Soergel bimodules and Kazhdan-Lusztig theory

The Hecke algebra is an algebraic deformation of the group algebra
of any Weyl group or Coxeter group, and can be studied combinatorially.
Kazhdan and Lusztig realized that a certain basis (the KL basis)
in the Hecke algebra of a Weyl group seemed to encode certain
multiplicities in the representation theory of the corresponding
Lie algebra, and later proved that this same basis also encodes
features of singularities in associated Schubert varieties.  This
observation spawned the flourishing field of geometric representation
theory, and the subfield of Kazhdan-Lusztig theory, which studies
the Hecke algebra and its two ``categorifications," one geometric
and one representation-theoretic. One also uses these results to
prove certain positivity properties of the KL basis itself, properties
which lack a direct algebraic proof. There is a lot of beautiful
and rich mathematics here, but it is also very technical and requires
a great deal of background.

However, a great miracle still needs to be explained: Hecke algebras
associated to arbitrary Coxeter groups still seem to possess the
same positivity properties and behavior as Hecke algebras for Weyl
groups, even though Coxeter groups do not have any associated
geometry or representation theory.  In an attempt to explain this
and find a simpler proof of Kazhdan and Lusztig's famous conjectures,
Soergel invented a certain monoidal category of bimodules (now known
as Soergel bimodules) which provides an algebraic categorification
of the Hecke algebra of any Coxeter group. Thankfully, Soergel
bimodules are very accessible, and various tools exist to study
them in a very concrete way. Recently, Geordie Williamson and I
proved Soergel's conjecture, which is the generalization of Kazhdan
and Lusztig's conjecture to arbitrary Coxeter groups, realizing
Soergel's dream. Our proof was an algebraic adaptation of de Cataldo
and Migliorini's Hodge-theoretic proof of the Decomposition Theorem
in geometry.

Our goal for this lecture series is to provide a thorough introduction
to Hecke algebras, Soergel bimodules, and the Hodge-theoretic
techniques which went into the proof of the Soergel conjecture. We
will also introduce the diagrammatic tools which are used to study
Soergel bimodules.