Affine Hecke algebras for Langlands parameters

TL;DR

This paper constructs affine Hecke algebras for enhanced Langlands parameters, establishing a bijection with Bernstein components and linking their representations to the local Langlands correspondence, thus extending Lusztig's work.

Contribution

It introduces a canonical association of affine Hecke algebras to Bernstein components of Langlands parameters, connecting representation theory and the local Langlands correspondence.

Findings

Bijection between irreducible representations of the algebra and Bernstein components.

Central characters correspond to cuspidal supports of Langlands parameters.

Morita equivalence with Hecke algebras for related Bernstein components.

Abstract

It is well-known that affine Hecke algebras are very useful to describe the smooth representations of any connected reductive p-adic group G, in terms of the supercuspidal representations of its Levi subgroups. The goal of this paper is to create a similar role for affine Hecke algebras on the Galois side of the local Langlands correspondence. To every Bernstein component of enhanced Langlands parameters for G we canonically associate an affine Hecke algebra (possibly extended with a finite R-group). We prove that the irreducible representations of this algebra are naturally in bijection with the members of the Bernstein component, and that the set of central characters of the algebra is naturally in bijection with the collection of cuspidal supports of these enhanced Langlands parameters. These bijections send tempered or (essentially) square-integrable representations to the expected kind of Langlands parameters. Furthermore we check that for many reductive p-adic groups, if a Bernstein component B for G corresponds to a Bernstein component B^\vee of enhanced Langlands parameters via the local Langlands correspondence, then the affine Hecke algebra that we associate to B^\vee is Morita equivalent with the Hecke algebra associated to B. This constitutes a generalization of Lusztig's work on unipotent representations. It might be useful to establish a local Langlands correspondence for more classes of irreducible smooth representations.