  We study the space $S(X)^I$ of smooth functions on a symmetric space $X=G/H$
invariant to the action of an Iwahori subgroup $I$, as a module over
$\mathcal{H}(G,I)$, the Iwahori Hecke algebra of a p-adic group $G$. We present
a description of this module that generalizes the description given to
$\mathcal{H}(G,I)$ by Iwahori and Matsumoto.
