A categorical approach to representation theory in positive characteristic

Vinoth Nandakumar – 7 November 2018

Classical questions about the representations of Lie algebra in positive characteristic are answered by Lusztig's conjectures, which have now been established by Bezrukavnikov-Mirkovic using geometric techniques. Lusztig's conjectures are phrased in terms of canonical bases, and we give a categorical approach to the proof in type A, which is more elementary in nature. In the case where the central character is a nilpotent whose Jordan type is a two-row partition, we give explicit combinatorial formulae for the dimensions of the irreducible modules. The categorical action of an affine braid group plays a key role here.