A new realization of the Langlands correspondence for PGL(2,F)

TL;DR

This paper introduces a novel approach to the local Langlands correspondence for PGL(2,F) over p-adic fields, using characters of covers of elliptic tori to parameterize supercuspidal representations without character twists.

Contribution

It proposes a new parameterization of supercuspidal representations via characters of covers of tori, inspired by real group theory, simplifying the local Langlands correspondence.

Findings

Defined a natural analogue of Harish-Chandra's character for PGL(2,F)

Established that this character corresponds to a unique supercuspidal representation

Eliminated the need for character twists in the correspondence

Abstract

In this paper, we give a new realization of the local Langlands correspondence for PGL(2,F), where F is a p-adic field of odd residual characteristic. In this case, supercuspidal representations of PGL(2,F) are parameterized by characters of elliptic tori. Taking a cue from real groups, we propose that supercuspidal representations are naturally parameterized by characters of covers of tori. Over the reals, Harish-Chandra defined the discrete series representations by specifying their characters restricted to an elliptic torus, and these characters may naturally be expressed in terms of characters of a cover of the torus. We write down a natural analogue of Harish-Chandra's character for PGL(2,F), and show that it is the character of a unique supercuspidal representation, on a canonical subset of the elliptic torus. This paves the way for a realization of the local Langlands correspondence for PGL(2,F) that eliminates the need for any character twists.