Iwahori Whittaker functions for unramified quasi-split groups

Abstract

Let $\mathbf{G}$ be a connected unramified quasi-split reductive group defined over a nonarchimedean local field $F$. This paper concerns the Iwahori Whittaker functions attached to the unramified principal series of $\mathbf{G}(F)$. We introduce Demazure-Whittaker operators and show that in the non-split case they generate a Hecke algebra with unequal parameters that are determined by $\mathbf{G}$. We then connect these operators to the Whittaker functions, establishing formulas for calculating all values for a basis of Iwahori Whittaker functions and giving a new proof of the Casselman-Shalika formula for the spherical Whittaker function. In doing so we build on the work for split groups of Brubaker, Bump and Licata; the new feature here is the appearance of a Hecke algebra with unequal parameters.