Quantum Category O and Categorification of Periodic Hecke Module, Lecture 1

The BGG category O plays an important role in the study of representations of semisimple Lie algebras. Its connection to the Hecke category is a starting point of Geometric Representation Theory. In this lecture, I will introduce a version of category O for quantum groups at roots of unity. I will explain a derived equivalence from (the principal block of) quantum category O to the affine Hecke category. Under the equivalence, the highest weight structure of quantum category O provides a categorification of the “periodic Hecke module”.

In the first lecture, I will briefly review the classical story for BGG category O. Then I will give an introduction to the quantum category O with motivations.