Soergel bimodules and Kazhdan-Lusztig conjectures
Course given with Ben Elias at the QGM Aarhus, 18th-22nd of March, 2013.

The first half of the course is based on this paper (which is still in a preliminary version). The second half is based on The Hodge theory of Soergel bimodules.

Monday: The Cast

Introduction: Category O and the Kazhdan-Lusztig conjectures
The happenings of 1979. The miracle of KL polynomials. Arbitrary Coxeter groups. The miracle of the localisation proof. Soergel's dream of an algebraic explanation... the deepening mystery of positivity.

Hecke algebras and Kazhdan-Lusztig polynomials
The Coxeter complex. The Hecke algebra of a Coxeter group. The presentation using standard generators. The standard basis. The Kazhdan-Lusztig basis and polynomials. The Kazhdan-Lusztig presentation. Products of Kazhdan-Lusztig generators and the defect formula. Slides.

Soergel bimodules
Invariant theory for finite reflection groups. Bimodules and monoidal categories. The category of Soergel bimodules. Singular Soergel bimodules. First examples.

How to draw monoidal categories
Higher algebra. Drawing adjunctions, cyclicity etc. Example: 2-groupoids. The Coxeter groupoid. The generalized Zamolodchikov relations.

Monday exercises

Tuesday: Getting to know Soergel bimodules

The classical apporach to Soergel bimodules
Standard bimodules. Support filtrations. Soergel's hom formula. Statement of Soergel's categorification theorem. Localization. Discussion.

The dihedral cathedral
Starting to draw Soergel bimodules. Soergel bimodules in rank 2. Jones-Wenzl projectors, connections to the Temperley-Lieb algebra and quantum groups. Categorification of the Kazhdan-Lusztig presentation.

Generators and relations, the light leaves basis
Generators and relations in general. Light leaves morphisms as a categorification of the defect formula. Double leaves give a basis for morphisms.

How to draw Soergel bimodules.
to draw Bott-Samelson bimodules, Soergel bimodules. Intersection forms. Discussion.

Tuesday Exercises

Wednesday: Soergel bimodules and glimpses of geometry

Soergel's categorification theorem
The cellular structure. A discussion of idempotent lifting. Generators and relations proof of Soergel's categorification theorem. Examples of intersection forms and idempotents.

Hodge theory and Lefschetz linear algebra
Review of the (real) Hodge theory of smooth projective algebraic varieties. A discussion of the weak and hard Lefschetz theorems. Lefschetz operators, Lefschetz forms and the Hodge-Riemann bilinear relations. Tricks establishing the Lefschetz package. The weak-Lefschetz substitute.

The Hodge theory of Soergel bimodules
Statement of the results and outline of the methods. The embedding theorem, the limit argument. The absence of the weak Lefschetz theorem.

Lightning introduction to IC, hypercohomology and Soergel bimodules Varieties stratified by affine spaces and the constructible derived category. How to compute stalks of a proper push-forward. Poincar\'e duality. Stalks definition of an IC sheaf. The connection to Kazhdan-Lusztig polynomials on the flag variety. Global sections and Soergel bimodules.

Wednesday Exercises

Thursday: Soergel's Conjecture and the Kazhdan-Lusztig conjecture

Rouquier complexes and homological algebra
The homotopy category of Soergel bimodules. Minimal complexes. Rouquier complexes. Examples.

Proof of hard Lefschetz
The perverse filtration on Soergel bimodules. The diagonal miracle. Factoring the Lefschetz operator. Hard Lefschetz.

Lightning introduction to category O and Soergel's V
Review of Verma modules, category $\OC$ and its block decomposition by central character. Statement of the Kazhdan-Lusztig conjecture. Soergel's functor $\mathbb{V}$. Soergel's conjecture implies the Kazhdan-Lusztig conjecture.

Overflow/ discussion session

Thursday Exercises

Friday: Discussion and Applications

Hecke algebras with unequal parameters and foldingi
(This talk ended up not getting given, as we discussed exercises instead!) Definition of Hecke algebras with unequal parameters. Equivariant K-theory. Categorification of unequal parameters in the quasi-split case.

The situation in characteristic p
Lusztig's conjecture. Intersection forms. The p-canonical basis. Examples. Many mysteries and open questions.

Categorifications of braid groups
Categorifying the braid group. Example of Rouquier complexes. Generators and relations for strict braid group actions. Deligne's theorem and the EW version.

Algebraic quantum geometric Satake
A discussion of ridiculous titles. Algebraizing the geometric Satake equivalence. Quantizing it in type A using Ben's favorite Cartan matrix.

Some research problems