Towards localization for affine Lie algebras at critical level

Abstract: I will report on an incomplete project, whose goal is to geometrically realize representations of critical level affine Lie algebras at all central characters. To do so, we study these categories of representations from the point of view of the local geometric Langlands program. As an accidental consequence, we will prove some new results in the local Langlands program, both in geometry and in arithmetic. No knowledge of affine Lie algebras or of Langlands will be assumed. Some parts are joint work with a number of other mathematicians, including Dhillon, Eteve, Gaitsgory, Raskin and Varshavsky.