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

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 $\overline{T^{\ast }(G/U)}$ of the cotangent bundle of the base affine space. We construct a surjective map $\mathbb{C}\left[{\overline{T^{\ast }(G/U)}}^{T\times B/U}\right]\twoheadrightarrow H^{\ast }\left(\widetilde{\mathcal{N}///T}\right)$ 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.

v2: 20 pages, 5 figures. Final version, to appear at Math. Z