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 $\{C_w\}$ 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 Bonnaf'e–Iancu: we show that $\{C_w\}$ is a ``cellular basis’’ in the sense of Graham–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 $B_n$.