Iwahori Spherical Whittaker Functions for Steinberg Representations

TL;DR

This paper explicitly determines Iwahori spherical Whittaker functions for Steinberg representations of split reductive p-adic groups, extending previous results from GL_n to more general groups.

Contribution

It generalizes the explicit computation of Iwahori spherical Whittaker functions from GL_n to all split reductive groups over p-adic fields.

Findings

Explicit formulas for Iwahori spherical Whittaker functions for Steinberg representations.

Determination of the Iwahori fixed vector via Hecke algebra characters.

Extension of previous GL_n results to general split reductive groups.

Abstract

Let $G(F)$ be a split reductive group over a $p$-adic field $F$ and let $(\pi_{St},V)$ be a (generalized) Steinberg representation of $G(F)$. It is known that the space of Iwahori fixed vectors in $V$ is one dimensional. The Iwahori Hecke algebra acts on this space via a character. We determine this fixed vector and use the Hecke algebra action on it to determine in full the Whittaker function associated with this Iwahori fixed vector. This generalizes our previous result for $GL_n(F)$.