on the choice of the Borel subgroup containing $T$; this is the only canonical way to do things. This standard normalisation is however not in general “defined over ${\mathbb{Q}}$”—for example if $k={\mathbb{Q}}_p$ and $G={\operatorname{GL}}_2$ and $K={\operatorname{GL}}_2({\mathbb{Z}}_p)$ then the Satake isomorphism sends the characteristic function of $K\smallmat{p}{0}{0}{1}K$ to a function on $T({\mathbb{Q}}_p)$ taking the value $\sqrt{p}$ on the matrix $\smallmat{p}{0}{0}{1}$. This square root of $p$ appears because the definition of the Satake isomorphism involves a twist by half the sum of the positive roots of $G$ (see formula (19) of section 4.2 of [@cartier:corvallis]) and because of this twist, the isomorphism does *not* in general induce a canonical isomorphism $H_{{\mathbb{Q}}}(G(k),K)={\mathbb{Q}}[X_*(T_d)]^{W_d}$.

In [@MR1729443] and [@MR1044819] this issue of square roots is avoided by renormalising the Satake isomorphism. Let us stress that we shall *not* do this here, and we shall think of $H_{{\mathbb{Q}}}(G(k),K)$ and ${\mathbb{Q}}[X_*(T_d)]^{W_d}$ as giving two possibly distinct ${\mathbb{Q}}$-structures on the complex algebraic variety ${\operatorname{Spec}}(H_{{\mathbb{C}}}(G(k),K))$ which shall perform two different functions—they will give us two (typically distinct) notions of being defined over a subfield of ${\mathbb{C}}$. We note however that if half the sum of the positive roots of $G$ is in the weight lattice $X^*(T)$ (this occurs for example 