Hikita conjecture for classical Lie algebras

Abstract

Let $G$ be $S{p}_{2n}$, $S{O}_{2n}$ or $S{O}_{2n+1}$ and let $G^{\vee }$ be its Langlands dual group. Barbasch and Vogan based on earlier work of Lusztig and Spaltenstein, define a duality map $D$ that sends nilpotent orbits ${\mathbb{O}}_{e^{\vee }}\subset {\mathfrak{g}}^{\vee }$ to special nilpotent orbits ${\mathbb{O}}_e\subset \mathfrak{g}$. In a work by Losev, Mason-Brown and Matvieievskyi, an upgraded version $\tilde{D}$ of this duality is considered, called the refined BVLS duality. $\tilde{D}({\mathbb{O}}_{e^{\vee }})$ is a $G$-equivariant cover ${\tilde{\mathbb{O}}}_e$ of ${\mathbb{O}}_e$. Let $S_{e^{\vee }}$ be the nilpotent Slodowy slice of the orbit ${\mathbb{O}}_{e^{\vee }}$. The two varieties $X^{\vee }=S_{e^{\vee }}$ and $X=$ Spec$(\mathbb{C}[{\tilde{\mathbb{O}}}_e])$ are expected to be symplectic dual to each other. In this context, a version of the Hikita conjecture predicts an isomorphism between the cohomology ring of the Springer fiber ${\mathcal{B}}_{e^{\vee }}$ and the ring of regular functions on the scheme-theoretic fixed point $X^T$ for some torus $T$. This paper verifies the isomorphism for certain pairs $e$ and $e^{\vee }$. These cases are expected to cover almost all instances in which the Hikita conjecture holds when $e^{\vee }$ regular in a Levi ${\mathfrak{l}}^{\vee }\subset {\mathfrak{g}}^{\vee }$. Our results in these cases follow from the relations of three different types of objects: generalized coinvariant algebras, equivariant cohomology rings, and functions on scheme-theoretic intersections. We also give evidence for the Hikita conjecture when $e^{\vee }$ is distinguished.