whenever $A\cong F^*A$.

\[r16\] Assume that $Z(G)$ is connected and that $G/Z(G)$ is simple; also recall that $F$ is assumed to act trivially on $W$. Let $A\in\hat{G}^{\operatorname{un}}$ and $x\in X(W)$ be such that $x=x_A$. Then $F^*A\cong A$ and $R_x$ is equal to $\chi_{A,\phi_A}$, up to a non-zero scalar multiple.

As already mentioned, Shoji’s results also apply to non-split groups and to non-unipotent characters. In [@S2], [@S3] it is assumed, however, that $p$ is “almost good”, that is, the following conditions hold. If $G$ is type $A_n$, $B_n$, $C_n$ or $D_n$, no condition. If $G$ is of type $E_6$, then $p\neq 2$. If $G$ is of type $G_2$, $F_4$ or $E_7$, then $p\neq 2,3$. If $G$ is of type $E_8$, then $p\neq 2,3,5$ (see [@L2e 23.0.1]). Since Lusztig [@L10] has established the “cleanness” of cuspidal character sheaves in full generality, the methods in [@S2], [@S3] which were used to prove Theorem \[r16\] for $G$ of exceptional type and almost good $p$ can now also be applied for any $p$. Consequently, as Shoji pointed out to the author, Theorem \[r16\] holds as stated above, without any condition on $p$.

\[s1\] In the setting of Theorem \[r16\], let $A\in 
\hat{G}^{\operatorname{un}}$ and $x\in X(W)$ 