Cuspidal character sheaves and small rank examples {#sec3}
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We keep the notation of the previous section; in particular, we label all objects by the parameter set $X(W)$ as in \[r18\]. By [@L3 3.5], the computation of the scalar $\zeta_x$ can be reduced to the case where $A_x$ is a cuspidal character sheaf (in the sense of [@L2a Def. 3.10]). So let us look in more detail at this case.

\[exp2\] Assume that $Z(G)=\{1\}$. Let $x\in X(W)$ be such that $A_x\in\hat{G}^{\text{un}}$ is cuspidal. Then there exists an $F$-stable conjugacy class $C$ of $G$ and an irreducible, $G$-equivariant $\overline{{{\mathbb{Q}}}}_\ell$-local system ${{\mathcal{E}}}$ on $C$ such that $F^*{{\mathcal{E}}}\cong 
{{\mathcal{E}}}$ and $A=\mbox{IC}(\overline{C},{{\mathcal{E}}})[\dim C]$; see [@L2a 3.12]. In particular, $\mbox{supp}(A_x)=\overline{C}$ and so $\hat{\varepsilon}_{A_x}=
(-1)^{\dim G-\dim C}$. Let us fix $g_1\in C^F$ and set $A(g_1):=C_G(g_1)/
C_G^\circ(g_1)$. Then $F$ induces an automorphism $\gamma\colon A(g_1)
\rightarrow A(g_1)$. We further assume that:

-   the local system ${{\mathcal{E}}}$ is one-dimensional and, hence, corresponds to a $\gamma$-invariant linear character $\lambda\colon 
    A(g_1) \rightarrow \overline{{{\mathbb{Q}}}}_\ell^\times$ (via [@Ldisc4 19.7]).

(This assumption will be satisfied in all examples that we consider.) We form the semidirect product $\tilde{A}(g_1)=A(g_1)\rtimes \langle 
\gamma\rangle$ such that, inside $\tilde{A}(g_1)$, we have the identity $\gamma(a)=\gamma a\gamma^{-1}$ for all $a\in A(g_1)$. By ($*$), 