To an effective local Langlands Correspondence

Source abstract

Let F F be a non-Archimedean local field. Let W F \mathcal {W}_{F} be the Weil group of F F and P F \mathcal {P}_{F} the wild inertia subgroup of W F \mathcal {W}_{F} . Let W ^ F \widehat {\mathcal {W}}_{F} be the set of equivalence classes of irreducible smooth representations of W F \mathcal {W}_{F} . Let A n 0 ( F ) \mathcal {A}^{0}_{n}(F) denote the set of equivalence classes of irreducible cuspidal representations of G L n ( F ) \mathrm {GL}_{n}(F) and set G L ^ F = ⋃ n ≥ 1 A n 0 ( F ) \widehat {\mathrm {GL}}_{F} = \bigcup _{n\ge 1} \mathcal {A}^{0}_{n}(F) . If σ ∈ W ^ F \sigma \in \widehat {\mathcal {W}}_{F} , let L σ ∈ G L ^ F ^{L}{\sigma }\in \widehat {\mathrm {GL}}_{F} be the cuspidal representation matched with σ \sigma by the Langlands Correspondence. If σ \sigma is totally wildly ramified, in that its restriction to P F \mathcal {P}_{F} is irreducible, we treat L σ ^{L}{\sigma } as known. From that starting point, we construct an explicit bijection N : W ^ F → G L ^ F \mathbb {N}:\widehat {\mathcal {W}}_{F} \to \widehat {\mathrm {GL}}_{F} , sending σ \sigma to N σ ^{N}{\sigma } . 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 \pi (of W F \mathcal {W}_{F} or G L n ( F ) \mathrm {GL}_{n}(F) ) by tame characters of a tamely ramified field extension of F F , canonically associated to π \pi . We show this operation is preserved by the Langlands correspondence.