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To an Effective Local Langlands Correspondence
Colin J. Bushnell, King's College London, United Kingdom, and Guy Henniart, Université Paris-Sud, Orsay, France
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Memoirs of the American Mathematical Society
2014; 88 pp; softcover
Volume: 231
ISBN-10: 0-8218-9417-X
ISBN-13: 978-0-8218-9417-0
List Price: US$71 Individual Members: US$42.60
Institutional Members: US\$56.80
Order Code: MEMO/231/1087

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 "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 $$\pi$$ (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.

• Introduction
• Representations of Weil groups
• Simple characters and tame parameters
• Action of tame characters
• Cuspidal representations
• Algebraic induction maps
• Some properties of the Langlands correspondence
• A naïve correspondence and the Langlands correspondence
• Totally ramified representations
• Unramified automorphic induction
• Discrepancy at a prime element
• Symplectic signs
• Main Theorem and examples
• Bibliography