  We determine the two-point invariants of the equivariant quantum cohomology
of the Hilbert scheme of points of surface resolutions associated to type A_n
singularities. The operators encoding these invariants are expressed in terms
of the action of the affine Lie algebra \hat{gl}(n+1) on its basic
representation. Assuming a certain nondegeneracy conjecture, these operators
determine the full structure of the quantum cohomology ring. A relationship is
proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas
theories of A_n x P^1. We close with a discussion of the monodromy properties
of the associated quantum differential equation and a generalization to
singularities of type D and E.
