An inversion formula for the Kazhdan-Lusztig polynomials for affine weyl groups

Abstract

Let W be a Coxeter group. In [10] Kazhdan and Lusztig defined for each y, w e W certain polynomials Py,w over 7/. They conjectured that in the case where W is the Weyl group corresponding to a complex semi-simple Lie algebra (5 these polynomials determine the characters of the irreducible IS-modules. (This conjecture has now been proved; see [3, 4].) Later G. Lusztig [11] formulated a corresponding conjecture for the irreducible modules for a semi-simple algebraic group G defined over an algebraically closed field of prime characteristic p involving the same polynomials except that W is now the affine Weyl group corresponding to G. Elementary properties of the polynomials Py,w show that there exist polynomials Q .... x, z ~ W such that the inversion formula