  We study moduli spaces of objects in the derived category of noncommutative
ruled surfaces over orbifold curves to find equivariant deformations of moduli
spaces of framed sheaves on equivariant elliptic surfaces. These derived
categories are a common generalization of the category of modules of the
preprojective algebra and of the category of twisted D-modules over a curve. We
show that the birational geometry of the Hilbert scheme of points on an
equivariant elliptic surface is controlled by an elliptic root system, and
conjecture in general and prove in special cases that moduli spaces of objects
in these derived categories provides a deformation of birational models for the
Hilbert scheme. We also study the applications of this deformation to the
geometric representation theory of the central fiber, specifically to produce
geometric correspondences giving rise to Lie algebra actions on the equivariant
cohomology of the central fiber and give formulas for the action of the
Namikawa-Markman Weyl group of monodromy reflections in terms of this Lie
algebra.
