On the Kazhdan-Lusztig polynomials for affine Weyl groups

Abstract

In 191, Kazhdan and Lusztig associated to a Coxeter group W certain polynomials P,,, (y, w E W). In the case where W is a Weyl group, these polynomials (or their values at 1, P,,,,(l)) give multiplicities of irreducible constituents in the Jordan-Holder series of Verma modules for a complex semisimple Lie algebra corresponding to W (see [ 3,4]). Also, in the case where W is an affine Weyl group, it has been conjectured by Lusztig [lo] that these polynomials or their values P,,,(l) enter the character formula of irreducible rational representations for a semisimple algebraic group (corresponding to W) over an algebraically closed field of prime characteristic. The purpose of this paper is to give some formula (4.2, see also 4.10) concerning the Kazhdan-Lusztig polynomials P,,, for affine Weyl groups which appear in this Lusztig conjecture; namely P,,, for y, w “dominant.” (This is a generalization of the q-analogue of Kostant’s weight multiplicity formula in [8].) As a corollary of the formula, we shall show that the Lusztig conjecture above is consistent with the Steinberg tensor product theorem [ 15 1. To be more precise, let P be the weight lattice of a root system R (II) R ‘, a positive root system). The Weyl group W of R acts on P. Hence we can define the semidirect product of W by P, p= W D( P. The element of @ corresponding to A E P is denoted by t,. The group @ contains an afline Weyl group W, = W D( Q as a normal subgroup, where Q is the root lattice. Although # is not a Coxeter group in general, we can define the Bruhat order >, the length function I: p+ La, and the Kazhdan-Lusztig polynomials P,., (y, w E I-V) as natural extensions of those for W,. The main result of this paper (under the specialization q --+ 1) is stated as follows (Corollary 4.10):