Maximality of hyperspecial compact subgroups avoiding Bruhat-Tits theory

TL;DR

This paper proves that hyperspecial subgroups of reductive groups over non-archimedean fields are maximal among bounded subgroups, using an approach inspired by GL_n that bypasses Bruhat-Tits theory.

Contribution

It introduces a novel proof demonstrating the maximality of hyperspecial subgroups without relying on Bruhat-Tits building theory.

Findings

Hyperspecial subgroups are maximal among bounded subgroups.

The proof is inspired by the case of GL_n.

Avoids traditional Bruhat-Tits considerations.

Abstract

Let $k$ be a complete non-archimedean field (non trivially valued). Given a reductive $k$-group $G$, we prove that hyperspecial subgroups of $G(k)$ (i.e. those arising from reductive models of $G$) are maximal among bounded subgroups. The originality resides in the argument: it is inspired by the case of $\textrm{GL}_n$ and avoids all considerations on the Bruhat-Tits building of $G$.