  Let $G=GL_{n}(F)$ and let $(\pi_{St},V)$ be a (generalized) Steinberg
representation of $G$. It is well 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 the value of this character on a particular Hecke
algebra element and use this action to determine in full the Whittaker function
associated with an Iwahori fixed vector generalizing a result of Baruch and
Purkait for $GL_2(F)$. We show that the Iwahori fixed vector is "new" in the
sense that it is not fixed by any larger parahoric. We also show that the
restriction of the (generalized) Steinberg representation to $SL_n(F)$ remains
irreducible hence we get the Whittaker function attached to a Steinberg
representation of $SL_n(F)$.
