Pramod Achar, Geometric Methods in Representation Theory Seminar

Title: Koszul duality and characters of tilting modules

Abstract: This talk is about the “Hecke category,” a monoidal category that appears in various incarnations in geometric representation theory. I will explain some of these incarnations and their roles in solving classical problems, such as the celebrated Kazhdan-Lusztig conjectures on Lie algebra representations. These conjectures (proved in 1981) hinge on the fact that the derived category of constructible sheaves on a flag variety is equipped with an obvious monoidal action of the Hecke category on the right.

It turns out that there is also a second, “hidden” action of the Hecke category on the left. The symmetry between the “hidden” left action and the “obvious” right action leads to the phenomenon known as Koszul duality. In the last part of the talk, I will discuss new results on Koszul duality with coefficients in a field of positive characteristic, with applications to characters of tilting modules for algebraic groups. This is joint work with S. Makisumi, S. Riche, and G. Williamson.