Soergel bimodules and Kazhdan-Lusztig polynomials

TL;DR

This paper provides an overview of Soergel bimodules and their application to Kazhdan-Lusztig theory, highlighting key results that offer alternative proofs of the Kazhdan-Lusztig conjectures.

Contribution

It summarizes Soergel's main results and their role in providing new proofs for Kazhdan-Lusztig conjectures, serving as a survey of the subject.

Findings

Soergel bimodules offer an algebraic framework for Kazhdan-Lusztig polynomials.

Alternative proofs of Kazhdan-Lusztig conjectures are achieved via Soergel's theory.

The paper serves as a condensed survey of recent advances in the field.

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.