To an effective local Langlands Corrspondence

الأصل · EN

Let F be a non-Archimedean local field. Let WF be the Weil group of F and PF the wild inertia subgroup of WF. Let WF be the set of equivalence classes of irreducible smooth representations of WF. Let A⁰ₙ(F) denote the set of equivalence classes of irreducible cuspidal representations of GLₙ(F) and set GLF = ₙ≥₁ A⁰ₙ(F). If σ∈ WF, let Lσ∈ GLF be the cuspidal representation matched with σ by the Langlands Correspondence. If σ is totally wildly ramified, in that its restriction to PF is irreducible, we treat Lσ as known. From that starting point, we construct an explicit bijection N: WF → GLF, sending σ to Nσ. We 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 π (of WF or GLₙ(F)) by tame characters of a tamely ramified field extension of F, canonically associated to π. We show this operation is preserved by the Langlands correspondence.