to $y_\chi$. It follows that if $r_\pi:W_k\to{}^LG$ is the representation of $W_k$ corresponding to $\pi$, then $r_{\pi\otimes\chi}(w)=\hat\xi(|w|^s)r_\pi(w)$, with $\hat\xi:{\mathbb{C}}^\times\to\widehat{T}\subseteq\widehat{G}$ the cocharacter of $\widehat{T}$ corresponding to the character $\xi$ of $T$.

We now come back to the $\mathbf{Q}$-structures on $H_{{\mathbb{C}}}(G(k),K)$. As we have already mentioned, we have a natural ${\mathbb{Q}}$-structure on $H_{{\mathbb{C}}}(G(k),K)$ coming from the ${\mathbb{Q}}$-valued functions $H_{{\mathbb{Q}}}(G(k),K)$, and we have another one coming from ${\mathbb{Q}}[X_*(T_d)]^{W_d}$ via the Satake isomorphism. This means that, for a smooth irreducible admissible representation $\pi$ of $G(k)$ with a $K$-fixed vector, there are two (typically distinct) notions of what it means to be “defined over $E$”, for $E$ a subfield of ${\mathbb{C}}$. Indeed, if $\pi$ is a smooth admissible irreducible representation of $G(k)$ with a $K$-fixed vector, then $\pi^K$ is a 1-dimensional complex vector space on which $H_{{\mathbb{C}}}(G(k),K)$ acts, and this action induces maps $$H_{{\mathbb{Q}}}(G(k),K)\to{\mathbb{C}}$$ and $${\mathbb{Q}}[X_*(T_d)]^{W_d}\to{\mathbb{C}}.$$

\[definedoverEdef\] Let $\pi$ be smooth, irreducible and admissible, with a $K$-fixed vector. Let $E$ be a subfield of ${\mathbb{C}}$.

\(i) We say that *$\pi$ is defined over $E$* if the induced map $H_{{\mathbb{Q}}}(G(k),K)\to{\mathbb{C}}$ has image lying in $E$.

\(ii) We say that *the Satake parameter of $\pi$ is defined over $E$* if the induced map ${\mathbb{Q}}[X_*(T_d)]^{W_d}\to{\mathbb{C}}$ has image lying 