Around Hikita-Nakajima conjecture for nilpotent orbits and parabolic Slodowy varieties

TL;DR

This paper investigates the Hikita conjecture for nilpotent Slodowy varieties, proposes a refined version due to counterexamples, and proves it for parabolic cases, linking symplectic duality, Lie theory, and combinatorics.

Contribution

The authors refine the Hikita conjecture for certain nilpotent orbit pairs and prove this refined version for parabolic Slodowy varieties, extending previous results.

Findings

Original Hikita conjecture does not hold for studied pairs.

Refined Hikita conjecture is proved for parabolic Slodowy varieties.

Connections established between Springer fibers and Kazhdan-Lusztig cells.

Abstract

Let $G$ be a complex reductive algebraic group. In arxiv:2108.03453 Ivan Losev, Lucas mason-Brown and the third-named author suggested a symplectic duality between nilpotent Slodowy slices in $\mathfrak{g}^\vee$ and affinizations of certain $G$-equivariant covers of special nilpotent orbits. In this paper, we study the various versions of Hikita conjecture for this pair. We show that the original statement of the conjecture does not hold for the pairs in question and propose a refined version. We discuss the general approach towards the proof of the refined Hikita conjecture and prove this refined version for the parabolic Slodowy varieties, which includes many of the cases considered in arxiv:2108.03453 and more. Applied to the setting of arxiv:2108.03453, the refined Hikita conjecture explains the importance of special unipotent ideals from the symplectic duality point of view. We also discuss applications of our results. In the appendices, we discuss some classical questions in Lie theory that relate the refined version and the original version. We also explain how one can use our results to simplify some proofs of known results in the literature. As a combinatorial application of our results we observe an interesting relation between the geometry of Springer fibers and left Kazhdan-Lusztig cells in the corresponding Weyl group.