Distinguished regular supercuspidal representations

TL;DR

This paper establishes a criterion for when regular supercuspidal representations of tamely ramified p-adic groups are distinguished, linking the distinction problem with Langlands functoriality and confirming a conjecture for specific cases.

Contribution

It provides a necessary and sufficient condition for distinguished regular supercuspidal representations, expanding understanding of their structure and relation to Langlands functoriality.

Findings

Derived a criterion for distinction of regular supercuspidal representations

Confirmed Lapid's conjecture for certain supercuspidal representations

Connected distinction problem with Langlands functoriality

Abstract

Based on recent work of Kaletha, we apply Hakim--Murnaghan's result to study distinguished regular supercuspidal representations of tamely ramified reductive $p$-adic groups. Assuming $p$ is sufficiently large, we obtain a necessary and sufficient condition for regular supercuspidal representations to be distinguished. We also investigate the relation between the distinction problem and the Langlands functoriality, and confirm a conjecture of Lapid for regular depth-zero or epipelagic supercuspidal representations.