Let \({\mathfrak {M}}_0\) be an affine Nakajima quiver variety, and let \({\mathcal {M}}\) be 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 \(H^*_{S}({\mathfrak {M}},{\mathbb {C}}) \simeq {\mathbb {C}}[{\mathcal {M}}_{{\mathfrak {s}}}^{{\mathbb {C}}^\times }]\) , where \(S \curvearrowright ~{\mathfrak {M}}_0\) 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 the 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 the degenerate cyclotomic Hecke algebra (generalizing some results of Shan, Varagnolo, and Vasserot).
