Let \(F\) be a non-Archimedean local field. Let \(\mathcal{W}_{F}\) be the Weil group of \(F\) and \(\mathcal{P}_{F}\) the wild inertia subgroup of \(\mathcal{W}_{F}\). Let \(\widehat {\mathcal{W}}_{F}\) be the set of equivalence classes of irreducible smooth representations of \(\mathcal{W}_{F}\). Let \(\mathcal{A}^{0}_{n}(F)\) denote the set of equivalence classes of irreducible cuspidal representations of \(\mathrm{GL}_{n}(F)\) and set \(\widehat {\mathrm{GL}}_{F} = \bigcup _{n\ge 1} \mathcal{A}^{0}_{n}(F)\). If \(\sigma \in \widehat {\mathcal{W}}_{F}\), let \(^{L}{\sigma }\in \widehat {\mathrm{GL}}_{F}\) be the cuspidal representation matched with \(\sigma \) by the Langlands Correspondence. If \(\sigma \) is totally wildly ramified, in that its restriction to \(\mathcal{P}_{F}\) is irreducible, the authors treat \(^{L}{\sigma}\) as known.

From that starting point, the authors construct an explicit bijection \(\mathbb{N}:\widehat {\mathcal{W}}_{F} \to \widehat {\mathrm{GL}}_{F}\), sending \(\sigma \) to \(^{N}{\sigma}\). The authors compare this “naïve correspondence” with the Langlands correspondence and so achieve an effective description of the latter, modulo the totally wildly ramified case. A key tool is a novel operation of “internal twisting” of a suitable representation \(\pi\) (of \(\mathcal{W}_{F}\) or \(\mathrm{GL}_{n}(F)\)) by tame characters of a tamely ramified field extension of \(F\), canonically associated to \(\pi \). The authors show this operation is preserved by the Langlands correspondence.