the group ${\operatorname{GL}}_n/k$ and let $\pi$ be an unramified representation of $G(k)$. Let $E$ be a subfield of ${\mathbb{C}}$. Then the following are equivalent:

\(i) The Satake parameter of $\pi$ is defined over $E$.

\(ii) The conjugacy class $S_{\pi}$ is defined over $E$.

\(iii) The conjugacy class $S_{\pi}$ contains an element of ${\operatorname{GL}}_n(E)$.

The statement that the Satake parameter is defined over $E$ is precisely the statement that the induced map ${\mathbb{Q}}[X_*(T)]^W\to{\mathbb{C}}$ takes values in $E$, which is the statement that the characteristic polynomial of an element of $S_{\pi}$ has coefficients in $E$. Hence (i) and (ii) are equivalent. Furthermore, (ii) is equivalent to (iii) because given a monic polynomial with coefficients in $E$ it is easy to construct a semisimple matrix with this polynomial as its characteristic polynomial.

We leave to the reader the following elementary checks. Let $G_1$ and $G_2$ be unramified connected reductive groups over $k$, and let $\pi_1$, $\pi_2$ be unramified representations of $G_1(k)$, $G_2(k)$. Then $\pi:=\pi_1\otimes\pi_2$ is an unramified representation of $(G_1\times G_2)(k)$. One can check that $\pi$ is defined over $E$ iff $\pi_1$ and $\pi_2$ are defined over $E$, and that $\pi$ has Satake parameter defined over $E$ iff $\pi_1$ and $\pi_2$ do. Now 