ON THE DISTINCTION OF IWAHORI-SPHERICAL DISCRETE SERIES REPRESENTATIONS

Let $E/F$ 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 ${\mathbb H} (F)$ -distinguished Iwahori-spherical representations of ${\mathbb H} (E)$ . For discrete series Iwahori-spherical representations of ${\mathbb H} (E)$ , we prove a numerical criterion of ${\mathbb H} (F)$ -distinction. As an application, we classify the ${\mathbb H} (F)$ -distinguished discrete series representations of ${\mathbb H} (E)$ corresponding to degree $1$ characters of the Iwahori-Hecke algebra.