A Note on the Spectral Transfer Morphisms for Affine Hecke Algebras

TL;DR

This paper clarifies and completes the proof of the uniqueness of spectral transfer morphisms for affine Hecke algebras, which are key tools in understanding unipotent representations and their classification.

Contribution

It provides detailed proof of the uniqueness property of spectral transfer morphisms and confirms three finite morphisms as spectral transfer morphisms.

Findings

Proved the uniqueness of spectral transfer morphisms for affine Hecke algebras.

Confirmed three finite morphisms of algebraic tori as spectral transfer morphisms.

Completed the theoretical framework connecting harmonic analysis and unipotent representations.

Abstract

E. Opdam introduced the tool of spectral transfer morphism (STM) of affine Hecke algebras to study the formal degrees of unipotent discrete series representations. He established a uniqueness property of STM for the affine Hecke algebras associated of unipotent discrete series representations. Based on this result, Opdam gave an explanation for Lusztig's arithmetic/geometric correspondence (in Lusztig's classification of unipotent representations of $p$-adic adjoint simple groups) in terms of harmonic analysis, and partitioned the unipotent discrete series representations into $L$-packets based on the Lusztig-Langlands parameters. The present paper provides some omitted details for the argument of the uniqueness property of STM. In the last section, we prove that three finite morphisms of algebraic tori are spectral transfer morphisms, and hence complete the proof of the uniqueness property.