Hermitian duals and generic representations for affine Hecke algebras

TL;DR

This paper advances the representation theory of affine Hecke algebras by establishing key duality and classification results, and introduces a new notion of genericity with partial proof of related conjectures.

Contribution

It develops the abstract theory for affine Hecke algebras with positive parameters, extending results known for p-adic groups, and proposes a new definition of generic modules.

Findings

Established relations between parabolic induction, Hermitian duals, and Bernstein's second adjointness.

Proved that known category equivalences preserve Hermitian duality.

Proposed a definition of genericity and proved cases of the generalized injectivity conjecture.

Abstract

We further develop the abstract representation theory of affine Hecke algebras with arbitrary positive parameters. We establish analogues of several results that are known for reductive p-adic groups. These include: the relation between parabolic induction/restriction and Hermitian duals, Bernstein's second adjointness and generalizations of the Langlands classification. We check that, in the known cases of equivalences between module categories of affine Hecke algebras and Bernstein blocks for reductive p-adic groups, such equivalences preserve Hermitian duality. We also initiate the study of generic representation of affine Hecke algebras. Based on an analysis of the Hecke algebras associated to generic Bernstein blocks for quasi-split reductive p-adic groups, we propose a fitting definition of genericity for modules over affine Hecke algebras. With that notion we prove special cases of the generalized injectivity conjecture, about generic subquotients of standard modules for affine Hecke algebras.