The Koszul duality of Beilinson-Ginzburg-Soergel is a derived equivalence involving the BGG category O, which plays a central role in the study of highest weight modules of a semisimple complex Lie algebra. Geometrically, this may be viewed as a derived equivalence relating Langlands dual flag varieties. In this talk, I will discuss a new approach to a modular (positive characteristic) analogue of this result proved by Pramod Achar and Simon Riche.
Prerequisites will be kept to a minimum: once I have motivated the result, I will not work with representations or perverse/parity sheaves, instead giving an algebraic/combinatorial model (moment graph sheaves, Soergel bimodules) for these objects, focusing on the case of SL_2. This will be enough for illustrating the key ingredient, a new construction of a left monodromy action.
I will also briefly report on joint work in progress with Pramod Achar, Simon Riche, and Geordie Williamson, in which we plan to extend the result to Kac-Moody flag varieties. The latter result would imply the Riche-Williamson conjecture on tilting modules of reductive groups.