On bounded elements of linear algebraic groups

Abstract:

Let $F$ be a local field of characteristic zero and ${\text {G }}$ a connected algebraic group defined over $F$. Let $G$ be the locally compact group of $F$-rational points. One characterizes the group $B(G)$ of $g \in G$ whose conjugacy class is relatively compact. For instance, if ${\text {G}}$ is $F$-split or reductive without anisotropic factors then $B(G)$ is the center of $G$. If $H$ is a closed subgroup of $G$ such that $G/H$ has finite volume, then the centralizer of $H$ in $G$ is contained in $B(G)$. If, moreover, $H$ is the centralizer of some $x \in G$ then $G/H$ is compact.