Soergel bimodules and Kazhdan-Lusztig polynomials

TL;DR

This paper surveys Soergel bimodules as the categorification of Hecke algebras and their application to Kazhdan-Lusztig theory. It outlines the construction of Bott-Samelson bimodules, the Grothendieck group framework, and Soergel’s categorification and Hom formulas, culminating in the strategy to prove the Kazhdan-Lusztig conjecture via the V-functor and translation functors. A central theme is that Soergel’s conjecture, relating bimodule characters to KL basis elements, implies KL positivity and the KL multiplicity formula, thereby connecting geometric representation theory to algebraic categorification. The appendix and references situate the approach within the broader landscape of Lie theory, category $\mathcal{O}$, and Schubert geometry, highlighting both the technical machinery and the historical motivation. Overall, the work demonstrates that categorification through Soergel bimodules yields a powerful, conceptually transparent route to KL-type results with deep geometric and representation-theoretic implications.

Abstract

This paper presents a brief exposition of Soergel bimodules with applications to some topics in Kazhdan-Lusztig theory. We ultimately exposit a few of Soergel's main results, which allowed him to give alternative proofs, using his theory, of the Kazhdan-Lusztig conjectures. This paper should be viewed as a (very) condensed outline following the work of Elias, Makisumi, Thiel, and Williamson in their lovely book Introduction to Soergel Bimodules, and is meant for a reader wishing to survey a quite vast subject.