such that, for any $i$ and $w$, all eigenvalues of $\psi_x\colon V_x\rightarrow V_x$ are equal to $\xi_x q^{(i-\dim G)/2}$. (The “cleanness” assumption in [@L2c 13.10] holds in general by [@L10].) It is then shown in [@L2c 14.14] that $$\rho_\epsilon=(-1)^{\dim G} \sum_{x\in X(W)} \xi_x(A_x:
K_\epsilon^{\mathcal{L}_0}) \chi_x \qquad \mbox{for all $\epsilon
\in{{\operatorname{Irr}}}(W)$}.$$ (There are also coefficients $\nu(A_x)$ in the formula in [@L2c 14.14] but these are all equal to $1$ in our situation.) Finally, a comparison with the formula in \[r18\] implies that $\xi_x=\zeta_x$ for all $x\in X(W)$.

\[r25\] [*Proof of Proposition \[p1\].*]{} Let $x\in X(W)$. We also fix $i,w$ and place ourselves in the setting of \[r23\]. The whole discussion there can be repeated with $F$ replaced by $F^m$. As in \[r22\], we have isomorphisms $$\tilde{\phi}_x^{(m)}\colon (F^*)^mA_x\stackrel{\sim}{\rightarrow} A_x
\qquad \mbox{and}\qquad {\phi}_x^{(m)}\colon (F^m)^*A_x
\stackrel{\sim}{\rightarrow} A_x.$$ Analogously, $\varphi\colon F^*K\stackrel{\sim}{\rightarrow} K$ induces isomorphisms $$\tilde{\varphi}^{(m)}\colon (F^*)^mK\stackrel{\sim}{\rightarrow} K\qquad 
\mbox{and}\qquad \varphi^{(m)}\colon (F^m)^*K\stackrel{\sim}{\rightarrow}
K.$$ Let us consider again the canonical isomorphism $$K\cong \bigoplus_{x\in X(W)} (A_x\otimes V_x)\qquad\mbox{where} \qquad
V_x=\mbox{Hom}(A_x,K).$$ Here, as before, each $V_x$ is endowed with a linear map $\psi_x^{(m)}\colon 
V_x \rightarrow V_x$ such that $$\psi_x^{(m)}(v)=\varphi^{(m)} \circ (F^m)^*(v)\circ (\phi_x^{(m)})^{-1} 
\qquad \mbox{for all $v\in V_x$}.$$ By Theorem \[r24\], the scalar $\zeta_x^{(m)}$ is determined by the eigenvalues of $\psi_x^{(m)}$.

Now, 