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 $widehat {mathrm{GL}}_{F} = bigcup _{nge 1} mathcal{A}^{0}_{n}(F)$. 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 ``naive 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.
