四元数辛群的等距作用
Isometric actions of quaternionic symplectic groups
摘要：
用$Sp(k,l)$表示签名$(k,l)$的四元数辛群。通过研究由$Sp(1) \backslash H$的仿射结构生成的自然非结合代数$\mathfrak{m}$，研究了嵌入$Sp(k,l) \times Sp(1) \hookrightarrow H$的形变刚性，其中$H$是$Sp(k+1,l)$或$Sp(k,l+1)$.我们计算了$\mathfrak{m}$的自同构群，作为一个召唤，我们能够计算$Sp(1) \backslash H$的等距群，至少到连通分量。利用这些结果，我们得到了关于$Sp(1) \backslash H$的结构以及等距左$Sp(k,l)$作用的唯一性结果，并将其有限体积子分类到有限覆盖.最后，我们将$Sp(k,l)$的任意等距作用分类为连通的、完备的、解析的、伪黎曼流形，该流形允许一个以$\textrm{dim}(Sp(1) \backslash H)$为界的维数稠密轨道。
英文摘要：
Denote by $Sp(k,l)$ the quaternionic symplectic group of signature $(k,l)$. We study the deformation rigidity of the embedding $Sp(k,l) \times Sp(1) \hookrightarrow H$, where $H$ is either $Sp(k+1,l)$ or $Sp(k,l+1)$, this is done by studying a natural non-associative algebra $\mathfrak{m}$ comming from the affine structure of $Sp(1) \backslash H$. We compute the automorphism group of $\mathfrak{m}$ and as a consecuence of this, we are able to compute the isometry group of $Sp(1) \backslash H$ at least up to connected components. Using these results, we obtain a uniqueness result on the structure of $Sp(1) \backslash H$ together with an isometric left $Sp(k,l)$-action and classify its finite volume quotients up to finite coverings. Finally, we classify arbitrary isometric actions of $Sp(k,l)$ into connected, complete, analytic, pseudo-Riemannian manifolds admitting a dense orbit of dimension bounded by $\textrm{dim}(Sp(1) \backslash H)$.
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