# Quantization of the minimal nilpotent orbits and the quantum Hikita conjecture

## TL;DR

This paper proves a quantum version of Hikita's conjecture by establishing an isomorphism between specialized quantum D-modules of equivariant quantum cohomology and D-modules of graded traces on minimal nilpotent orbits for ADE singularities, extending to BCFG types.

## Contribution

It generalizes recent results to verify the quantum Hikita conjecture for ADE and BCFG singularities, linking quantum cohomology and nilpotent orbit structures.

## Key findings

- Isomorphism between quantum D-modules and graded trace D-modules for ADE singularities
- Verification of the quantum Hikita conjecture in this setting
- Extension of results to BCFG singularities

## Abstract

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 [Hikita conjecture for the minimal nilpotent orbit, to appear in Proc. AMS, https://doi.org/10.1090/proc/15281] and hence verifies in this case the quantum version of Hikita's conjecture, proposed by Kamnitzer, McBreen and Proudfoot [The quantum Hikita conjecture, Advances in Mathematics 390 (2021) 107947]. We also show analogous isomorphisms for singularities of BCFG type.