Lusztig's a-function in type Bₙ in the asymptotic case

الأصل · EN

In this paper, we study Lusztig's a-function for a Coxeter group with unequal parameters. We determine that function explicitly in the ``asymptotic case'' in type Bₙ, where the left cells have been determined in terms of a generalized Robinson--Schensted correspondence by Bonnafé and the second author. As a consequence, we can also show that all of Lusztig's conjectural properties (P1)--(P15) hold in this case, except possibly (P9), (P10) and (P15). Our methods rely on the ``leading matrix coefficients'' introduced by the first author. We also interprete the ideal structure defined by the two-sided cells in the associated Iwahori--Hecke algebra ₙ in terms of the Dipper--james--Murphy basis of ₙ.