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**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 *sl*_{2} 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.