$G$ be of (split) classical type. Then Shoji has shown that we always have $\zeta_x=1$ for cuspidal $A_x\in
\hat{G}^{\text{un}}$; see [@Sclass Prop. 6.7] for $p\neq 2$, and [@S7 Theorem 6.2] for $p=2$. Note that this involves, in each case, the choice of a particular representative in the conjugacy class supporting $A_x$. Since classical groups of low rank appear as Levi subgroups in groups of exceptional type, it will be useful to work out explicitly the relevant identities $R_x=(-1)^{\dim C} \zeta_x\chi_{(g_1,\lambda)}$ for $G$ of type $C_2$, $D_4$ and $p=2$. This also provides a good illustration for: (a) the role of the choice of a class representative as above and (b) the strategy that we will employ when dealing with groups of exceptional type.

\[sp4\] Let $G=\mbox{Sp}_4(k)$ be the $4$-dimensional symplectic group. Then $G=\langle x_\alpha(t)\mid \alpha\in \Phi, t\in k
\rangle$ with root system $\Phi=\{\pm a,\pm b,\pm (a+b),\pm (2a+b)\}$. The Weyl group $W=\langle s_a,s_b\rangle$ is dihedral of order $8$ and we have $${{\operatorname{Irr}}}(W)=\{1_W,\text{sgn},\text{sgn}_a,\text{sgn}_b,r\}$$ where $1_W$ is the trivial character, $\text{sgn}$ is the sign character, $r$ has degree $2$, and $\text{sgn}_a,\text{sgn}_b$ are linear characters such that $\text{sgn}_a(s_a)=1$, $\text{sgn}_a(s_b)=-1$, $\text{sgn}_b
(s_a)=-1$, $\text{sgn}_b(s_b)=1$. By [@C2 p. 468], we have $$X(W)=\{x_1, x_{\text{sgn}}, x_{\text{sgn}_a}, x_{\text{sgn}_b}, x_{r},
x_0\}$$ where 