# memo_has_moved_text();To an effective local Langlands Correspondence

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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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 $F$ 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, we treat $^{L}{\sigma }$ as known. From that starting point, we construct an explicit bijection $\mathbb {N}:\widehat {\mathcal {W}}_{F} \to \widehat {\mathrm {GL}}_{F}$, sending $\sigma$ to $^{N}{\sigma }$. 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 $\pi$ (of $\mathcal {W}_{F}$ or $\mathrm {GL}_{n}(F)$) by tame characters of a tamely ramified field extension of $F$, canonically associated to $\pi$. We show this operation is preserved by the Langlands correspondence.