On the local Langlands correspondence

2003/04/22 by Michael Harris
Mathematics · #math.NT #math.RT #msc:11 #msc:14 #msc:22

paper · pdf

published as Proceedings of the ICM, Beijing 2002, vol. 2, 583--598

arxiv created 2003/04/22 · arxiv updated 2009/11/30

Abstract

The local Langlands correspondence for GL(n) of a non-Archimedean local field F parametrizes irreducible admissible representations of GL(n,F) in terms of representations of the Weil-Deligne group WD_F of F. The correspondence, whose existence for p-adic fields was proved in joint work of the author with R. Taylor, and then more simply by G. Henniart, is characterized by its preservation of salient properties of the two classes of representations. The article reviews the strategies of the two proofs. Both the author's proof with Taylor and Henniart's proof are global and rely ultimately on an understanding of the ℓ-adic cohomology of a family of Shimura varieties closely related to GL(n). The author's proof with Taylor provides models of the correspondence in the cohomology of deformation spaces, introduced by Drinfeld, of certain p-divisible groups with level structure.

Related

• The local Langlands conjecture for GL(n) over a p-adic field, n < p
• The Local Langlands correspondence for \GL_n over p-adic fields
• The local Langlands correspondence for GL(n) over p-adic fields
• Geometric approach to the local Jacquet-Langlands correspondence
• Geometric realization of special cases of local Langlands and\n Jacquet-Langlands correspondences