On the Local Langlands Correspondences of DeBacker/Reeder and Reeder for $GL(\ell,F)$, where $\ell$ is prime

TL;DR

This paper verifies that the conjectural local Langlands correspondences of DeBacker/Reeder and Reeder match the established Moy correspondences for $GL( ext{prime},F)$, confirming their consistency in depth zero and positive depth cases.

Contribution

It proves the equivalence of the conjectural and established local Langlands correspondences for $GL( ext{prime},F)$, clarifying their agreement in depth zero and positive depth scenarios.

Findings

Depth zero correspondence matches Moy's description.

Positive depth correspondence aligns under certain compatibility assumptions.

Constructs detailed for $GL( ext{prime},F)$, confirming conjectural frameworks.

Abstract

We prove that the conjectural depth zero local Langlands correspondence of DeBacker/Reeder agrees with the depth zero local Langlands correspondence as described by Moy, for the group $GL(\ell,F)$, where $\ell$ is prime and F is a local non-archimedean field of characteristic 0. We also prove that if one assumes a certain compatibility condition between Adler's and Howe's construction of supercuspidal representations, then the conjectural positive depth local Langlands correspondence of Reeder also agrees with the positive depth local Langlands correspondence as described by Moy, for $GL(\ell,F)$. Specifically, we first work out in detail the construction of DeBacker/Reeder for $GL(\ell,F)$, we then restate the constructions in the language of Moy, and finally prove that the correspondences agree. Up to a compatibility between Adler's and Howe's constructions, we then do the same for Reeder's positive depth construction.