To an effective local Langlands correspondence

Let $F$ be a non-Archimedean local field. Let $mathcalWF$ be the Weil group of $F$ and $mathcalPF$ the wild inertia subgroup of $mathcalWF$. Let $widehat mathcalWF$ be the set of equivalence classes of irreducible smooth representations of $mathcalWF$. Let $mathcalA0n(F)$ denote the set of equivalence classes of irreducible cuspidal representations of $mathrmGLn(F)$ and set $widehat mathrmGLF = bigcup nge 1 mathcalA0n(F)$. If $sigma in widehat mathcalWF$, let $Lsigma in widehat mathrmGLF$ be the cuspidal representation matched with $sigma$ by the Langlands Correspondence. If $sigma$ is totally wildly ramified, in that its restriction to $mathcalPF$ is irreducible, the authors treat $Lsigma$ as known. From that starting point, the authors construct an explicit bijection $ mathcalWF to widehat mathrmGLF$, sending $sigma$ to $Nsigma$. 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 $mathcalWF$ or $mathrmGLn(F)$) 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.