Minicourse given at Higher Structures in China II, August 2011.

Lecture 1: A geometric realization of the regular representation of the symmetric group

In this lecture we examine the action of the symmetric group on the cohomology of the flag variety, either via the (antiholomorphic) action of the group on the variety, or via the convolution construction.

Lecture 2: Categorifying the regular representation of the symmetric group

In this lecture we use D-modules on the flag variety to promote this action to an action of the braid group on a regular block of BGG category O.

Lecture 3: Geometric category O and symplectic duality

Last, we explain the speaker's joint work with Braden, Licata, and Webster, in which the cotangent bundle of the flag variety is replaced with other interesting symplectic varieties, and new categories are obtained that share many of the beautiful properties of BGG category O.