La théorie de Hodge des bimodules de Soergel (d'après Soergel et Elias-Williamson) - Archive ouverte HAL Access content directly
Journal Articles Asterisque Year : 2019

La théorie de Hodge des bimodules de Soergel (d'après Soergel et Elias-Williamson)

Simon Riche
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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.
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Dates and versions

hal-01632321 , version 1 (10-11-2017)

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Simon Riche. La théorie de Hodge des bimodules de Soergel (d'après Soergel et Elias-Williamson). Asterisque, 2019, Séminaire Bourbaki 2017/2018, 414, pp.125-166. ⟨hal-01632321⟩
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