To an effective local Langlands correspondence
著者
書誌事項
To an effective local Langlands correspondence
（Memoirs of the American Mathematical Society, no. 1087）
American Mathematical Society, 2014, c2013
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注記
Includes bibliographical references (p. 87-88)
"September 2014, volume 231, number 1087 (fourth of 5 numbers)."
内容説明・目次
内容説明
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.
目次
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
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