Travaux de Frenkel, Gaitsgory et Vilonen sur la correspondance de Drinfeld-Langlands

TL;DR

This paper discusses the advancements made by Frenkel, Gaitsgory, and Vilonen in establishing parts of the Drinfeld-Langlands correspondence for algebraic curves and the general linear group, a key aspect of the geometric Langlands program.

Contribution

It reviews and synthesizes recent progress in proving the Drinfeld-Langlands correspondence for GL(n) over projective curves, extending geometric Langlands theory.

Findings

Significant parts of the correspondence have been established for GL(n).

The work bridges automorphic and Galois representations in the geometric setting.

Progress supports the broader Langlands conjectures in algebraic geometry.

Abstract

In 1967, Langlands conjectured a natural correspondence between automorphic representations and Galois representations, over number fields as well as over function fields. In 1983, Drinfeld discovered a geometric analog of the Langlands correspondence in the function field case, which extends the geometric class field theory of Lang and Rosenlicht. The so called Drinfeld-Langlands correspondence is a conjectural duality between two moduli spaces that are naturally associated to an algebraic curve X and a reductive group G. When X is projective and G is the full linear group GL(n), a large part of this correspondence has recently been established by E. Frenkel, D. Gaitsgory et K. Vilonen.