On the number of simple modules of Iwahori--Hecke algebras of finite Weyl groups

Let $H_k(W,q)$ be the Iwahori--Hecke algebra associated with a finite Weyl group $W$, where $k$ is a field and $0 \neq q \in k$. Assume that the characteristic of $k$ is not ``bad'' for $W$ and let $e$ be the smallest $i \geq 2$ such that $1+q+q^2+... +q^{i-1}=0$. We show that the number of simple $H_k(H,q)$-modules is ``generic'', i.e., it only depends on $e$. The proof uses some computations in the {\sf CHEVIE} package of {\sf GAP} and known results due to Dipper--James, Ariki--Mathas, Rouquier and the author.