$B$, also defined over $k$, and let $T_d$ be the maximal $k$-split sub-$k$-torus of $T$. Let $W_d$ be the subgroup of the Weyl group of $G$ consisting of elements which map $T_d$ to itself. Let ${}^oT$ denote the maximal compact subgroup of $T(k)$. It follows from an easy cohomological calculation (done for example in §9.5 of [@MR546608]) that the inclusion $T_d\to T$ induces an isomorphism of groups $T_d(k)/T_d({\mathcal{O}})\to T(k)/{}^oT$. We normalise Haar measure on $G(k)$ so that $K$ has measure 1 (and remark that by 3.8.2 of [@MR546588] this normalisation is independent of the choice of hyperspecial maximal compact $K$). If $R$ is a field of characteristic zero then let $H_R(G(k),K)$ denote the Hecke algebra of bi-$K$-invariant $R$-valued functions on $G(k)$ with compact support, and with multiplication given by convolution. Similarly let $H_R(T(k),{}^oT)$ denote the analogous Hecke algebra for $T(k)$ (with Haar measure normalised so that ${}^oT$ has measure 1).

The Satake isomorphism (see for example §4.2 of [@cartier:corvallis]) is a canonical isomorphism $H_{{\mathbb{C}}}(G(k),K)={\mathbb{C}}[X_*(T_d)]^{W_d}=H_{{\mathbb{C}}}(T_d(k),T_d({\mathcal{O}}))^{W_d}=H_{{\mathbb{C}}}(T(k),{}^oT)^{W_d}$, where $X_*(T_d)$ is the cocharacter group of $T_d$ (the Satake isomorphism is the first of these equalities; the others are easy). We normalise the Satake isomorphism in the usual way, so that it does not depend 