Question

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 $$ C_{G_v}(\gamma) = \{ g \in G_v : g\gamma = \gamma g \}?$$

Answer 1

The centralizer of $\gamma$ is a torus of the form $E^\times$, where $E/F$ is a quadratic field extension. Assume moreover that $\gamma$ is regular (so that $E/F$ is separable). The centralizer of $\gamma$ in $G_v$ is $(E\otimes_F F_v )^\times$. The algebra $E\otimes F_v$ is either a field (in that case $\gamma$ is elliptic in $G_v)$) or a sum of two fields isomorphic to $F_v$, according to whether the prime $v$ splits in $E$ or not. In the latter case $\gamma$ is a regular element lying in a split torus of $G_v$.