论文标题
扩展的Siegel-Jacobi上半部空间上不变的度量
Invariant metric on the extended Siegel-Jacobi upper half space
论文作者
论文摘要
真正的jacobi组$ g^j_n（\ mathbb {r}）$，定义为Heisenberg Group $ {\ rm h} _n（\ r）$的半领产品，sympletic group $ {\ mr {sp mr {sp}}}（n，emped $ \ text {sp}（n+1，\ mathbb {r}）$。修改的$ \ rm {sp}（n，\ mathbb {r}）$的修改后的iwasawa分解使我们能够引入一个方便的坐标$ s_n $ of $ g^j_n（\ mathbb {r}）$，对于$ g^j_1（rmathiate）$ g^j__1（\ mathbbbbbbbbbb {\ mathbbbb coint）确定了$ g^j_n（\ mathbb {r}）$上不变的一形形式。 $ g^j_1（\ r）$上的4参数不变度度量的公式扩展到$ g^j_n（\ r）$，$ n \ in \ mathbb {n} $。我们在扩展的siegel-jacobi上半空间上获得了三个参数不变的度量，半空间$ {\ MATHCAL {X}}^J_n = \ frac {g^j_n（\ Mathbb {r}）}} {\ MR {u}（n）\ times \ times \ times \ mathbb {r {r}} $。
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}}$.
