Kazhdan–Lusztig coefficients for an affine Weyl group of type [formula omitted

Abstract

In this paper we compute the leading coefficients μ ( u , w ) of the Kazhdan–Lusztig polynomials P u , w for an affine Weyl group of type B ˜ 2 . By using the a-function of a Coxeter group defined by Lusztig, we compute most μ ( u , w ) explicitly. With part of these values μ ( u , w ) , we show that a conjecture of Lusztig on distinguished involutions is true for an affine Weyl group of type B ˜ 2 . We also show that the conjectural formula concerning some semilinear equations needs a modification.