Geometric representation theory, canonical bases, and explicit character formulas for affine Lie algebras

I will begin with a tour of classical ideas in geometric representation theory including Kazhdan-Lusztig polynomials and Beilinson-Bernstein localization. Then I will delve into a logical continuation of these ideas in the work of Bezrukavnikov on exotic t-structures. Finally I will explain an application of these ideas to finding explicit character formulas for affine Lie algebras in our joint work with Andrei Ionov and Vasily Krylov.