Congruences of parahoric group schemes

Abstract

Let $F$ be a nonarchimedean local field and let $T$ be a torus over $F$. With ${\mathcal{T}}^{NR}$ denoting the Néron–Raynaud model of $T$, a result of Chai and Yu asserts that the model ${\mathcal{T}}^{NR}{\times }_{{\mathfrak{O}}_F}{\mathfrak{O}}_F∕{\mathfrak{p}}_F^m$ is canonically determined by $\left({Tr}_l\left(F\right),\mathrm{\Lambda }\right)$ for $l\gg m$, where ${Tr}_l\left(F\right)=\left({\mathfrak{O}}_F∕{\mathfrak{p}}_F^l,{\mathfrak{p}}_F∕{\mathfrak{p}}_F^{l+1},ϵ\right)$ with $ϵ$ denoting the natural projection of ${\mathfrak{p}}_F∕{\mathfrak{p}}_F^{l+1}$ on ${\mathfrak{p}}_F∕{\mathfrak{p}}_F^l$, and $\mathrm{\Lambda }:=X_{\ast }\left(T\right)$. In this article we prove an analogous result for parahoric group schemes attached to facets in the Bruhat–Tits building of a connected reductive group over $F$.