霍奇的苏尔格尔双模理论（苏尔格尔与埃利亚斯-威廉森之后）
La théorie de Hodge des bimodules de Soergel (d'après Soergel et Elias-Williamson)
摘要：
Soergel双模是多项式代数上的某些双模，与Coxeter群有关，由Soergel在20世纪90年代研究复半单李代数的O范畴时引入。尽管它们的定义是代数的，而且相当初等，但直到最近，它们的一些关键性质才在结晶学考克斯特群的情况下被发现，在那里这些双模可以用舒伯特簇的等变上同调来解释。在最近的工作埃利亚斯和威廉姆森证明了这些性质在充分的一般性，表明这些双模具有“霍奇型”的性质。这些结果充分地揭示了Kazhdan-Lusztig多项式的正性，并为Kazhdan-Lusztig猜想提供了一个代数证明.
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
