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

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.