Category O, symplectic duality, and the Hikita conjecture

Minicourse given at the Summer School on Geometric Representation Theory, July 2018.

Abstract: The category O for a semisimple Lie algebra can be interpreted geometrically in terms of the Springer resolution, and then generalized to other symplectic resolutions. These resolutions tend to come in dual pairs whose associated categories are related by Koszul duality. After introducing these concepts, I will discuss various versions of the Hikita conjecture, which relates the cohomology of one resolution to the coordinate ring of its dual.

Lecture 1: Category O
Lecture notes, Exercises

Lecture 2: The Hikita conjecture
Lecture notes, Exercises

Lecture 3: The equivariant Hikita conjecture
Lecture notes