Quantization of the minimal nilpotent orbits and the quantum Hikita conjecture

We show that the specialized quantum D-module of the equivariant quantum cohomology ring of the minimal resolution of an ADE singularity is isomorphic to the D-module of graded traces on the minimal nilpotent orbit in the Lie algebra of the same type. This generalizes a recent result of Shlykov [to appear in Proc. Amer. Math. Soc.] and hence verifies in this case the quantum version of Hikita’s conjecture, proposed by Kamnitzer, McBreen and Proudfoot [Adv. Math. 390 (2021)]. As a corollary, we get the Hikita-Nakajima conjecture in this case. We also show analogous isomorphisms for singularities of BCFG type.