[@cartier:corvallis], and the arguments in §4.4 of [@cartier:corvallis] show that $\pi\cong V_0\otimes_E{\mathbb{C}}$.

\[Carith\_well\_defined\] If $\pi$ is a smooth irreducible admissible unramified representation of $G(k)$, then the notion of being “defined over $E$” is independent of the choice of hyperspecial maximal compact $K$ for which $\pi^K$ is non-zero.

This is because condition (iii) of Lemma \[definedoverE\] is independent of this choice.

We now prove the analogous result for Satake parameters.

\[Larith\_well\_defined\] If $\pi$ is a smooth irreducible admissible unramified representation of $G(k)$, then the notion of $\pi$ having Satake parameter being defined over $E$ is independent of the choice of hyperspecial maximal compact $K$ for which $\pi^K\not=0$.

Say $\pi$ is an unramified smooth irreducible admissible representation of $G(k)$ with a $K$-fixed vector. The Satake isomorphism associated to $K$ gives us a character of the algebra $H_{{\mathbb{C}}}(T(k),{}^oT)^{W_d}$ and hence a $W_d$-orbit of complex characters of $T(k)$. Now by p45 of [@MR546608] and sections 3 and 4 of [@cartier:corvallis], $\pi$ is a subquotient of the principal series representation attached to any one of these characters, and Theorem 2.9 of [@MR0579172] then implies that this $W_d$-orbit of complex characters are the only characters for which $\pi$ occurs as a subquotient of the corresponding induced 