Hikita-Nakajima conjecture for the Gieseker variety

Abstract

Let $\mathfrak{M}_0$ be an affine Nakajima quiver variety, and $\mathcal{M}$ is the corresponding BFN Coulomb branch. Assume that $\mathfrak{M}_0$ can be resolved by the (smooth) Nakajima quiver variety $\mathfrak{M}$. The Hikita-Nakajima conjecture claims that there should be an isomorphism of (graded) algebras $H^*_{S}(\mathfrak{M},\mathbb{C}) \simeq \mathbb{C}[\mathcal{M}_{\mathfrak{s}}^{\mathbb{C}^\times}]$, here $S \curvearrowright \mathfrak{M}_0$ is a torus acting on $\mathfrak{M}_0$ preserving the Poisson structure, $\mathcal{M}_{\mathfrak{s}}$ is the (Poisson) deformation of $\mathcal{M}$ over $\mathfrak{s}=\operatorname{Lie} (S)$, $\mathbb{C}^\times$ is a generic one-dimensional torus acting on $\mathcal{M}$, and $\mathbb{C}[\mathcal{M}_{\mathfrak{s}}^{\mathbb{C}^\times}]$ is the algebra of schematic $\mathbb{C}^\times$-fixed points of $\mathcal{M}_{\mathfrak{s}}$. We prove the Hikita-Nakajima conjecture for $\mathfrak{M}=\mathfrak{M}(n,r)$ Gieseker variety ($ADHM$ space). We produce the isomorphism explicitly on generators. We also describe the Hikita-Nakajima isomorphism above using the realization of $\mathcal{M}_{\mathfrak{s}}$ as the spectrum of the center of rational Cherednik algebra corresponding to $S_n \ltimes (\mathbb{Z}/r\mathbb{Z})^n$ and identify all the algebras that appear in the isomorphism with the center of degenerate cyclotomic Hecke algebra (generalizing some results of Shan, Varagnolo, and Vasserot).