finally, induces an isomorphism $$\phi_x^{(m)} \colon (F^m)^*A_x\stackrel{\sim}{\rightarrow} A_x\qquad
\mbox{(see \cite[1.1]{S2})}.$$ The latter isomorphism again satisfies the conditions in \[r15\]; we denote the corresponding characteristic function by $\chi_x^{(m)}\colon G^{F^m}\rightarrow \overline{{{\mathbb{Q}}}}_\ell$. Note that, if $g$ is an element in $G^F$ (and not just in $G^{F^m}$), then $$\chi_x^{(m)}(g)=\sum_i (-1)^i\mbox{Trace}\bigl((\phi_x)_{i,g}^m,
{{\mathcal{H}}}_g^i(A_x)\bigr).$$ (See again [@S2 1.1].) As in Definition \[s1\], we define $0\neq 
\zeta_x^{(m)}\in \overline{{{\mathbb{Q}}}}_\ell$ by the condition that $$R_x^{(m)}=(-1)^{\dim G}\hat{\varepsilon}_x\zeta_x^{(m)}\chi_x^{(m)}.$$ (Thus, if $m=1$, then $\zeta_x^{(1)}=\zeta_x$.) We can now state the main result of this section.

\[p1\] In the setting of \[r21\], \[r22\], we have $\zeta_x^{(m)}=\zeta_x^m$ for all $x\in X(W)$.

A result of this kind is implicitly contained in Lusztig [@L2c §14] and Shoji [@S2 §2 and 5.19]; the proof will be given in \[r25\]. First, we need some preparations.

\[r23\] We recall some constructions from [@L2c §12, §13]. For any $w\in W$, we assume chosen once and for all a representative $\dot{w}\in N_G(T)$. There is a corresponding complex $\bar{K}_{\dot{w}}^{\mathcal{L}_0}\in \mathcal{D}G$ as defined in [@L2c 12.1]. Then $\hat{G}^{\text{un}}$ can also be characterized as the set of isomorphism classes of simple perverse sheaves on $G$ which occur as constituents of a perverse cohomology sheaf ${^p\!H}^i(
\bar{K}_{\dot{w}}^{\mathcal{L_0}})$ for some $w\in W$ and some $i\in{{\mathbb{Z}}}$. Furthermore, for each 