vector, $\pi\otimes\chi$ is also irreducible with a $K$-fixed vector. The map $\chi$ induces an automorphism of $H_{\mathbb{C}}(G(k),K)$ sending a function $f$ to the function $g\mapsto \chi(g)f(g)$, and similarly it induces automorphisms of $H_{\mathbb{C}}(T(k)/{}^oT)$ and $H_{\mathbb{C}}(T_d(k)/T_d({\mathcal{O}}))$. These latter two automorphisms are equivariant for the action of $W_d$, because $\chi$ is constant on $G(k)$-conjugacy classes. Furthermore, the Satake isomorphism commutes with these automorphisms—this follows without too much trouble from the formula defining the Satake isomorphism in §4.2 of [@cartier:corvallis] and the observation that the kernel of $\xi$ contains the unipotent radical of any Borel in $G$. Now if $Y={\operatorname{Spec}}(H_{\mathbb{C}}(T_d(k),T_d({\mathcal{O}}))$ is the complex torus dual to $T_d$, then given an irreducible $\pi$ with a $K$-fixed vector the Satake isomorphism produces a $W_d$-orbit of elements of $Y$. Twisting $\pi$ by $\chi$ amounts to changing this orbit by adding to it the $W_d$-stable element $y_\chi$ of $Y$ corresponding to the character of $T_d(k)/T_d({\mathcal{O}})$ induced by $\chi$. Let us define an element $t_\chi$ of $\widehat{T}=X^*(T)\otimes{\mathbb{C}}^\times$, the complex torus dual to $T$, by $t_\chi=\xi\otimes q^{-s}=\xi\otimes|\varpi|^s$ (with $q$ the size of the residue field of $k$ and $\varpi$ a uniformiser). The inclusion $T_d\to T$ induces a surjection $\widehat{T}\to Y$ and some elementary unravelling shows that it sends $t_\chi$ 