  In this paper, we consider how to express an Iwahori--Whittaker function
through Demazure characters. Under some interesting combinatorial conditions,
we obtain an explicit formula and thereby a generalization of the
Casselman--Shalika formula. Under the same conditions, we compute the
transition matrix between two natural bases for the space of Iwahori fixed
vectors of an induced representation of a p-adic group; this generalizes a
result of Bump--Nakasuji.
