eBook

To an effective local Langlands Correspondence

Bushnell, Colin J., Henniart, Guy

2014

Overview

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.

• Publisher: American Mathematical Society
• Subject: Automorphic forms / Local fields (Algebra) / Representations of groups
• ISBN: 082189417X, 9780821894170