We use the p-adic local Langlands correspondence for GL2(Qp) to explicitly compute the reduction modulo p of certain 2-dimensional crystalline representations of small slope, and give applications to modular forms.

over a field F of characteristic 0. The methods of [A3] are by comparison with representations of the group GL(N). This eventually leads to results on the representations of the full orthogonal group O(N). But it is the connected subgroup G = SO(N) of O(N) that is the ultimate object of interest. In [A3, §8.4], we were able to characterize certain representations of G from the results obtained ...

The classical Gindikin-Karpelevich formula appears in Langlands’ calculation of the constant terms of Eisenstein series on reductive groups and in Macdonald’s work on p-adic groups and affine Hecke algebras. The formula has been generalized in the work of Garland to the affine Kac-Moody case, and the affine case has been geometrically constructed in a recent paper of Braverman, Finkelberg, and ...

Twenty-five years ago R. Langlands proposed [L] a “fantastic generalization” of Artin-Hasse reciprocity law in the classical class field theory. He conjectured the existence of a correspondence between automorphic irreducible infinite-dimensional representations of a reductive groupG over a global number field on the one hand, and (roughly speaking) finite dimensionsional representations of the...

0.1. Background. Let X be a smooth, complete, geometrically connected curve over the finite field Fq. Denote by F the field of rational functions on X and by A the ring of adèles of F . The Langlands conjecture, recently proved by L. Lafforgue [Laf], establishes a correspondence between cuspidal automorphic forms on the group GLn(A) and irreducible, almost everywhere unramified, n–dimensional `...

A combinatorial description of the crystal B(∞) for finitedimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig’s parametrization, and obtain a combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum over the set of Young ta...

The purpose of this article is to state a strong form of the conjecture of Langlands and Rapoport (1987, §5), and to outline a proof of it. For Shimura varieties of PEL type, the outline is complete, and for Shimura varieties of Hodge type it becomes complete if one accepts certain statements, stated as conjectures in the original version of this article, whose proof has been announced by Vasiu...

Let K be a local non-Archimedean field of positive characteristic and let L be the degree-n unramified extension of K. Let θ be a smooth character of L× such that for each nontrivial γ ∈ Gal(L/K), θ and θ/θ have the same level. Via the local Langlands and Jacquet-Langlands correspondences, θ corresponds to an irreducible representation ρθ of D×, where D is the central division algebra over K wi...

We explain (following V. Drinfeld) how the G(C[[t]]) equivariant derived category of the affine Grassmannian can be described in terms of coherent sheaves on the Langlands dual Lie algebra equivariant with respect to the adjoint action, due to some old results of V. Ginzburg. The global cohomology functor corresponds under this identification to restriction to the Kostant slice. We extend this ...

We provide a motivated introduction to the theory of categorical actions groups and local geometric Langlands program. Along way, we emphasize applications, old new, usual representation reductive affine Lie algebras.