Matrix Coefficients and Iwahori-Hecke Algebra Modules

TL;DR

This paper links unique models of p-adic group representations with Hecke algebra characters, enabling explicit evaluation of spherical vectors and their images in unramified principal series representations.

Contribution

It introduces a uniform method to evaluate spherical vectors using Hecke algebra characters and provides explicit formulas in terms of dual group representation theory.

Findings

Explicit evaluation formulas for spherical vectors

Connection between models and Hecke algebra characters

Representation-theoretic expression for vector images

Abstract

We establish a connection between certain unique models, or equivalently unique functionals, for representations of p-adic groups and linear characters of their corresponding Hecke algebras. This allows us to give a uniform evaluation of the image of spherical and Iwahori-fixed vectors in the unramified principal series for this class of models. We provide an explicit alternator expressionfor the image of the spherical vectors under these functionals in terms of the representation theory of the dual group.