one has to establish certain identities of class functions on $G(q)$ of the form $R_x=\xi \chi_A$, where $R_x$ denotes an “almost character” (that is, an explicitly known linear combination of irreducible characters) and $\chi_A$ denotes the characteristic function of a suitable “character sheaf” on the underlying algebraic group $G$; furthermore, $\xi$ is a scalar of absolute value $1$. This program has been successfully carried out in many cases, see, e.g., Bonnafé [@Bo3], Lusztig [@L3], [@L7], Shoji [@Sclass], [@S7] and Waldspurger [@wald], but not in complete generality.

In this paper, we will assume that $G(q)$ is of split type and only consider the above problem as far as unipotent characters of $G(q)$ are concerned, as defined by Deligne and Lusztig [@DeLu]. By Shoji’s work [@S2], [@S3], we know that the desired identities $R_x=\xi \chi_A$ as above hold, but the scalars $\xi$ are not yet determined in all cases. And even in those cases where they are known, this often required elaborate computations. In such cases, the scalars then turn out to behave rather uniformly as $q$ varies (see, e.g., Shoji [@Sclass], [@S7]). The main theoretical result of this paper, Proposition \[p1\], provides a partial, [*a priori*]{} explanation for this phenomenon; the 