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

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