La th\'eorie de Hodge des bimodules de Soergel (d'apr\`es Soergel et Elias-Williamson)

TL;DR

This paper discusses the extension of Hodge theory to Soergel bimodules associated with Coxeter groups, proving key properties that lead to a proof of the Kazhdan-Lusztig conjecture.

Contribution

It generalizes Hodge-theoretic properties of Soergel bimodules to all Coxeter groups, establishing positivity of Kazhdan-Lusztig polynomials and proving the conjecture.

Findings

Proved Hodge-type properties for all Coxeter groups.

Established positivity of Kazhdan-Lusztig polynomials.

Provided an algebraic proof of the Kazhdan-Lusztig conjecture.

Abstract

Soergel bimodules are certain bimodules over polynomial algebras, associated with Coxeter groups, and introduced by Soergel in the 1990's while studying the category O of complex semisimple Lie algebras. Even though their definition is algebraic and rather elementary, some of their crucial properties were known until recently only in the case of crystallographic Coxeter groups, where these bimodules can be interpreted in terms of equivariant cohomology of Schubert varieties. In recent work Elias and Williamson have proved these properties in full generality by showing that these bimodules possess "Hodge type" properties. These results imply positivity of Kazhdan-Lusztig polynomials in full generality, and provide an algebraic proof of the Kazhdan-Lusztig conjecture.