Representations of affine hecke algebras of type

Let k: be a field and q a nonzero element in k such that the square roots of q are in k. We use Hq to denote an affine Hecke algebra over k of type with parameter q. The purpose of this paper is to study representations of Hq by using based rings of two-sided cells of an affine Weyl group W of type . We shall give the classification of irreducible representations of Hq. We also remark that a calculation in [11] actually shows that Theorem 2 in [1] needs a modification, a fact is known to Grojnowski and Tanisaki long time ago. In this paper we also show an interesting relation between Hq and an Hecke algebra corresponding to a certain Coxeter group. Apparently the idea in this paper works for all affine Weyl groups, but that is the theme of another paper.