Relative Kazhdan-Lusztig cells

In this paper, we study the Kazhdan-Lusztig cells of a Coxeter group $W$ in a ``relative'' setting, with respect to a parabolic subgroup $W_I \subseteq W$. This relies on a factorization of the Kazhdan-Lusztig basis $\{\mathbf{C}_w\}$ of the corresponding (multi-parameter) Iwahori-Hecke algebra with respect to $W_I$. We obtain two applications to the ``asymptotic case'' in type $B_n$, as introduced by Bonnafé and Iancu: we show that $\{\mathbf{C}_w\}$ is a ``cellular basis'' in the sense of Graham and Lehrer, and we construct an analogue of Lusztig's canonical isomorphism from the Iwahori-Hecke algebra to the group algebra of the underlying Weyl group of type $B_n$.