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To an Effective Local Langlands Correspondence
 
Colin J. Bushnell King’s College London, London, United Kingdom
Guy Henniart Université Paris-Sud, Orsay, France
Front Cover for To an Effective Local Langlands Correspondence
Available Formats:
Electronic ISBN: 978-1-4704-1723-9
Product Code: MEMO/231/1087.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
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To an Effective Local Langlands Correspondence
Colin J. Bushnell King’s College London, London, United Kingdom
Guy Henniart Université Paris-Sud, Orsay, France
Available Formats:
Electronic ISBN:  978-1-4704-1723-9
Product Code:  MEMO/231/1087.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2312014; 88 pp
    MSC: Primary 22;

    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
     
     
    • Chapters
    • Introduction
    • 1. Representations of Weil groups
    • 2. Simple characters and tame parameters
    • 3. Action of tame characters
    • 4. Cuspidal representations
    • 5. Algebraic induction maps
    • 6. Some properties of the Langlands correspondence
    • 7. A naïve correspondence and the Langlands correspondence
    • 8. Totally ramified representations
    • 9. Unramified automorphic induction
    • 10. Discrepancy at a prime element
    • 11. Symplectic signs
    • 12. Main Theorem and examples
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Volume: 2312014; 88 pp
MSC: Primary 22;

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.

  • Chapters
  • Introduction
  • 1. Representations of Weil groups
  • 2. Simple characters and tame parameters
  • 3. Action of tame characters
  • 4. Cuspidal representations
  • 5. Algebraic induction maps
  • 6. Some properties of the Langlands correspondence
  • 7. A naïve correspondence and the Langlands correspondence
  • 8. Totally ramified representations
  • 9. Unramified automorphic induction
  • 10. Discrepancy at a prime element
  • 11. Symplectic signs
  • 12. Main Theorem and examples
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