Introduction to Soergel Bimodules

Preface
How to Read This Book
About This Book and Acknowledgements
Leitfaden
Contents
Contributors
Part I The Classical Theory of Soergel Bimodules
	1 How to Think About Coxeter Groups
		1.1 Coxeter Systems and Examples
			1.1.1 Definition of a Coxeter System
			1.1.2 Example: Type A
			1.1.3 Example: Type B
			1.1.4 Example: Type D
			1.1.5 Example: Dihedral Groups
			1.1.6 Coxeter Groups and Reflections
			1.1.7 The Geometric Representation and the Classification of Finite Coxeter Groups
			1.1.8 Crystallographic Coxeter Systems
		1.2 Coxeter Group Fundamentals
			1.2.1 The Length Function
			1.2.2 The Descent Set
			1.2.3 The Exchange Condition
			1.2.4 The Longest Element
			1.2.5 Matsumoto's Theorem
			1.2.6 Bruhat Order
			1.2.7 Additional Exercises
	2 Reflection Groups and Coxeter Groups
		2.1 Reflections and Affine Reflections
		2.2 Affine Reflection Groups
		2.3 Affine Reflection Groups are Coxeter Groups
		2.4 Expressions and Strolls
		2.5 Classification of Affine Reflection Groups
		2.6 The Coxeter Complex
	3 The Hecke Algebra and Kazhdan–Lusztig Polynomials
		3.1 The Hecke Algebra
			3.1.1 The Standard Basis
			3.1.2 Inversion
		3.2 The Kazhdan–Lusztig Basis
			3.2.1 The Standard Form on H
		3.3 Existence of the Kazhdan–Lusztig Basis
			3.3.1 A Motivating Example
			3.3.2 Construction of the Kazhdan–Lusztig Basis
			3.3.3 The Kazhdan–Lusztig Presentation
			3.3.4 Deodhar's Formula
	4 Soergel Bimodules
		4.1 Gradings
		4.2 Polynomials
			4.2.1 Invariant Polynomials
		4.3 Demazure Operators
		4.4 Bimodules and Tensor Products
		4.5 Bott–Samelson Bimodules
		4.6 Soergel Bimodules
		4.7 Examples of Soergel Bimodules
		4.8 A First Glimpse of Categorification
	5 The ``Classical'' Theory of Soergel Bimodules
		5.1 Twisted Actions
		5.2 Standard Bimodules
		5.3 Soergel Bimodules and Standard Filtrations
		5.4 Localization
		5.5 Soergel's Categorification Theorem
		5.6 A Technical Wrinkle
		5.7 Realizations of a Coxeter System
		5.8 More Technicalities
	6 Sheaves on Moment Graphs
		6.1 Roots in the Geometric Representation
		6.2 Bruhat Graphs
		6.3 Moment Graphs
		6.4 Sheaves on Moment Graphs
		6.5 The Braden–MacPherson Algorithm
		6.6 Stalks and Standard Bimodules
		6.7 A Functor to Sheaves on Moment Graphs
		6.8 Soergel's Conjecture and the Braden–MacPherson Algorithm
Part II Diagrammatic Hecke Category
	7 How to Draw Monoidal Categories
		7.1 Linear Diagrams for Categories
		7.2 Planar Diagrams for 2-Categories
		7.3 Drawing Monoidal Categories
		7.4 The Temperley–Lieb Category
		7.5 More About Isotopy
	8 Frobenius Extensions and the One-Color Calculus
		8.1 Frobenius Structures
			8.1.1 Frobenius Algebra Objects
			8.1.2 Diagrammatics for Frobenius Algebra Objects
			8.1.3 Playing with Isotopy
			8.1.4 Frobenius Extensions
		8.2 A Tale of One Color
			8.2.1 Frobenius Structure
			8.2.2 Additional Generators and Relations
			8.2.3 The Moral of the Tale
			8.2.4 A Direct Sum Decomposition, Diagrammatically
	9 The Dihedral Cathedral
		9.1 A Tale of Two Colors
		9.2 The Temperley–Lieb 2-Category
		9.3 Jones–Wenzl Projectors
		9.4 Two-color Relations
	10 Generators and Relations for Bott–Samelson Bimodules and the Double Leaves Basis
		10.1 Why Present BSBim?
		10.2 Generators and Relations
			10.2.1 A Diagrammatic Reminder
			10.2.2 An Isotopy Presentation of HBS
			10.2.3 Examples and Exercises
			10.2.4 A Presentation of HBS
			10.2.5 The Functor to Bimodules
			10.2.6 General Realizations
		10.3 Rex Moves and the 3-color Relations
		10.4 Light Leaves and Double Leaves
			10.4.1 Overview
			10.4.2 The Algorithm
			10.4.3 Diagrammatics and Bimodules
			10.4.4 Light Leaves and Localization
	11 The Soergel Categorification Theorem
		11.1 Introduction
		11.2 Prelude: From HBS to H
			11.2.1 Graded Categories
			11.2.2 Additive Closure
			11.2.3 Karoubian Closure
			11.2.4 Karoubi Envelopes Are Krull–Schmidt
			11.2.5 Diagrammatics and Karoubi Envelopes
		11.3 Grothendieck Groups of Object-Adapted Cellular Categories
			11.3.1 Object-Adapted Cellular Categories
			11.3.2 First Properties
			11.3.3 The Main Example
			11.3.4 Classifying Indecomposables in Object-Adapted Cellular Categories
		Appendix 1: Krull–Schmidt Categories
			Categories with Unique Decompositions
			Krull–Schmidt Categories
			The Karoubian Property
			Semiperfect Rings
			The Split Grothendieck Group of a Category with Shift Functor
		Appendix 2: Composition Forms, Cellular Forms, and Local Intersection Forms
	12 How to Draw Soergel Bimodules
		12.1 The 01-Basis
		12.2 Commutative Ring Structure on a Bott–Samelson Bimodule
		12.3 Trace and the Global Intersection Form
		12.4 Bott–Samelson Bimodules and the Light Leaves Basis
		12.5 Light Leaves Basis and the Standard Filtration on Bott–Samelson Bimodules
		Appendix 1: A Crucial Positivity Result
Part III Historical Context: Category Oand the Kazhdan–Lusztig Conjectures
	13 Category O and the Kazhdan–Lusztig Conjectures
		13.1 Introduction
		13.2 The Verma Problem
			13.2.1 Verma Modules
			13.2.2 Category O and Its Mysteries
		13.3 The Kazhdan–Lusztig Conjectures
			13.3.1 The Multiplicity Conjecture
			13.3.2 Positivity and Schubert Varieties
		13.4 Two Proofs of the Kazhdan–Lusztig Conjecture
	14 Lightning Introduction to Category O
		14.1 Lie Algebra Basics
		14.2 Category O
		14.3 Duality in O
			14.3.1 Standard Filtrations and BGG Reciprocity
		14.4 Blocks of Category O
		14.5 Example: g = sl3(C)
	15 Soergel's V Functor and the Kazhdan–Lusztig Conjecture
		15.1 Brief Reminder on Category O
		15.2 Translation Functors
			15.2.1 Tensor Products
			15.2.2 Definition of Translation Functors and First Properties
			15.2.3 Effect on Verma Modules
			15.2.4 Wall-Crossing Functors
			15.2.5 Effect on Projective Modules
		15.3 Soergel Modules
		15.4 Soergel's V Functor
		15.5 Soergel's Approach to the Kazhdan–Lusztig Conjecture
	16 Lightning Introduction to Perverse Sheaves
		16.1 Motivation
		16.2 Stratified Spaces and Examples
			16.2.1 Stratified Resolutions and Schubert Varieties
			16.2.2 Constructible Sheaves and Pushforwards
			16.2.3 Perverse Sheaves
			16.2.4 The Decomposition Theorem
			16.2.5 Connection to the Hecke Algebra
Part IV The Hodge Theory of Soergel Bimodules
	17 Hodge Theory and Lefschetz Linear Algebra
		17.1 Introduction
		17.2 Hard Lefschetz
		17.3 Hodge–Riemann Bilinear Relations
		17.4 Lefschetz Lemmas
	18 The Hodge Theory of Soergel Bimodules
		18.1 Introduction
		18.2 Overview and Preliminaries
			18.2.1 The Conjectures of Soergel and Kazhdan–Lusztig
			18.2.2 Duality and Invariant Forms
				18.2.2.1 Morphisms Between Soergel Bimodules
				18.2.2.2 Invariant Forms
				18.2.2.3 Invariant Forms on Soergel Bimodules
				18.2.2.4 Lefschetz Forms and Positivity
				18.2.2.5 Key Statements in the Induction
				18.2.2.6 Induced Forms
			18.2.3 The Main Theorem
		18.3 Outline of the Proof
			18.3.1 Step 1
				18.3.1.1 Deforming the Lefschetz Operator
				18.3.1.2 Flowchart for Step 1
			18.3.2 Step 2
				18.3.2.1 The Local Intersection Form
				18.3.2.2 HR(x,s) Versus HR(x s)
				18.3.2.3 Flowchart for Step 2
		18.4 The Weak Lefschetz Problem
		18.5 From zeta = 0 to zeta >> 0
		18.6 From Local to Global Intersection Forms
		18.7 Hodge Theory of Matroids
	19 Rouquier Complexes and Homological Algebra
		19.1 Motivation
		19.2 Some Homological Algebra
			19.2.1 Complexes and Homotopies
			19.2.2 Gaussian Elimination and Minimal Complexes
			19.2.3 Grothendieck Groups
		19.3 Rouquier Complexes and Categorification of the Braid Group
		19.4 Cohomology of Rouquier Complexes
		19.5 Perversity
		19.6 The Diagonal Miracle
		Appendix: More Homological Algebra
			Triangulated Structure
			Triangulated Grothendieck Groups
			Perverse t-Structure
	20 Proof of the Hard Lefschetz Theorem
		20.1 Introduction
		20.2 Preliminaries
		20.3 The Hodge–Riemann Relations for the Rouquier Complex
		20.4 Positivity of Breaking
		20.5 Sketch of the Proof of Hard Lefschetz
		Appendix: Some Historical Context and Geometric Intuition for the Proof of Soergel's Conjecture
			The Kazhdan–Lusztig Conjecture for Weyl Groups and the Decomposition Theorem
			What We Need I: Hard Lefschetz in a Family
			What We Need II: A Substitute for Hyperplane Sections
Part V Special Topics
	21 Connections to Link Invariants
		21.1 Temperley–Lieb Algebra
		21.2 Schur–Weyl Duality
		21.3 Trace and Link Invariants
		21.4 Quantum Groups and Link Invariants
		21.5 Ocneanu Trace and HOMFLYPT Polynomial
		21.6 Categorification of Braids and of the HOMFLYPT Invariant
			21.6.1 Rouquier Complexes
			21.6.2 Hochschild Homology
			21.6.3 Categorifying the Standard Trace
	22 Cells and Representations of the Hecke Algebra in Type A
		22.1 Cells
			22.1.1 Cells for a Monoidal Category
			22.1.2 Cell Module Categories
			22.1.3 Cells for a Based Algebra
		22.2 Cells in Type A
			22.2.1 Young Diagrams and Tableaux
			22.2.2 The Robinson–Schensted Correspondence
			22.2.3 Cells in Type A
			22.2.4 The k-row Quotient of the Hecke Algebra
		22.3 Representations of the Hecke Algebra in Type A
	23 Categorical Diagonalization
		23.1 Classical Linear Algebra
		23.2 Categorified Linear Algebra
			23.2.1 Eigenobjects
			23.2.2 Prediagonalizability
			23.2.3 Twisted Complexes
			23.2.4 Diagonalizability
			23.2.5 Smallness
			23.2.6 Lagrange Interpolation
		23.3 A Toy Example
		23.4 Diagonalizing the Full Twist
			23.4.1 Type A1
			23.4.2 Type A
			23.4.3 The General Case
	24 Singular Soergel Bimodules and Their Diagrammatics
		24.1 The Classical Theory of Singular Soergel Bimodules
			24.1.1 Bimodules and Functors
			24.1.2 Singular Soergel Bimodules
			24.1.3 Categorification Theorems
		24.2 One-Color Singular Diagrammatics
			24.2.1 Diagrammatics for a Frobenius Extension
			24.2.2 Relationship to the One-Color Soergel Calculus
			24.2.3 Relationship with the Temperley–Lieb 2-Category
		24.3 Singular Soergel Diagrammatics in General
			24.3.1 The Upgraded Chevalley Theorem, Part I
			24.3.2 The Upgraded Chevalley Theorem, Part II
			24.3.3 Diagrammatics for a Cube of Frobenius Extensions
			24.3.4 The Jones–Wenzl Relation
			24.3.5 Additional Relations, Applications, and Future Work
			24.3.6 Other Realizations
	25 Koszul Duality I
		25.1 Introduction
			25.1.1 Morita Theory
		25.2 dg-Algebras
		25.3 dg-Morita Theory
		25.4 Koszul Duality for Polynomial Rings
		25.5 Review of the Kazhdan–Lusztig Conjecture
		25.6 Evidence of Koszul Duality in Category O
	26 Koszul Duality II
		26.1 Introduction
		26.2 Graded Category O0
			26.2.1 Desiderata
			26.2.2 Motivation from Soergel's V Functor
			26.2.3 Definition of Soergel Category O0
			26.2.4 Example: Soergel O0 in Type A1
		26.3 Homological Properties of Soergel O0
			26.3.1 Highest Weight Structure
			26.3.2 Tilting Objects and the Realization Functor
			26.3.3 Ringel Duality
		26.4 Koszul Duality
			26.4.1 Statement
			26.4.2 Monodromy Action
			26.4.3 Wall-Crossing Functors
			26.4.4 Outline of the Proof of Theorem 26.26
		26.5 Some Odds and Ends
	27 The p-Canonical Basis
		27.1 Introduction
		27.2 Definition of the p-Canonical Basis
		27.3 Computing the p-Canonical Basis
		27.4 Geometric Incarnation of the Hecke Category
			27.4.1 Parity Complexes on Flag Varieties
			27.4.2 Parity Complexes and the Hecke Category
		27.5 Modular Representation Theory of Reductive Groups
			27.5.1 Soergel's Modular Category O
			27.5.2 The Riche–Williamson Conjecture
			27.5.3 This Is Not the End
References
Index