Spectral correspondences for affine Hecke algebras

TL;DR

This paper introduces spectral transfer morphisms between affine Hecke algebras, showing they preserve spectral measures and impact the understanding of Langlands parameters for unipotent representations.

Contribution

It defines spectral transfer morphisms and a partial order on affine Hecke algebras, advancing the study of their spectral and representation-theoretic properties.

Findings

Spectral transfer morphisms induce measure-preserving correspondences.

A partial order on affine Hecke algebras is established.

Implications for Langlands parameters of unipotent representations.

Abstract

We introduce the notion of spectral transfer morphisms between normalized affine Hecke algebras, and show that such morphisms induce spectral measure preserving correspondences on the level of the tempered spectra of the affine Hecke algebras involved. We define a partial ordering on the set of isomorphism classes of normalized affine Hecke algebras, which plays an important role for the Langlands parameters of Lusztig's unipotent representations.