Distinction and quadratic base change for regular supercuspidal representations

TL;DR

This paper investigates Prasad's conjecture for regular supercuspidal representations, providing new interpretations, reducing the proof to tori, and confirming the conjecture in specific cases.

Contribution

It offers novel insights into the numerical quantities in Prasad's formula, simplifies the proof to tori, and proves the conjecture for unramified extensions and quasi-split groups.

Findings

Reduced proof of Prasad's conjecture to tori cases.

Identified conditions for the coincidence of quadratic characters.

Proved Prasad's conjecture for unramified extensions of quasi-split groups.

Abstract

In this article, we study Prasad's conjecture for regular supercuspidal representations based on the machinery developed by Hakim and Murnaghan to study distinguished representations, and the fundamental work of Kaletha on parameterization of regular supercuspidal representations. For regular supercuspidal representations, we give some new interpretations of the numerical quantities appearing in Prasad's formula, and reduce the proof to the case of tori. The proof of Prasad's conjecture then reduces to a comparison of various quadratic characters appearing naturally in the above process. We also have some new observations on these characters and study the relation between them in detail. For some particular examples, we show the coincidence of these characters, which gives a new purely local proof of Prasad's conjecture for regular supercuspidal representations of these groups. We also prove Prasad's conjecture for regular supercuspidal representations of G(E), when E/F is unramified and G is a general quasi-split reductive group.