Iwahori Spherical Whittaker Functions for Steinberg Representations

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 $GL_n(F)$.