We prove that the Hilbert scheme of $k$ points on $mathbb{C}^2$ (Hilb$^k[mathbb{C}^2]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kahler and equivariant parameters as well as inverting the weight of the $mathbb{C}^times_hbar$-action. First, we find a two-parameter family $X_{k,l}$ of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of Hilb$^k[mathbb{C}^2]$ is obtained via direct limit $ltoinfty$ and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted $hbar$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-$N$ sheaves on $mathbb{P}^2$ with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.
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