We introduce the Kazhdan-Lusztig conjectures, and describe why they are really asking a precise question about morphisms in category O. Instead of working with category O, however, we will work with Soergel bimodules, because they are easier to define and to calculate with (the exact connection between category O and Soergel bimodules will be explained on Tuesday). Then we develop the background and techniques needed to work with Soergel bimodules.
Lecture 1.1 ::: 9-1030 ::: Ben ::: The Kazhdan-Lusztig conjectures
Projectives in O, KL Conjectures, Coxeter groups, Hecke algebras
Lecture 1.2 ::: 1045-1130 ::: Ben ::: The reflection representation and Frobenius extensions
Reflection rep, Chevalley theorem, Frobenius extensions, demazure operators
Exercises 1.1 ::: 1130-1215 ::: Exercise hints ::: 1215
Exercise discussion ::: 2-215
Lecture 1.3 ::: 215-315 ::: Ben ::: Bott-Samelson bimodules, Soergel bimodules, and Soergel's categorification theorem
'Nuff said. Also, Deodhar defect formula.
Lecture 1.4 ::: 330-415 ::: Ben ::: Diagrammatics for 2-categories
Diagrammatics and some philosophy, adjunction and isotopy, examples
Lecture 1.5 ::: 430-515 ::: Alex Ellis ::: The Coxeter 2-groupoid
2-groupoids attached to a group. Group actions on categories. 2-Presentations. The Coxeter 2-presentation. Zamolodchikov relations.
Also Ben's supplementary notes.
Today we develop the diagrammatic approach to Soergel bimodules, expressing the subcategory of Bott-Samelson bimodules by generators and relations. Interwoven with this, we develop some of the abstract theory of Soergel bimodules: standard filtrations and localization. Finally, we explain the connection with category O.
Triage ::: 9-920 ::: Ben
Lecture 2.1 ::: 930- 1045 ::: Ben ::: Diagrammatic calculus for Bott-Samelson bimodules in ranks less than or equal to 2
One-color calculus and Frobenius extensions, Temperley-Lieb algebra, 2-color calculus
Lecture 2.2 ::: 11-1145 ::: Jacob Matherne ::: Standard bimodules and Localization
Parts of the classical approach (i.e. Soergel's approach) to Soergel bimodules are discussed.
Exercises 2.1 ::: 1145-1230 ::: Exercise hints ::: 1230
Exercise discussion ::: 215-230
Lecture 2.3 ::: 230-315 ::: Ben ::: Generators and relations, and a basis.
We discuss path morphisms attached to paths in the reduced expression graph. These lead to the rank 3 relations. We discuss Libedinsky's Light Leaves basis for morphism spaces between Bott-Samelson bimodules.
Lecture 2.4 ::: 330-415 ::: Ben ::: Introduction to category O
The basics of category O
Lecture 2.5new ::: 430-515 ::: Ben ::: Category O and Soergel bimodules
Translation functors and projective functors, Bott-Samelson projectives, the Soergel functor.
Intersection Form Day! We describe the local intersection form and the global intersection form. Then we discuss Hodge-theoretic aspects of intersection forms, and state our main results.
Triage ::: 9 - 920 ::: Ben
Lecture 3.1 ::: 930-1015 ::: Ben ::: Cellular algebras and the local intersection form
Bott-Samelson bimodules are actually an object-adapted cellular category, which allows one an elegant proof of Soergel's categorification theorem. We also introduce local intersection forms, which govern the multiplicities of summands.
Lecture 3.2 ::: 1030-1115 ::: Ben ::: Elements in Bott-Samelson bimodules and the global intersection form
Having spent much time on morphisms between Bott-Samelson bimodules, we now view them once more as modules, and investigate some of their properties. We introduce the global intersection form.
Lecture 3.3 ::: 1130-1230 ::: Ben ::: Lefschetz linear algebra
We introduce the hard Lefschetz property and the Hodge-Riemann bilinear relations. We state what Hodge-theoretic properties Bott-Samelson bimodules should have, and why these properties should imply the Soergel conjecture.
Hike ::: 130
Picnic ::: 630 ::: University Park (5 blocks south of campus on University Avenue). Pizza and salads will be provided.
Today we discuss Rouquier complexes, which are certain complexes of Soergel bimodules which have independent interest. We reduce the question of the Hodge-Riemann bilinear relations to a fact about Rouquier complexes, which we finally prove on Friday.
In the afternoon, we provide the connection between geometric constructions (perverse sheaves on the flag variety) and Soergel bimodules. We also provide diagrammatic tools for singular Soergel bimodules.
Triage ::: 9 - 920 ::: Ben
Lecture 4.1 ::: 930 - 1030 ::: Matt Hogancamp ::: Introduction to Rouquier complexes
Rouquier complexes are certain complexes of Soergel bimodules which induce a braid group action under tensor product. Rouquier complexes for reduced expressions possess a strong property known as the "diagonal miracle."
Lecture 4.2 ::: 1045-1145 ::: Ben ::: The Hodge theory of Soergel bimodules
We outline the proof of the Soergel conjecture, via the fact that Soergel modules possess the Hodge-Riemann bilinear relations. We prove the embedding theorem and set up the final inductive proof.
Exercises 4.1 ::: 1145-1230 ::: Exercise hints ::: 1230
Lecture 4.3 ::: 215 - 315 ::: Ben ::: Perverse sheaves on the flag variety
Perverse sheaves are complicated to define, but easy to work with. We give the layman's introduction to perverse sheaves, and discuss the connection between Soergel bimodules and perverse sheaves on the flag variety.
Lecture 4.4 ::: 330-430 ::: Ben ::: More fun with diagrammatics
Anyone serious about playing with Soergel bimodules morphism-theoretically will benefit from a few more tools. We introduce diagrammatics for singular Soergel bimodules. These singular bimodules correspond to perverse sheaves on partial flag varieties, or to category O at singular weights.
We finish the proof of the Soergel conjecture with a technical inductive argument. Fun! Then, we spend the rest of the day on some interesting topics, some of which will be determined by public opinion.
Triage ::: 9 - 920 ::: Ben
Lecture 5.1 ::: 930-1030 ::: Ben ::: Proof of hard Lefschetz for Soergel bimodules
There it is.
Lecture 5.2 ::: 1045-1145 :::
Geometric Satake for dummies
Going over previous exercises ::: 1145-1230
Possible topics include, but are not limited to:
A) A diagrammatic calculus for localized Soergel bimodules and Geordie's local hard Lefschetz theorem
B) Bott-Samelson bimodules and their Hodge theory: the relative Hodge-Riemann bilinear relations
C) Characteristic p: why people care.
D) Diagonalization and categorical representation theory in type A
E) Eidempotents for longest elements, and higher Bruhat orders in type A
F) Folding, and how to categorify Hecke algebras with unequal parameters
G) Going over exercises