On the local Langlands correspondences of DeBacker-Reeder and Reeder for \(\text{GL}(\ell ,F)\), where \(\ell \) is prime (Q429097)

Harris-Taylor and Henniart have independently proven the existence of the local Langlands correspondence for all \(p\)-adic general linear groups \(\mathrm{GL}(n,F)\). That is, they construct a correspondence between the set of irreducible, admissible representations of these groups, and the set of Langlands parameters. Meanwhile, DeBacker and Reeder have constructed a concrete and explicit correspondence between certain depth-zero, supercuspidal representations of more-or-less general reductive \(p\)-adic groups and certain Langlands parameters. For the representations that both correspondences consider, are the correspondences the same? When \(n\) is prime, the paper under review shows that they are.NEWLINENEWLINEMoreover, the paper shows the same for Reeder's conjectural Langlands correspondence involving certain positive-depth supercuspidal representations, provided that Howe's and the reviewer's constructions of these representations parametrize them in the same way.