Kazhdan–Lusztig coefficients for an affine Weyl group of type

In this paper we compute the leading coefficients μ(u,w) of the Kazhdan–Lusztig polynomials Pu,w for an affine Weyl group of type . 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 . We also show that the conjectural formula concerning some semilinear equations needs a modification.