On the local Langlands correspondence

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 $\ell$-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.