Memoirs of the American Mathematical Society 2014; 88 pp; softcover Volume: 231 ISBN10: 082189417X ISBN13: 9780821894170 List Price: US$71 Individual Members: US$42.60 Institutional Members: US$56.80 Order Code: MEMO/231/1087
 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  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 naïve correspondence and the Langlands correspondence
 Totally ramified representations
 Unramified automorphic induction
 Discrepancy at a prime element
 Symplectic signs
 Main Theorem and examples
 Bibliography
