Title: On smooth-group actions on reductive groups and spherical buildings Show Chinese title

Title: 关于约化群和球面建筑上的光滑群作用

Abstract: Let $k$ be a field, and suppose that $\Gamma$ is a smooth $k$-group that acts on a connected, reductive $k$-group $\widetilde G$. Let $G$ denote the maximal smooth, connected subgroup of the group of $\Gamma$-fixed points in $\widetilde G$. Under fairly general conditions, we show that $G$ is a reductive $k$-group, and that the image of the functorial embedding $\mathscr{S}(G) \longrightarrow \mathscr{S}(\widetilde G)$ of spherical buildings is the set of ``$\Gamma$-fixed points in $\mathscr{S}(\widetilde G)$'', in a suitable sense. In particular, we do not need to assume that $\Gamma$ has order relatively prime to the characteristic of $k$ (nor even that $\Gamma$ is finite), nor that the action of $\Gamma$ preserves a Borel-torus pair in $\widetilde G$. Show Chinese abstract

Abstract: 设 $k$ 为一个域，并假设 $\Gamma$ 是作用在一个连通可还原的 $k$-群 $\widetilde G$ 上的光滑 $k$-群。 设 $G$ 表示 $\widetilde G$ 中 $\Gamma$- 固定点群的最大光滑连通子群。 在相当一般的条件下，我们证明了$G$是一个约化$k$-群，并且函子嵌入$\mathscr{S}(G) \longrightarrow \mathscr{S}(\widetilde G)$的球状建筑的像是“$\mathscr{S}(\widetilde G)$中的$\Gamma$-不动点”集合，以某种意义而言。 特别地，我们不需要假设 $\Gamma$ 的阶与 $k$ 的特征数互素（甚至不需要假设 $\Gamma$ 是有限的），也不需要假设 $\Gamma$ 的作用在 $\widetilde G$ 中保持一个 Borel-环面对。