Hikita surjectivity for $\mathcal N /// T$

TL;DR

This paper constructs a surjective algebra map related to the Hikita conjecture for a specific Nakajima quiver variety, providing evidence for the conjecture and proposing a broader inheritance framework.

Contribution

The authors explicitly construct a surjective map between graded algebras in the context of the Hikita conjecture for $ abla///T$, linking Coulomb branches and symplectic resolutions.

Findings

Constructed a surjective algebra map matching the Hikita conjecture prediction.

The map factors through Kirwan surjectivity for quiver varieties.

Proposed that many Hikita maps can be inherited from dual pairs.

Abstract

The Hamiltonian reduction $\mathcal N///T$ of the nilpotent cone in $\mathfrak{sl}_n$ by the torus of diagonal matrices is a Nakajima quiver variety which admits a symplectic resolution $\widetilde{\mathcal N///T}$, and the corresponding BFN Coulomb branch is the affine closure $\overline{T^*(G/U)}$ of the cotangent bundle of the base affine space. We construct a surjective map $\mathbb C\left[\overline{T^*(G/U)}^{T\times B/U}\right] \twoheadrightarrow H^*\left(\widetilde{\mathcal N /// T}\right)$ of graded algebras, which the Hikita conjecture predicts to be an isomorphism. Our map is inherited from a related case of the Hikita conjecture and factors through Kirwan surjectivity for quiver varieties. We conjecture that many other Hikita maps can be inherited from that of a related dual pair.