Given an element $\gamma$ in $GL(n,F)$, where $F$ is either a global field or a non archimedean local field.
Assume $\gamma$ is elliptic, i.e. its characteristic polynomial irreducible. Let $Z(F)$ be the center of $GL(n,F)$.
Is $\gamma$ conjugated to an element in $Z(F)GL(n, o)$, where $o$ is the ring of integers of $F$?
Heuristic: The centralizer of $\gamma$ is compact modulo the centrum.
What about $GL(n ,\mathbb{R})$ and $GL(n, \mathbb{C})$, where $GL(n,o)$ is replaced by $O(n)$ respective $U(n)$?