To an Effective Local Langlands Correspondence

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 ``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.