Orbits of a symmetric subgroup in the Bruhat-Tits building

TL;DR

This paper studies how the symmetric subgroup of a reductive algebraic group acts on the chambers of its associated Bruhat-Tits building over a non-archimedean field, providing a detailed orbit description.

Contribution

It provides a new description of the orbits of symmetric subgroups on the chambers of the Bruhat-Tits building for groups over non-archimedean fields.

Findings

Orbit classification of $H_F$ in the building's chambers

Explicit description of symmetric subgroup actions

Insights into the structure of reductive groups over local fields

Abstract

Let $G$ be a connected reductive algebraic group defined over a non-archimedean locally compact field $F$ of odd residue characteristic. Let $\theta$ be an $F$-rational involution of $G$ and $H$ be the reductive $F$-group $G^\theta$. We describe the orbits of the symmetric subgroup $H_F$ in the set of chambers of the reduced building of $G$ over $F$.