Hikita conjecture for classical Lie algebras

TL;DR

This paper verifies the Hikita conjecture for certain pairs of nilpotent orbits in classical Lie algebras, connecting cohomology rings and functions on fixed point schemes, and provides evidence for broader cases.

Contribution

It confirms the Hikita conjecture for specific nilpotent orbit pairs in classical Lie algebras, expanding understanding of symplectic duality and scheme-theoretic fixed points.

Findings

Verified the Hikita conjecture for certain orbit pairs

Established relations among coinvariant algebras, cohomology rings, and functions

Provided evidence for the conjecture when the dual orbit is distinguished

Abstract

Let $G$ be $Sp_{2n}$, $SO_{2n}$ or $SO_{2n+1}$ 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.