On the distinction of Iwahori-spherical representations

Let $E\mathrm{/}F$ be a quadratic unramified extension of non-archimedean local fields and $\mathbb{H}$ a simply connected semisimple algebraic group defined and split over $F$. We establish general results (multiplicities, test vectors) on $\HH (F)$-distinguishedIwahori-sphericalrepresentationsof$\HH (E)$. For discrete series Iwahori-spherical representations of $\text{HH}(E)$, we prove a numerical criterion of $\text{HH}(F)$-distinction. As an application, we classify the $\HH (F)$-distinguisheddiscreteseriesrepresentationsof$\HH (E)$ corresponding to degree $1$ characters of the Iwahori-Hecke algebra.