On the Local Langlands Correspondence

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 WDF 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. The general local Langlands correspondence replaces GL(n, F ) by an arbitrary reductive group G over F , whose representations are conjecturally grouped in packets parametrized by homomorphisms from WDF to the Langlands dual group G. The article describes partial results in this direction for certain classical groups G, due to Jiang-Soudry and Fargues. The bulk of the article is devoted to motivating problems that remain open even for GL(n). Foremost among them is the search for a purely local proof of the correspondence, especially the relation between the Galoistheoretic parametrization of representations ofGL(n, F ) and the group-theoretic parametrization in terms of Bushnell-Kutzko types. Other open questions include the fine structure of the cohomological realization of the local Langlands correspondence: does the modular local Langlands correspondence of Vigneras admit a cohomological realization? 2000 Mathematics Subject Classification: 11, 14, 22. *Institut de Mathématiques de Jussieu-UMR CNRS 7586, Université Paris 7. Membre, Institut Universitaire de France, France. E-mail: [email protected]