Hecke algebras and local Langlands correspondence for non-singular depth-zero representations

TL;DR

This paper establishes a Local Langlands Correspondence for non-singular depth-zero representations of reductive groups over non-archimedean fields, linking them to modules over affine Hecke algebras and demonstrating compatibility with induction.

Contribution

It introduces a categorical framework connecting non-singular depth-zero representations with affine Hecke algebra modules, extending the LLC to this class.

Findings

Establishes LLC for non-singular depth-zero representations.

Constructs an equivalence of categories with affine Hecke algebra modules.

Shows compatibility of LLC with parabolic induction.

Abstract

Let G be a connected reductive group over a non-archimedean local field. We say that an irreducible depth-zero (complex) G-representation is non-singular if its cuspidal support is non-singular. We establish a Local Langlands Correspondence for all such representations. We obtain it as a specialization from a categorical version: an equivalence between the category of finite-length non-singular depth-zero G-representations and the category of finite-length right modules of a direct sum of twisted affine Hecke algebras constructed from Langlands parameters. We also show that our LLC and our equivalence of categories have several nice properties, for example compatibility with parabolic induction.