Generalised Springer correspondence for Z/m-graded Lie algebras

TL;DR

This paper proves a conjecture linking simple perverse sheaves on graded Lie algebra nilpotent cones to modules of a degenerate double affine Hecke algebra, extending the Deligne--Langlands correspondence.

Contribution

It establishes a bijection between simple perverse sheaves in a fixed block and simple modules of a degenerate double affine Hecke algebra, generalizing prior work.

Findings

Proved the conjecture of Lusztig and Yun on bijectivity.

Extended the Deligne--Langlands correspondence to graded Lie algebra settings.

Connected geometric and algebraic structures via spiral inductions.

Abstract

Let $G$ be a simple simply connected complex algebraic group and let $\mathfrak{g}_*$ be a $\mathbf{Z}/m$-grading on its Lie algebra $\mathfrak{g}$. In a recent series of articles, G. Lusztig and Z. Yun, studied the classification of simple $G_0$-equivariant perverse sheaves on the nilpotent cone of $\mathfrak{g}_i$ for $i\in \mathbf{Z}/m$, where $G_0$ is the exponentiation of the degree zero piece $\mathfrak{g}_0$. They proved a decomposition of the equivariant derived category of $\ell$-adic sheaves on the nilpotent cone of $\mathfrak{g}_i$ into blocks, each generated by a certain cuspidal local system via {\itshape spiral inductions}. We prove a conjecture of them, which predicts the bijectivity of a map from 1) the set of simple perverse sheaves in a fixed block to 2) the set of simple modules of a block of a (trigonometric) degenerate double affine Hecke algebra (dDAHA). This is a dDAHA analogue of the Deligne--Langlands correspondence for affine Hecke algebras proven by Kazhdan--Lusztig. Our results generalise a previous work of E. Vasserot, where the perverse sheaves in the principal block were considered.