在 $\text{Sp}(n+1,\mathbb{R})$ 中嵌入矩阵。$\rm{Sp}(n,\mathbb{R})$ 的改进的 pre-Iwasawa 分解允许我们引入一个方便的 $G^J_n(\mathbb{R})$ 的协调 $S_n$，对于 $ G^J_1(\mathbb{R})$ 与 $S$ 坐标重合。确定 $G^J_n(\mathbb{R})$ 上的不变单形式。$G^J_1(\R)$ 上的 4 参数不变度量公式作为 6 个不变单形式的平方和获得，扩展为 $G^J_n(\R)$, $n\in\mathbb{ N}$。