On the distinction of Iwahori-spherical representations

TL;DR

This paper investigates Iwahori-spherical representations of a semisimple algebraic group over a quadratic unramified extension, establishing criteria for distinction and classifying certain distinguished discrete series representations.

Contribution

It provides general results on multiplicities and test vectors, and introduces a numerical criterion for distinction of discrete series Iwahori-spherical representations.

Findings

Established multiplicity results for distinguished representations.

Proved a numerical criterion for $ ext{H}(F)$-distinction.

Classified degree 1 character distinguished discrete series representations.

Abstract

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