representations. Hence the $W_d$-orbit of characters, and hence the map ${\mathbb{Q}}[X_*(T_d)]^{W_d}\to{\mathbb{C}}$ attached to $\pi$, does not depend on the choice of $K$ in the case when $\pi$ has fixed vectors for more than one conjugacy class of hyperspecial maximal compact. In particular the image of ${\mathbb{Q}}[X_*(T_d)]^{W_d}$ in ${\mathbb{C}}$ is well-defined independent of the choice of $K$, and hence the notion of having Satake parameter defined over $E$ is also independent of the choice of $K$.

To clarify the meaning of having a Satake parameter defined over $E$, we now explain that in the case of $G={\operatorname{GL}}_n$ the notion becomes a more familiar one. If $\pi$ is an unramified representation of ${\operatorname{GL}}_n(k)$ and we choose our Borel to be the upper triangular matrices and our torus to be the diagonal matrices, then the formalism above associates to $\pi$ an algebra homomorphism ${\mathbb{C}}[X_*(T_d)]^{W_d}\to{\mathbb{C}}$. But here $T=T_d$ is the diagonal matrices, and $W_d$ is the usual Weyl group $W$ of $G$. The ring ${\mathbb{C}}[X_*(T_d)]={\mathbb{C}}[X_*(T)]$ is then just the ring of functions on the dual torus ${{\widehat{T}}}$, and hence an unramified $\pi$ gives rise to a $W_d$-orbit on ${{\widehat{T}}}$, which can be interpreted as a semisimple conjugacy class $S_{\pi}$ in ${\operatorname{GL}}_n({\mathbb{C}})$.

Let $G$ be 