The based ring of the lowest generalized two-sided cell of an extended affine Weyl group

Let c0 be the lowest generalized two-sided cell of an extended affine Weyl group W with unequal parameters. We first prove that certain conjectures of Lusztig (called P1-P15) hold for c0, which implies the existence of the based ring of c0, and then we determine the structure of the based ring. As an application, we use the structure of the based ring to study certain simple modules of the Hecke algebra with unequal parameters associated to W, namely those attached to c0. Further we give a set of prime ideals p of the center Z of the affine Hecke algebra H such that the reduced algebra kpH is simple over kp, where kp=Frac(Z/p) is the residue field of Z at p. In particular, we show that the algebra H⊗ZFrac(Z) is split simple over the field Frac(Z).