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

Let $\mathbf{c}_0$ be the lowest generalized two-sided cell of an extended affine Weyl group W. We determine the structure of the based ring of $\mathbf{c}_0$. For this we show that certain conjectures of Lusztig on generalized cells (called P1-P15) hold for $\mathbf{c}_0$. As an application, we use the structure of the based ring to study certain simple modules of Hecke algebras of $W$ with unequal parameters, namely those attached to $\mathbf{c}_0$. Also we give a set of prime ideals $\mathfrak{p}$ of the center $\mathcal{Z}$ of the generic affine Hecke algebra $\mathcal{H}$ such that the reduced affine Hecke algebra $k_\mathfrak{p}\mathcal{H}$ is simple over $k_\mathfrak{p}$, where $k_\mathfrak{p}=\rm{Frac}(\mathcal{Z}/\mathfrak{p})$ is the residue field of $\mathcal{Z}$ at $\mathfrak{p}$. In particular, we show that the algebra $\mathcal{H}\otimes_\mathcal{Z}\rm{Frac}(\mathcal{Z})$ is a split simple algebra over the field $\rm{Frac}(\mathcal{Z})$.