Matrix coefficients and Iwahori-Hecke algebra modules

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 expression for the image of spherical vectors under these functionals in terms of the representation theory of the dual group.