Lecture III: Introduction to localization in representation theory

The Localization Theorem of Beilinson-Bernstein realizes representations of a complex semisimple Lie algebra via sheaves-with-flat-connection (made precise using D-modules) on the flag variety. The theorem made possible the application of deep geometric and topological tools (especially, Hodge theory) to purely representation-theoretic questions and revolutionized the field. This lecture will provide a low-tech introduction to localization, emphasizing a point of view that comfortably generalizes to a broader symplectic setting, with an indication of just a few of the many applications.