Relative Kazhdan–Lusztig cells

Abstract

In this paper, we study the Kazhdan–Lusztig cells of a Coxeter groupWWin a “relative” setting, with respect to a parabolic subgroupWI⊆WW_I \subseteq W. This relies on a factorization of the Kazhdan–Lusztig basis{Cw}\{\mathbf {C}_w\}of the corresponding (multi-parameter) Iwahori–Hecke algebra with respect toWIW_I. We obtain two applications to the “asymptotic case” in typeBnB_n, as introduced by Bonnafé and Iancu: we show that{Cw}\{\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 typeBnB_n.