Leading coefficients and cellular bases of Hecke algebras

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.