if $G$ is semi-simple and simply connected, or a torus) then the map $\delta^{1/2}:T(k)\to{\mathbb{R}}_{>0}$ mentioned in formula (19) of [@cartier:corvallis] is ${\mathbb{Q}}$-valued (see formula (4) of [@cartier:corvallis] for the definition of $\delta$) and the proof of Theorem 4.1 of [@cartier:corvallis] makes it clear that in this case the Satake isomorphism does induce an isomorphism $H_{{\mathbb{Q}}}(G(k),K)={\mathbb{Q}}[X_*(T_d)]^{W_d}$.

Next we recall how the Satake isomorphism above leads us to an unramified local Langlands correspondence. Say $\pi$ is a smooth admissible irreducible complex representation of $G(k)$, and assume that $\pi^K\not=0$ for our given choice of $K$ (so in particular, $\pi$ is unramified). Then $\pi^K$ is a 1-dimensional representation of $H_{{\mathbb{C}}}(G,K)$ and hence gives rise to a ${\mathbb{C}}$-valued character of $H_{{\mathbb{C}}}(G,K)$. Now results of Gantmacher and Langlands in section 6 of [@MR546608] enable one to canonically associate to $\pi$ an unramified continuous admissible representation $r_{\pi}:W_k\to{{{}^{L}G}}({\mathbb{C}})$, where “admissible” in this context means a group homomorphism such that if one composes it with the canonical map ${{{}^{L}G}}({\mathbb{C}})\to\Gamma_k$ then one obtains the canonical map $W_k\to\Gamma_k$ (part of the definition of the Weil group) and “unramified” means that the resulting 1-cocycle $W_k\to{{\widehat{G}}}({\mathbb{C}})$ is trivial on the inertia subgroup of $W_k$. In fact there are two ways of normalising the 