a simple induction on $m$ shows that $$\psi_x^m(v)=\tilde{\varphi}^{(m)} \circ (F^*)^m(v)\circ 
(\tilde{\phi}_x^{(m)})^{-1} \qquad \mbox{for all $v\in V_x$}.$$ Next, we use again that we have canonical isomorphisms $(F^*)^mA_x\cong 
(F^m)^*A_x$ and $(F^*)^mK\cong (F^m)^*K$. Under these isomorphisms, the above map $(F^*)^m(v)$ corresponds to the map $(F^m)^*(v)\colon 
(F^m)^*A_x \rightarrow (F^m)^*K$. Hence, we also have $$\psi_x^m(v)=\varphi^{(m)} \circ (F^m)^*(v)\circ (\phi_x^{(m)})^{-1} 
\qquad \mbox{for all $v\in V_x$}.$$ Thus, we have $\psi_x^{(m)}=\psi_x^m$ for all $x\in X(W)$ and so the eigenvalues of $\psi_x^{(m)}$ are obtained by raising the eigenvalues of $\psi_x$ to the $m$-th power. It remains to use Theorem \[r24\]. This completes the proof of Proposition \[p1\].

\[exp1\] We can, and will assume that $G$ is defined and split over the prime field ${{\mathbb{F}}}_p$ of $k$. Let $F_0\colon G\rightarrow G$ be the corresponding Frobenius map. If $q=p^f$ where $f{\geqslant}1$, then $F=
F_0^f$. Hence, Proposition \[p1\] means that it will be sufficient to determine the scalars $\zeta_x$ $(x\in X(W)$) for the group $G^{F_0}=
G({{\mathbb{F}}}_p)$. For specific values of $p$ (e.g., bad primes $p=2,3,5$), we may then use ad hoc information which is available, for example, in the Cambridge [ATLAS]{} [@atl], or via computer algebra methods (using [GAP]{} [@gap], [CHEVIE]{} [@jmich]). This is the basis for the discussion of the examples below.

