Affine Hecke algebras for Langlands parameters

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.

V2: some parts concerning essentially square-integrable representations were corrected. V3: to repair problems with Lemma 3.10 from the previous versions, a new paragraph 3.2 was created. Version 4: new Definition 3.11. V5 and V6: additional results in section 1, to fix problems from [AMS2]. Minor corrections in each version

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Cited by in corpus (7)

• On formal degrees of unipotent representations
• Affine Hecke algebras and the conjectures of Hiraga, Ichino and Ikeda
• Affine Hecke algebras for classical p-adic groups
• Parameters of Hecke algebras for Bernstein components of p-adic groups
• A nonabelian Fourier transform for tempered unipotent representations
• Correspondences between affine Hecke algebras and applications
• Jacquet modules and local Langlands correspondence