# To an Effective Local Langlands Correspondence

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*Colin J. Bushnell; Guy Henniart*

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

# Table of Contents

## To an Effective Local Langlands Correspondence

- Cover Cover11 free
- Title page i2 free
- Introduction 18 free
- Chapter 1. Representations of Weil groups 916 free
- Chapter 2. Simple characters and tame parameters 1522
- Chapter 3. Action of tame characters 2330
- Chapter 4. Cuspidal representations 3340
- Chapter 5. Algebraic induction maps 3542
- Chapter 6. Some properties of the Langlands correspondence 4552
- Chapter 7. A naïve correspondence and the Langlands correspondence 4956
- Chapter 8. Totally ramified representations 5360
- Chapter 9. Unramified automorphic induction 6168
- Chapter 10. Discrepancy at a prime element 6976
- Chapter 11. Symplectic signs 7986
- Chapter 12. Main Theorem and examples 8592
- Bibliography 8794
- Back Cover Back Cover1100