Constructing and characterizing a local Langlands correspondence

Friday, November 16, 2018 - 1:30pm - 2:30pm
Keller 3-180
Michael Harris (Columbia University)
A Langlands correspondence for a reductive group G over a local field F is a partial or complete classification of representations of the group G(F) in terms of Galois parameters for F with values in the Langlands dual group of G. The model for this correspondence is the parametrization of irreducible representations of GL(n,F) by n-dimensional representations of the Weil-Deligne group of F. The first complete proofs of this classification were obtained more than 20 years ago; the talk will review these and more recent proofs, practically all of which are obtained by global means.

Alongside Vincent Lafforgue's construction of parameters for automorphic representations of general reductive groups over function fields, Genestier and Lafforgue have defined Langlands parameters for reductive groups over local fields of positive characteristic. Work in progress of Fargues and Scholze promises to extend the Genestier-Lafforgue construction to general non-archimedean local fields, using the geometry of new kinds of moduli spaces. Many questions remain open, most notably: do all local Galois parameters arise in this way? What is the structure of the fibers of these parametrizations? And, perhaps most fundamentally, can these correspondences be characterized independently of their construction?