An approach to modular representation theory of Lie algebras via categorification and knot theory

Abstract:

The irreducible representations of the general linear Lie algebra in positive characteristic are not well-understood. Lusztig's conjectures, now a theorem of Bezrukavnikov-Mirkovic, describe their dimensions using techniques from geometric representation theory. We give an approach to Lusztig's conjectures using higher representation theory, which is easier to compute with. In the case with a central p-character given by a nilpotent whose Jordan type is a two-row partition, we give combinatorial dimension formulae for the irreducible modules, composition multiplicities of the simples in the baby Vermas, and a description of the Ext spaces. The key technical step is a categorical action of the affine tangle calculus, building on results of Cautis-Kamnitzer and Bezrukavnikov-Mirkovic-Rumynin. This Ext algebra is an "annular" analogue of Khovanov's arc algebra, and can be used to give an extension of Khovanov homology to links in the annulus. This is joint work with Rina Anno and David Yang.