Generic representations for symmetric spaces

Abstract

For a connected quasi-split reductive algebraic group $G$ over a field $k$, which is either a finite field or a non-archimedean local field, $\theta$ an involutive automorphism of $G$ over $k$, let $K =G^\theta$. Let $K^1=[K^0,K^0]$, the commutator subgroup of $K^0$, the connected component of identity of $K$. In this paper, we provide a simple condition on $(G,\theta)$ for there to be an irreducible admissible generic representations $\pi$ of $G$ with ${\rm Hom}_{K^1}[\pi,{\mathbb C}] \not = 0$. The condition is most easily stated in terms of a real reductive group $G_\theta({\mathbb R})$ associated to the pair $(G,\theta)$ being quasi-split.