Department of Mathematics - Seminar on Pure Mathematics - Soergel Bimodules and Kazhdan-Lusztig Polynomials

Let g be a semisimple Lie algebra. The Kazhdan-Lusztig conjecture expresses the multiplicity of a simple g-module in a Verma module in terms of the Kazhdan-Lusztig polynomials. These polynomials are encoded in the combinatorics of the Hecke algebra of the Weyl group of g, but the Hecke algebra is invisible from the representations of g. This conjecture was proved first by Beilinson-Bernstein and Brylinski-Kashiwara using geometry. More recently Soergel found a more algebraic approach, using so-called Soergel bimodules. In this (series of) talk(s) we briefly introduce the Kazhdan-Lusztig conjecture, and Soergel's approach to this conjecture.