On smooth-group actions on reductive groups and spherical buildings
Jeffrey D. Adler, Joshua M. Lansky, Loren Spice
TL;DR
The paper shows that for a smooth $k$-group $oldsymbol{ m \Gamma}$ acting on a connected reductive $k$-group $ ilde{G}$, the fixed-point subgroup $G=( ilde{G}^{oldsymbol{ m \Gamma}})_{ m sm}^{ ext{0}}$ is reductive under general hypotheses, and the spherical-building embedding identifies $ extbf{S}(G)$ with the $oldsymbol{ m \Gamma}$-fixed points in $ extbf{S}( ilde{G})$. It introduces a robust framework of reductive data and an induction mechanism to transfer root-system information between $ ilde{G}$ and $G^{oldsymbol{ m \Gamma}}$, leveraging root data, BdS theory, and properties of fixed points. The main theorems—quasisemisimple, ka-quass, and loc-quass—provide precise criteria for reductivity and smoothability of fixed points, including subtle phenomena in bad characteristics and for outer automorphisms, with implications for lifting representations and parahoric-like structures in $p$-adic contexts. The results yield a comprehensive toolkit for descent and lifting in reductive group settings, clarifying when fixed-point subgroups retain reductive structure and how their buildings reflect symmetry.
Abstract
Let $k$ be a field, and suppose that $Γ$ 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 $Γ$-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 ``$Γ$-fixed points in $\mathscr{S}(\widetilde G)$'', in a suitable sense. In particular, we do not need to assume that $Γ$ has order relatively prime to the characteristic of $k$ (nor even that $Γ$ is finite), nor that the action of $Γ$ preserves a Borel-torus pair in $\widetilde G$.