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We keep the basic assumptions of the previous section; we also assume that $Z(G)$ is connected and $G/Z(G)$ is simple. We fix a square root of $q$ in $\overline{{{\mathbb{Q}}}}_\ell$. For any $A\in\hat{G}^{\text{un}}$, we know by Theorem \[r16\] that $F^*A\cong A$; we assume that an isomorphism $\phi_A\colon F^*A \stackrel{\sim}{\rightarrow} A$ has been chosen as in \[r15\]. Our aim is to get hold of the coefficients $\zeta_A$ in Definition \[s1\].

\[r18\] For the further discussion, it will be convenient to change the notation and label everything by elements of $X(W)$. Thus, via the bijection ${{\mathfrak{U}}}(G^F)\leftrightarrow X(W)$ in \[r11\] (arranged as in Remark \[r11a\]), we can write $${{\mathfrak{U}}}(G^F)=\{\rho_x \mid x \in X(W)\} \qquad \mbox{where}\qquad
\rho_\epsilon=\rho_{x_\epsilon} \mbox{ for $\epsilon \in {{\operatorname{Irr}}}(W)$}.$$ For $x \in X(W)$, we write $\delta_x:=\delta_{\rho_x}$. Then $R_y=\sum_{x 
\in X(W)}\delta_x \{x,y\}\rho_x$ for all $y \in X(W)$. Next, via the bijection $\hat{G}^{\text{un}}\leftrightarrow X(W)$ in \[r14\], we can write $$\hat{G}^{\text{un}}=\{A_x \mid x \in X(W)\}.$$ For $x \in X(W)$, we denote an isomorphism $F^*A_x \cong A_x$ as in \[r15\] by $\phi_x$ and the corresponding characteristic function simply by $\chi_x$. Then the relation in Definition \[s1\] is rephrased as $$R_x=(-1)^{\dim G} \hat{\varepsilon}_x \zeta_x\chi_x \qquad \mbox{where}
\qquad \hat{\varepsilon}_x:=\hat{\varepsilon}_{A_x} \mbox{ and } \zeta_x:=
\zeta_{A_x}.$$ For $\epsilon\in{{\operatorname{Irr}}}(W)$, the 