  Let $\bH$ be the generic Iwahori--Hecke algebra associated with a finite
Coxeter group $W$. Recently, we have shown that $\bH$ admits a natural cellular
basis in the sense of Graham--Lehrer, provided that $W$ is a Weyl group and all
parameters of $\bH$ are equal. The construction involves some data arising from
the Kazhdan--Lusztig basis $\{\bC_w\}$ of $\bH$ and Lusztig's asymptotic ring
$\bJ$. This article attemps to study $\bJ$ and its representation theory from a
new point of view. We show that $\bJ$ can be obtained in an entirely different
fashion from the generic representations of $\bH$, without any reference to
$\{\bC_w\}$. Then we can extend the construction of the cellular basis to the
case where $W$ is not crystallographic. Furthermore, if $\bH$ is a
multi-parameter algebra, we will see that there always exists at least one
cellular structure on $\bH$. Finally, one may also hope that the new
construction of $\bJ$ can be extended to Hecke algebras associated to complex
reflection groups.
