Lecture 11: Soergel Bimodules

In this lecture we continue to study the category O0 and explain some ideas towards the proof of the Kazhdan-Lusztig conjecture. We start by introducing projective functors Pi : O0 → O0 that act by w 7→ w(1 + si) on K0(O0). Using these functors we produce a projective generator of O0. In Section 2 we explain some of the work of Soergel that ultimately was used by Elias and Williamson to give a relatively elementary proof of the Kazhdan-Lusztig conjecture. In order to relate the category O0 to the Hecke algebra Hq(W ) one needs to produce a graded lift of that category. In order to do that, Soergel constructed a functor O0 → C[h] -mod, where C[h] is the so called coinvariant algebra. He proved that this functor is fully faithful on the projective objects and has described the image of a projective generator that turns out to be a graded module. This gives rise to a graded lift of O0. Also these results of Soergel lead to the notion of Soergel (bi)modules that are certain (bi)modules over C[h]. They are of great importance for modern Representation theory. We finish by briefly describing some related constructions: Kazhdan-Lusztig bases for Hecke algebras with unequal parameters and multiplicities for rational representations of semisimple algebraic groups in positive characteristic.