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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
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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.

Table of Contents

  • 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
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