Leading coefficients and cellular bases of Hecke algebras

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