A distinction criterion for Iwahori-spherical representations

Abstract

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 $\Gamma =\Gamma (G/H)$ 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 "$\Gamma$-distinguished".

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