be such that $x=x_A$. Recall that $\phi_A\colon F^*A \stackrel{\sim}{\rightarrow} A$ is assumed to be chosen as in \[r15\]. Then we define $0\neq \zeta_A\in\overline{{{\mathbb{Q}}}}_\ell$ by the condition that $R_x=(-1)^{\dim G}\hat{\varepsilon}_A\zeta_A\chi_{A,
\phi_A}$.

\[r17\] In the setting of \[r11a\], let $\epsilon\in
{{\operatorname{Irr}}}(W)$ and consider the corresponding character $\rho_\epsilon$. We have $\delta_{\rho_\epsilon}=1$ and $x_\epsilon=x_{\rho_\epsilon}$. So, using the formula in \[r12\], we obtain: $$\begin{aligned}
\rho_\epsilon&=\sum_{x\in X(W)} \{x, x_{\rho_\epsilon}\}R_x=
\sum_{x\in X(W)} \{x, x_\epsilon\}R_x\\&=(-1)^{\dim G}\sum_{A\in 
\hat{G}^{\text{un}}} \hat{\varepsilon}_A\zeta_A\{x_A, x_\epsilon\}
\chi_{A,\phi_A}\qquad \mbox{(see Def.~\ref{s1})}\\&=(-1)^{\dim G}
\sum_{A\in \hat{G}^{\text{un}}} \zeta_A(A:K_\epsilon^{\mathcal{L}_0})
\chi_{A,\phi_A} \qquad \mbox{(see \ref{r14})}.\end{aligned}$$ Such an expression for $\rho_\epsilon$ as a linear combination of characteristic functions first appeared in [@L2c 14.14]; it is actually an important ingredient in the proof of Theorem \[r16\]. On the other hand, the argument in [@L2c 14.14] relies on an alternative interpretation of the coefficients $\zeta_A$, which we will consider in more detail in the following section. Note that, for a given $A\in\hat{G}^{\text{un}}$, there always exists some $\epsilon\in{{\operatorname{Irr}}}(W)$ such that $(A:K_\epsilon^{\mathcal{L}_0})\neq 0$; see [@L2c 14.12]. Furthermore, by [@L2e 25.2] (and the comments in [@L10 3.10]), the functions $\{\chi_{A,\phi_A}\mid A\in \hat{G}^{\text{un}}\}$ are linearly independent. It follows that the coefficients $\zeta_A$ are uniquely determined by the above system of equations, where $\epsilon$ runs over ${{\operatorname{Irr}}}(W)$.

The scalars $\zeta_A$ {#sec2}
