Consider a global field $F$ and the group $\Gamma =GL(2,F)$. An element $\gamma \in \Gamma$ is called elliptic, if its eigenvalues do not lie in $F$. Now consider a completion $F_v$ of $F$ and $G_v = GL(2,F_v)$. Now the eigenvalues of $\gamma$ may or may not lie in $F_v$. What is the centralizer $$C_{G_v}(\gamma) = \{ g \in G_v : g\gamma = \gamma g \}?$$
# Centralizer of elliptic elements in $GL(2)$
Consider a global field $F$ and the group $\Gamma =GL(2,F)$. An element $\gamma in \Gamma$ is called elliptic, if its eigenvalues do not lie in $F$. Now consider a completion $F_v$ of $F$ and $G_v = GL(2,F_v)$. Now the eigenvalues of $\gamma$ may or may not lie in $F_v$. What is the centralizer $$C_{G_v}(\gamma) = \{ g \in G_v : g\gamma = \gamma g \}?$$