Dubrovin duality and mirror symmetry for ADE resolutions

We show that, under Dubrovin’s notion of ‘‘almost’’ duality, the Frobenius manifold structure on the orbit spaces of the extended affine Weyl groups of type $\mathrm{A}\mathrm{D}\mathrm{E}$ is dual, for suitable choices of weight markings, to the equivariant quantum cohomology of the minimal resolution of the du Val singularity of the same Dynkin type. We also provide a uniform Lie-theoretic construction of Landau-Ginzburg mirrors for the quantum cohomology of $\mathrm{A}\mathrm{D}\mathrm{E}$ resolutions. The mirror B-model is described by a one-dimensional LG superpotential associated to the spectral curve of the $\widehat{\mathrm{A}\mathrm{D}\mathrm{E}}$ affine relativistic Toda chain.