  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.
