Quasi-parabolic Kazhdan-Lusztig bases and reflection subgroups

Recently, Wang and the second author constructed a bar involution and canonical basis for a quasi-permutation module of the Hecke algebra associated to a type B Weyl group $W$, where the basis is parameterized by left cosets of a quasi-parabolic reflection subgroup in $W$. In this paper we provide an alternative approach to these constructions, and then generalize to Coxeter groups which contain a product of type B Weyl groups as a parabolic subgroup.