| Given two elements y≤w in a Coxeter group W with S the set of simple reflections, we have a Kazhdan-Lusztig polynomial Py,w in an indeterminate q. The degree of Py,w is less than or equal to 1/2(l(w) - l(y) -1) if y < w and Pw,w = 1. Particular interests are focused on the coefficientμ(y,w) of the term q1/2(l(w)-l(y)-1), since it plays a key role in understanding Kazhdan-Lusztig polynomials . Moreover, the coefficients for Weyl groups or affine Weyl groups are important in representation theory and in Lie theory, which are related to some cohomology groups and some difficult irreducible characters.However, it is in general hard to compute the leading coefficients. In this thesis we compute the leading coefficients of Kazhdan-Lusztig polynomials for an affine Weyl group of type (?) or type (?). For any y≤w∈W such that a(y)≤a(w), we compute allμ(y,w) clearly, where a : W→N is the a-function introduced by Lusztig (see Section 2.2). Moreover, for an affine Weyl group of type (?), we also computeμ(y,w) for those y≤w∈W such that a(y) = 2 and a(w) = 1 and give some results when a(y) = 4 and a(w) = f or 2. With these valuesμ(y,w), we can show that a conjecture of Lusztig proposed in 1987 on distinguished involutions is true for an affine Weyl group of type (?). Moreover, we show that a conjectural formula in [L3] is not true (see Remark 5.2.5 (2)). In order to study the nonlocal finiteness of a W-graph, Lusztig introduced some semilinear equations to compute certain leading coefficients for an affine Weyl group in the 1996's paper [L3]. Our computations in this thesis mainly rely on these semilinear equations (see Section 5.2) and a formula of Springer (see Section 2.4). In the last section of this thesis, we give some interesting formulas, which indicate some relations between the leading coefficients and representations of a related algebraic group and the structure of the based ring of W (see Section 5.9).
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