  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.
