David Jordan (University of Edinburgh) An introduction to geometric representation theory for quantum groups. One of the most important developments in representation theory of algebraic groups and Lie algebras in the last 30 years was the introduction of methods from algebraic geometry. By studying line bundles, differential operators, orbits, slices and other geometric constructions on spaces with group actions, such as flag varieties $G/B$, or coadjoint quotients $\mathfrak{g}/G$, we obtain deep insight into the representation theory of these structures. More recently, starting with works of Tanisaki, Backelin–Kremnitzer, Arkhipov–Gaitsgory, Varangolo–Vasserot a program of geometric representation theory for quantum groups was founded. I will survey these works and then outline some more recent developments.

Assistants

Juan Guzmán was the assistant to David Jordan for this course’s tutorials.

Juan received his PhD in Mathematics in December 2022 from the FaMAF (Universidad Nacional de Córdoba), where he also completed his undergraduate studies: Licenciatura and Profesorado in Mathematics. His interests include vertex algebras and conformal Lie algebras, as well as chiral and factorization algebras.

Reading material for the course

1. Introduction to geometric representation theory. The course will cover material from Chapters 2 through Chapter 6. The topics covered in Chapter 1 (Lie algebra representations of $\mathfrak{sl}(2,\mathbb C)$) a are prerequisite for the course.

Exercises

You can find the exercises here.