Generic representations for symmetric spaces

TL;DR

This paper establishes a criterion for the existence of certain irreducible admissible generic representations of a reductive group over fields like finite or non-archimedean local fields, based on the quasi-split property of an associated real group.

Contribution

It provides a simple condition linking the existence of generic representations to the quasi-split nature of an associated real reductive group.

Findings

Existence of generic representations characterized by quasi-split condition

Condition applies to groups over finite and non-archimedean local fields

Connects algebraic group automorphisms to representation theory

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.