Soergel bimodules and representation theory
Course given at the University of Sydney from October until December, 2012.

Abstract: This course will be an introduction to Soergel bimodules, with emphasis on applications in representation theory. The main topics will be generators and relations for Soergel bimodules, the proof of Soergel's conjecture using ideas from Hodge theory, and the use of Soergel bimodules to produce counterexamples to conjectures of Lusztig and James. Much of what I will talk about is either due to Soergel, or is joint work with Ben Elias. Throughout the course I will try to emphasise unsolved problems and aspects where new research can be done.

Thanks to Anthony Henderson for taking notes!

4/10: Lecture 1: (Anthony's notes.) Motivation: basic questions in representation theory where Soergel bimodules are useful. Overview of category O and why Soergel bimodules might help to understand the Kazhdan-Lusztig conjecture.

8/10: Lecture 2: (Anthony's notes.) Introduction to the "classical" theory of Soergel bimodules. Soergel bimodules in low rank. Costandard filtrations. The character of a Soergel bimodule. Soergel's categorification theorem.

11/10: Lecture 3: (Anthony's notes.) Introduction to string diagrams. Adjunctions, biadjointness. Frobenius extensions of rings. Generators and relations for Soergel bimodules and singular Soergel bimodules in rank 1.

15/10: Lecture 4: (Anthony's notes.) Examples of diagrammatic presentations with sl2 and the Temperly-Lieb category. Jones-Wenzl projectors. Generators and relations in rank 1 again. Singular Soergel bimodules.

18/10: Lecture 5: (Anthony's notes.) The Schur algebroid. The Satake isomorphism.

22/10: Lecture 6: (Anthony's notes.) Singular Soergel bimodules. Explicit geometric equivalence in rank 2. Rank 2 generators and relations. Zamolodchikov relations.

25/10: Lecture 7: (Anthony's notes.) Soergel's hom formula. Deodhar's defect.

29/10: Lecture 8: (Anthony's notes.) Light leaves maps. Double leaves theorem. Main theorems. Canonical basis and p-canonical basis.

No lectures on 1/10, 5/10 or 8/10.

12/11: Lecture 9: (Anthony's notes.) Lefschetz linear algebra. Invariant forms on Soergel and Bott-Samelson bimodules.

The week before I gave a related lecture in Melbourne. (Here the focus is more geometric, and I state some open problems at the end.)

15/11: Lecture 10: (Anthony's notes.) Outline of the proof of Soergel's conjecture.

19/11: Lecture 11: (Anthony's notes.) Rouquier complexes, sketch of proof of hard Lefschetz.

29/11: Lecture 12: (Anthony's notes.) How to calculate the p-canonical basis. Intersection forms. Two examples.

3/12: Lecture 13: (Anthony's notes.) More examples of intersection forms. Formula for entries in the intersection form in terms of the nil Hecke ring. Super-linear growth of torsion.
Sample A7 calculation.