Root systems and the quantum cohomology of ADE resolutions

We compute the [math] -equivariant quantum cohomology ring of [math] , the minimal resolution of the DuVal singularity [math] where [math] is a finite subgroup of [math] . The quantum product is expressed in terms of an ADE root system canonically associated to [math] . We generalize the resulting Frobenius manifold to nonsimply laced root systems to obtain an [math] parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold Gromov–Witten potential of [math] .