Unitary equivalences for reductive p-adic groups

TL;DR

This paper demonstrates a transfer of unitarity from reductive p-adic groups to affine Hecke algebras within the framework of types, revealing relations between the unitary duals of different groups, akin to endoscopy.

Contribution

It establishes a transfer of unitarity for Bernstein components from p-adic groups to affine Hecke algebras with geometric parameters, expanding the understanding of unitary duals.

Findings

Transfer of unitarity established for certain Bernstein components.

Relations between unitary duals of different groups are derived.

Applicable to a large class of affine Hecke algebras with geometric parameters.

Abstract

We establish a transfer of unitarity for a Bernstein component of the category of smooth representations of a reductive p-adic group to the associated Hecke algebra, in the framework of the theory of types, whenever the Hecke algebra is an affine Hecke algebra with geometric parameters, in the sense of Lusztig (possibly extended by a group of automorphisms of the root datum). It is known that there is a large class of such examples (detailed in the paper). As a consequence, we establish relations between the unitary duals of different groups, in the spirit of endoscopy.