we can canonically extend $\lambda$ to a linear character $$\tilde{\lambda}\colon \tilde{A}(g_1) \rightarrow 
\overline{{{\mathbb{Q}}}}_\ell^\times,\qquad  a\gamma\mapsto \lambda(a).$$ For each $a\in A(g_1)$ we have a corresponding element $g_a\in C^F$, well-defined up to conjugation within $G^F$. (We have $g_a=hg_1h^{-1}$ where $h\in G$ is such that $h^{-1}F(h)\in C_G(g_1)$ has image $a\in 
A(g_1)$.) We define a class function $\chi_{g_1,\lambda} \colon G^F
\rightarrow \overline{{{\mathbb{Q}}}}_\ell$ by $$\chi_{g_1,\lambda}(g)=\left\{\begin{array}{cl} q^{(\dim G-\dim C)/2}
\lambda(a) & \quad\mbox{if $g=g_a$ for some $a\in A(g_1)$},\\ 0 & \quad
\mbox{if $g\not\in C^F$}.\end{array}\right.$$ Now, we can choose an isomorphism $F^*{{\mathcal{E}}}\stackrel{\sim}{\rightarrow}{{\mathcal{E}}}$ such that the induced map on the stalk ${{\mathcal{E}}}_{g_1}$ is scalar multiplication by $q^{(\dim G-\dim C)/2}$. Then this isomorphism canonically induces an isomorphism $\phi_x\colon F^*A_x \stackrel{\sim}{\rightarrow} A_x$ which satisfies the requirements in \[r15\], and we have $\chi_x=\chi_{A_x,
\phi_x}=\chi_{g_1,\lambda}$. (This follows from the fact that $A_x$ is “clean” [@L10], using the construction in [@Ldisc4 19.7].) With this choice of $\phi_x$, we also have for all $m{\geqslant}1$: $$\chi_x^{(m)}(g_a^{(m)})=\left\{\begin{array}{cl} q^{m(\dim G-\dim C)/2}
\lambda(a) & \quad\mbox{if $g=g_a^{(m)}$ for some $a\in A(g_1)$},\\ 0 & 
\quad \mbox{if $g\not\in C^{F^m}$}.\end{array}\right.$$ (Here, $g_a^{(m)}=hg_1h^{-1}$ where now $h\in G$ is such that $h^{-1}F^m(h)
\in C_G(g_1)$ has image $a\in A(g_1)$; see again [@Ldisc4 19.7].) The identity in Definition \[s1\] now reads: $$R_x=\sum_{y\in X(W)} \delta_y\{y,x\}\rho_y=(-1)^{\dim C}
\zeta_x\chi_{g_1,\lambda}.$$

\[rem1\] Let 