Some Kazhdan–Lusztig Coefficients of Affine Weyl Group of Type B~ 2

Abstract

Let [Formula: see text] be the affine Weyl group of type [Formula: see text], on which we consider the length function [Formula: see text] from [Formula: see text] to [Formula: see text] and the Bruhat order [Formula: see text]. For [Formula: see text] in [Formula: see text], let [Formula: see text] be the coefficient of [Formula: see text] in Kazhdan–Lusztig polynomial [Formula: see text]. We determine some [Formula: see text] for [Formula: see text] and [Formula: see text], where [Formula: see text] is the lowest two-sided cell of [Formula: see text] and [Formula: see text] is the higher one. Furthermore, we get some consequences using left or right strings and some properties of leading coefficients.