Gamma factors and root numbers of pairs for the Galois and the linear model

TL;DR

This paper extends a local converse theorem for p-adic GLn to square integrable representations using harmonic analysis, and proves related results for linear models, gamma factors, and root numbers.

Contribution

It advances the understanding of gamma factors and local converse theorems for a broader class of representations, including non-cuspidal ones, using harmonic analysis techniques.

Findings

Extended local converse theorem to square integrable representations

Proved a variant of the theorem for linear models

Verified triviality of gamma and epsilon factors at s=1/2

Abstract

Using harmonic analysis on Harish-Chandra Schwartz spaces of various spherical spaces, we extend a relative local converse theorem of Youngbin Ok for the Galois model of p-adic GLn, from the class of cuspidal representations to that of square integrable representations, which is its optimal form. We also prove a variant of this result for linear models by the same method. The above statements are luckily non empty as we verify triviality results for gamma and epsilon factors of pairs of distinguished representations at the central value s=1/2. Along the way, we offer a new proof of conjectures of D. Prasad and D. Ramakrishnan on local components of symplectic cuspidal automorphic representations, and root numbers of pairs of symplectic representations.