Local Langlands correspondence for regular supercuspidal representations of GL(n)

TL;DR

This paper proves that Kaletha's recent construction of the local Langlands correspondence for regular supercuspidal representations of GL(n) aligns with Harris--Taylor's established correspondence, using known tame cases and root-theoretic computations.

Contribution

It establishes the equivalence of two major constructions of the local Langlands correspondence for regular supercuspidal representations of GL(n).

Findings

Kaletha's construction coincides with Harris--Taylor's correspondence.

Reduction to root-theoretic computation simplifies the proof.

Utilizes Bushnell--Henniart's tame correspondence and Tam's rectifiers.

Abstract

In this paper, we prove the coincidence of Kaletha's recent construction of the local Langlands correspondence for regular supercuspidal representations with Harris--Taylor's one in the case of general linear groups. The keys are Bushnell--Henniart's essentially tame local Langlands correspondence and Tam's result on Bushnell--Henniart's rectifiers. By combining them, our problem is reduced to an elementary root-theoretic computation on the difference between Kaletha's and Tam's $\chi$-data.