The cocenter-representation duality

TL;DR

This paper explores an algebraic approach using cocenter-representation duality to study finite dimensional representations of affine Hecke algebras, broadening the scope beyond traditional geometric or analytic methods.

Contribution

It introduces a new algebraic framework via cocenter-representation duality for analyzing affine Hecke algebra representations, applicable to arbitrary isogeny classes and generic complex parameters.

Findings

Provides a new algebraic method for representation analysis.

Works with complex parameters beyond equal or positive parameters.

Potentially applicable to modular cases and positive characteristic fields.

Abstract

Affine Hecke algebras arise naturally in the study of smooth representations of reductive $p$-adic groups. Finite dimensional complex representations of affine Hecke algebras (under some restriction on the isogeny class and the parameter function) has been studied by many mathematicians, including Kazhdan-Lusztig \cite{KL}, Ginzburg \cite{CG}, Lusztig \cite{L1}, Reeder \cite{Re}, Opdam-Solleveld \cite{OS}, Kato \cite{Kat}, etc. The approaches are either geometric or analytic. In this note, we'll discuss a different route, via the so-called "cocenter-representation duality", to study finite dimensional representations of affine Hecke algebras (for arbitrary isogeny class and for a generic complex parameter). This route is more algebraic, and allows us to work with complex parameters, instead of equal parameters or positive parameters. We also expect that it can be eventually applied to the "modular case" (for representations over fields of positive characteristic and for parameter equal to a root of unity).