Gamma factors root numbers and distinction

TL;DR

This paper investigates the relationship between distinguished representations of GL over quadratic extensions and their local and global root numbers, establishing triviality results for Rankin-Selberg and gamma factors.

Contribution

It demonstrates that the local Rankin-Selberg root number is trivial for distinguished representations and explores the triviality of gamma factors at 1/2, providing new insights into their properties.

Findings

Rankin-Selberg root number is trivial for distinguished pairs

Global root number is trivial for distinguished cuspidal representations

Gamma factor at 1/2 is often trivial for distinguished representations

Abstract

We study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of $p$-adic fields. We show that the local Rankin-Selberg root number of any pair of distinguished representation is trivial and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at $1/2$ is trivial for distinguished representations as well as the converse problem.