This book provides a comprehensive introduction to Soergel bimodules. First introduced by Wolfgang Soergel in the early 1990s, they have since become a powerful tool in geometric representation theory. On the one hand, these bimodules are fairly elementary objects and explicit calculations are possible. On the other, they have deep connections to Lie theory and geometry. Taking these two aspects together, they offer a wonderful primer on geometric representation theory. In this book the reader is introduced to the theory through a series of lectures, which range from the basics, all the way to the latest frontiers of research.
This book serves both as an introduction and as a reference guide to the theory of Soergel bimodules. Thus it is intended for anyone who wants to learn about this exciting field, from graduate students to experienced researchers.
Introduction to Soergel Bimodules
- Part I The Classical Theory of Soergel Bimodules. - How to Think About Coxeter Groups. - Reflection Groups and Coxeter Groups. - The Hecke Algebra and Kazhdan-Lusztig Polynomials. - Soergel Bimodules. - The "Classical" Theory of Soergel Bimodules. - Sheaves on Moment Graphs. - Part II Diagrammatic Hecke Category. - How to Draw Monoidal Categories. - Frobenius Extensions and the One-Color Calculus. - The Dihedral Cathedral. - Generators and Relations for Bott-Samelson Bimodules and the Double Leaves Basis. - The Soergel Categorification Theorem. - How to Draw Soergel Bimodules. - Part III Historical Context: Category O and the Kazhdan-Lusztig Conjectures. - Category O and the Kazhdan-Lusztig Conjectures. - Lightning Introduction to Category O. - Soergel's V Functor and the Kazhdan-Lusztig Conjecture. - Lightning Introduction to Perverse Sheaves. - Part IV The Hodge Theory of Soergel Bimodules - Hodge Theory and Lefschetz Linear Algebra. - The Hodge Theory of Soergel Bimodules. - Rouquier Complexes and Homological Algebra. - Proof of the Hard Lefschetz Theorem. - Part V Special Topics. - Connections to Link Invariants. - Cells and Representations of the Hecke Algebra in Type A. - Categorical Diagonalization. - Singular Soergel Bimodules and Their Diagrammatics. - Koszul Duality I. - Koszul Duality II. - The p-Canonical Basis.