Ordering Lusztig’s families in type Bn
Let W be a finite Coxeter group and L be a weight function on W in the sense of Lusztig. We have recently introduced a pre-order relation ⪯L on the set of irreducible characters of W which extends Lusztig’s definition of “families” and which, conjecturally, corresponds to the ordering given by Kazhdan–Lusztig cells. Here, we give an explicit description of ⪯L for W of type Bn and any L. (All other cases are known from previous work.) This crucially relies on some new combinatorial constructions around Lusztig’s “symbols”. Combined with previous work, we deduce general compatibility results between ⪯L and Lusztig’s a-function, valid for any W,L.