proof is merely an elaboration of ideas which are already contained in Lusztig’s and Shoji’s papers. The fact that we can formulate this result without any assumptions on $q$ essentially relies on Lusztig [@L10], where the “cleanness” of cuspidal character sheaves is established in complete generality and, consequently, the principal results of [@L2a]–[@L2e] (e.g. [@L2e Theorems 23.1 and 25.2]) hold unconditionally.

The main observation of this paper is that the statement of Proposition \[p1\] can also be exploited in a different way, as follows. For a given type of group, we consider the base case where $q=p$ is a prime. For a specific value of $p$, we can use ad hoc methods and/or computer algebra systems like [GAP]{} [@gap] to perform all kinds of computations within the fixed finite group $G(p)$. If we succeed in this way to determine the scalars $\xi$ for $G(p)$, then Proposition \[p1\] tells us that the analogous result will hold for any power of $p$. This is particularly relevant for “bad” primes $p=2,3,5$ which, typically, are known to cause additional complications and require separate arguments. We illustrate this procedure with a number of examples. In particular, we determine the scalars $\zeta$ in two cases, where 