Electronic ISBN:  9781470417239 
Product Code:  MEMO/231/1087.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 231; 2014; 88 ppMSC: Primary 22;
Let \(F\) be a nonArchimedean 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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Let \(F\) be a nonArchimedean 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