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

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