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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 Click above image for expanded view 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.

• 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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