  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 $GL_n(F)$.
