Orbits of a symmetric subgroup in the Bruhat-Tits building

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 $F$.