Whittaker functions for Steinberg representations of $GL(n)$ over a $p$-adic field

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 $SL_n(F)$.