The Hodge theory of Soergel bimodules

TL;DR

This paper proves Soergel's conjecture on the characters of indecomposable bimodules, establishing positivity of Kazhdan-Lusztig polynomials for all Coxeter systems and providing an algebraic proof of the Kazhdan-Lusztig conjecture.

Contribution

It confirms Soergel's conjecture and demonstrates positivity of Kazhdan-Lusztig polynomials in a general setting, advancing representation theory.

Findings

Proof of Soergel's conjecture on bimodule characters

Positivity of Kazhdan-Lusztig polynomials for all Coxeter systems

Algebraic proof of the Kazhdan-Lusztig conjecture

Abstract

We prove Soergel's conjecture on the characters of indecomposable Soergel bimodules. We deduce that Kazhdan-Lusztig polynomials have positive coefficients for arbitrary Coxeter systems. Using results of Soergel one may deduce an algebraic proof of the Kazhdan-Lusztig conjecture.