Around the Langlands program
This survey article gives a wide-ranging introduction to the Langlands program, and will be of use to anyone seeking an overview of this topic. After an introduction, the article presents basic information about number and function fields, even defining the \(p\)-adic numbers. Next representations of locally profinite groups are treated, and then \(p\)-adic reductive groups are discussed. The fourth section concerns aspects of harmonic analysis over non-Archimedean local fields \(F\), defining the \(L\)- and epsilon-factors for a character of \(F^\times\), defining the Weyl group \(W_F\), and extending the notions of \(L\)- and epsilon-factors to representations of \(W_F\). In the following section the \(L\)-group, a stable orbital integral, and the local \(L\)-packet are defined. Then the local Langlands correspondence (LLC) is described, enhanced Langlands parameters are defined, and the cuspidality conjecture relating cuspidal enhanced Langlands parameters to irreducible supercuspidal representations is formulated in the split case. The strategy to reduce the LLC for irreducible smooth representations of \(G\) to the LLC for supercuspidal representations of Levi subgroups of \(G\) is also explained. After these local considerations, the sixth section introduces the adèle ring \(\mathbb{A}_K\) of a global field \(K\) and the group \(G(\mathbb{A}_K)\) where \(G\) is a connected reductive group defined over \(K\), and presents the strong approximation theorem for \(G(\mathbb{A}_\mathbb{Q})\). Then modular and automorphic forms are defined and the Hilbert space decomposition of the space \(L^2_{\text{cusp}}(G(K)\backslash G({\mathbb{A}}_K),\omega)\) of cuspidal automorphic functions with central character \(\omega\) is described. The eighth section briefly introduces Eisenstein series. Then the ninth presents the principle of functoriality and briefly describes trace formulae and the fundamental lemma. This is followed by short sections on the hypothetical automorphic Langlands group, the mod \(\ell\) local Langlands correspondence, the mod \(p\) and \(p\)-adic local Langlands correspondence, the geometric Langlands program, and the geometrization of the local Langlands correspondence.
scientific article; zbMATH DE number 6830365
scientific article