An approach to the local geometric Langlands conjectures via higher representation theory

The local geometric Langlands conjectures, formulated by Frenkel and Gaitsgory, give a geometric description of the representation categories of affine Lie algebras at the critical level. Bezrukavnikov-Mirkovic simplify their results, and express their conjectures using representation theory of Lie algebras in positive characteristic. We give an approach to their work using higher representation theory, and reformulate Lusztig's conjectures for modular representations using ``exotic canonical bases" in tensor product representations. The main technical tool is the categorical framework developed by Bernstein-Frenkel-Khovanov, Cautis-Kamnitzer-Licata and Losev-Webster. This is joint work with Rina Anno, Galyna Dobrovolska, David Yang and Gufang Zhao.