On two questions concerning representations distinguished by the Galois involution

TL;DR

This paper explores two methods for characterizing distinguished representations of GL(n,E) by GL(n,F), relating base change lifts and symmetry conditions, and examines their limitations through examples.

Contribution

It demonstrates the close relationship between base change lifts and symmetry conditions in distinguished representations, and highlights their limitations with new examples.

Findings

The union of base change images relates to symmetry conditions.

Base change and symmetry approaches are closely connected.

Examples show limitations of symmetry conditions in detecting distinction.

Abstract

Let E/F be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the smooth irreducible representations of GL(n,E) that are distinguished by its subgroup GL(n,F). One relates this class to representations which come as base change lifts from a quasi-split unitary group F, while another deals with a certain symmetry condition. By characterizing the union of images of the base change maps we show that these two approaches are closely related. Using this observation, we are able to prove a statement relating base change and distinction for ladder representations. We then produce a wide family of examples in which the symmetry condition does not impose GL(n,F)-distinction, and thus exhibit the limitations of these two approaches.