A distinction criterion for Iwahori-spherical representations

Let $G/H$ 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 Γ = Γ (G/H) of the group of invertible elements of the Iwahori-Hecke algebra H of G such that an Iwahori-spherical representation of G is H-distinguished if and only if the corresponding Iwahori-Hecke module is "Γ-distinguished".