Some Non-Trivial Kazhdan-Lusztig Coefficients of an Affine Weyl Group of Type ${\tilde{A}}_n$

In this paper we show that the leading coefficient $\mu (y,w)$ of some Kazhdan-Lusztig polynomials $P_{y,w}$ with $y,w$ in an affine Weyl group of type ${\tilde{A}}_n$ is $n+2$. This fact has some consequences on the dimension of first extension groups of finite groups of Lie type with irreducible coefficients.