A distinction criterion for Iwahori-spherical representations

Let $G\mathrm{/}H$ be a Galois symmetric space for an unramified quadratic extension of a locally compact field $F$, where the group $H$ is semisimple, simply connected, defined and split over $F$. We prove that there exists a subgroup $\mathrm{\Gamma }=\mathrm{\Gamma }(G\mathrm{/}H)$ of the group of invertible elements of the Iwahori-Hecke algebra $\mathcal{H}$ of $G$ such that an Iwahori-spherical representation of $G$ is $H$-distinguished if and only if the corresponding Iwahori-Hecke module is “$\mathrm{\Gamma }$-distinguished”.