  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})$.

Representation Theory