Unique functionals and representations of hecke algebras

Rogawski (1985) used the affine Hecke algebra to model the intertwining operators of unramified principal series representations of p-adic groups. On the other hand, a representation of this Hecke algebra in which the standard generators act by Demazure-Lusztig operators was introduced by Lusztig (1989) and applied by Kazhdan and Lusztig (1987) to prove the Deligne-Langlands conjecture. These operators appear in various other contexts. Ion (2006) used them to express matrix coefficients of principal series representations in terms of nonsymmetric Macdonald polynomials, while Brubaker, Bump and Licata (2011) found essentially the same operators underlying recursive relationships for Whittaker functions. Here we explainthe role of unique functionals and Hecke algebras in these contexts and revisit the results of Ion from the point of view of Brubaker et al.