Formality in the Deligne-Langlands correspondence

TL;DR

This paper establishes a categorical equivalence linking irreducible representations of affine Hecke algebras with perverse sheaves, using formality results for Springer sheaves to deepen the understanding of the Deligne-Langlands correspondence.

Contribution

It constructs an equivalence of triangulated categories for affine Hecke algebra representations and perverse sheaves, extending the classical correspondence through a formality theorem.

Findings

Constructed an equivalence of triangulated categories for each central character.

Proved a formality result for a broad class of Springer sheaves.

Enhanced the categorical understanding of the Deligne-Langlands correspondence.

Abstract

The Deligne-Langlands correspondence parametrizes irreducible representations of the affine Hecke algebra $\mathcal{H}^{\text{aff}}$ by certain perverse sheaves. We show that this can be lifted to an equivalence of triangulated categories. More precisely, we construct for each central character $\chi$ of $\mathcal{H}^{\text{aff}}$ an equivalence of triangulated categories between a perfect derived category of dg-modules $D_{\text{perf}}(\mathcal{H}^{\text{aff}}/(\text{ker}(\chi)) - \text{dgMod})$ and the triangulated category generated by the corresponding perverse sheaves. The main step in this construction is a formality result that we prove for a wide range of `Springer sheaves'.