To an Effective Local Langlands Correspondence (Memoirs of the American Mathematical Society)

To an Effective Local Langlands Correspondence (Memoirs of the American Mathematical Society)

By: Guy Henniart (author), Colin J. Bushnell (author)Paperback

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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 ``naive 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.

About Author

Colin J. Bushnell, King's College London, UK Guy Henniart, Universite Paris-Sud, Orsay, France


Introduction Representations of Weil groups Simple characters and tame parameters Action of tame characters Cuspidal representations Algebraic induction maps Some properties of the Langlands correspondence A naive correspondence and the Langlands correspondence Totally ramified representations Unramified automorphic induction Discrepancy at a prime element Symplectic signs Main Theorem and examples Bibliography

Product Details

  • ISBN13: 9780821894170
  • Format: Paperback
  • Number Of Pages: 88
  • ID: 9780821894170
  • ISBN10: 082189417X

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