‎In The endoscopic classification of representations‎‎, ‎J‎. ‎Arthur has proved the Langlands' classification for discrete series of p-adic classical groups‎. ‎This uses endoscopy and twisted endoscopy‎. ‎In this very short note‎, ‎we remark that the normalization ‎$‎rm‎grave{a}‎$‎ la Langlands-Shahidi of the intertwining operators‎, ‎allows to avoid endoscopy‎. ‎This is based on the intertwini...

In this essay I hope to explain the classification of characters of algebraic tori defined over R, particularly Langlands’ parametrization in terms of homomorphisms from the Weil group into the associated L-group. I’ll follow more or less the treatment in §§2 and 3 of [Langlands:1974/1989], avoiding most subsequent ‘improvements’. The only slightly new thing is how I relate Langlands’ formula f...

Representations of affine and graded Hecke algebras associated to Weyl groups play an important role in the Langlands correspondence for the admissible representations of a reductive p-adic group. We work in the general setting of a graded Hecke algebra associated to any real reflection group with arbitrary parameters. In this setting we provide a classification of all irreducible representatio...

Our goal is to formulate a theorem that is part of a recent classification of automorphic representations of orthogonal and symplectic groups. To place it in perspective, we devote much of the paper to a historical introduction to the Langlands program. In our attempt to make the article accessible to a general mathematical audience, we have centred it around the theory of L-functions, and its ...

Let F be a non-Archimedean local field of characteristic 0 and G be an inner form of the general linear group G = GLn over F . We show that the rectifying character appearing in the essentially tame Jacquet-Langlands correspondence of Bushnell and Henniart for G and G can be factorized into a product of some special characters, called zeta-data in this paper, in the theory of endoscopy of Langl...

We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an “interpolating quantum group” depending on two parameters which interpolates bet...

This is an introduction to the work of Beilinson and Drinfeld [6] on the Langlands program. Mathematics Subject Classification (2000). Primary 11R39; Secondary 14D20.