The relation between Parabolic Hecke modules and $W$-graph ideal modules in Kazhdan-Lusztig theory

In 2011, Howlett and Nguyen \cite{r1} introduced the concept of a $W$-graph ideal $E_J$ in $\left ( W,\leqslant_{L} \right )$ with respect to $J$ (a subset of $S$), where $\leqslant _{L}$ is the left weak order on $W$. They proved that one can construct a $W$-graph from a given $W$-graph ideal by constructing a Hecke module structure on $E_J$, where the $W$-graph was introduced by Kazhdan and Lusztig in \cite{d1}. In this paper, we give the relation between Hecke modules on $E_J$ and general Hecke algebras by considering the relation between Hecke modules on $E_J$ and parabolic Hecke modules. And inspired by Lusztig \cite{g3}, we show that the parabolic Hecke module is isomorphic to a left ideal of the Hecke algebra. Lastly, we give the relation between $R$-polynomials on $E_J$ and parabolic $R$-polynomials as an application of the main results.