M ay 2 00 8 Leading coefficients of the Kazhdan - Lusztig polynomials for an Affine Weyl group of type

In this paper we compute the leading coefficients μ(y,w) of the Kazhdan-Lusztig polynomials Py,w for an affineWeyl group of type B̃2. When a(y) ≤ a(w) or a(y) = 2 and a(w) = 1, we compute all μ(y,w) clearly, where a(y) is the a-function of a Coxeter group defined by Lusztig (see [L1]). With these values μ(y,w), we are able to show that a conjecture of Lusztig on distinguished involutions is true for an affine Weyl group of type B̃2. We also show that the conjectural formula in [L3, (12)] needs a modification. Moreover, in the last section, we give some interesting formulas on CrtSλ, which imply some relations between μ(y,w) and representations of some algebraic groups. The Kazhdan-Lusztig polynomials of a Coxeter group W play a central role in Kazhdan-Lusztig theory. From [KL1] one sees that the leading coefficients μ(y, w) of some Kazhdan-Lusztig polynomials Py,w are very important in understanding the Kazhdan-Lusztig polynomials. Moreover, the coefficients are of great importance in representation theory and Lie theory, which are related to some cohomology groups and some difficult irreducible characters (see [A,C,S]). However, it is in general hard to compute the leading coefficients. In [L3] Lusztig computes the leading coefficients for some Kazhdan-Lusztig polynomials of an affine Weyl group of type B̃2 (see Remark 4.1.5 (2)). In [S] for