Iwahori Matsumoto presentation for modules of Iwahori fixed functions on symmetric spaces

We study the space $S(X{)}^I$ of smooth functions on a symmetric space $X=G\mathrm{/}H$ 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.