  We give a construction of an affine Hecke algebra associated to any Coxeter
group acting on an abelian variety by reflections; in the case of an affine
Weyl group, the result is an elliptic analogue of the usual double affine Hecke
algebra. As an application, we use a variant of the $\tilde{C}_n$ version of
the construction to construct a flat noncommutative deformation of the $n$th
symmetric power of any rational surface with a smooth anticanonical curve, and
give a further construction which conjecturally is a corresponding deformation
of the Hilbert scheme of points.
