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To an effective local Langlands Correspondence


About this Title

Colin J. Bushnell and Guy Henniart

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 231, Number 1087
ISBNs: 978-0-8218-9417-0 (print); 978-1-4704-1723-9 (online)
DOI: http://dx.doi.org/10.1090/memo/1087
Published electronically: February 10, 2014
Keywords:Explicit local Langlands correspondence, automorphic induction

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Table of Contents


Chapters

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

Abstract


Let be a non-Archimedean local field. Let be the Weil group of and the wild inertia subgroup of . Let be the set of equivalence classes of irreducible smooth representations of . Let denote the set of equivalence classes of irreducible cuspidal representations of and set . If , let be the cuspidal representation matched with by the Langlands Correspondence. If is totally wildly ramified, in that its restriction to is irreducible, we treat as known. From that starting point, we construct an explicit bijection , sending to . We compare this “na\i ve 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 (of or ) by tame characters of a tamely ramified field extension of , canonically associated to . We show this operation is preserved by the Langlands correspondence.

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