# To an Effective Local Langlands Correspondence

4.11 - 1251 ratings - Source

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 qnaAmve correspondenceq 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 qinternal twistingq 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.Colin J. Bushnell, Guy Henniart. AMS Author Handbook and the Instruction Manual are available in PDF format from the author package link. The author package can also be obtained free of charge by sending email toanbsp;...

 Title : To an Effective Local Langlands Correspondence Author : Colin J. Bushnell, Guy Henniart Publisher : American Mathematical Soc. - 2014-08-12

You must register with us as either a Registered User before you can Download this Book. You'll be greeted by a simple sign-up page.