Types and unitary representations of reductive p-adic groups

TL;DR

This paper establishes a bijection between irreducible unitary representations and modules over Hecke algebras for certain types in reductive p-adic groups, generalizing previous unitarity criteria.

Contribution

It extends the unitarity criterion to a broader class of Bushnell-Kutzko types satisfying a rigidity condition, linking Bernstein components to Hecke algebra modules.

Findings

Bijection between irreducible unitary representations and Hecke algebra modules.

Generalization of Barbasch-Moy unitarity criterion.

Applicability to types satisfying a specific rigidity assumption.

Abstract

We prove that for every Bushnell-Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors.