Kazhdan-Lusztig polynomials are fascinating objects in representation theory. They have interesting connections to the geometry of Schubert varieties and to combinatorics. This course will provide an introduction to Coxeter groups, their Hecke algebras, its Kazhdan-Lusztig basis, and Kazhdan-Lusztig polynomials. The second half of the course will focus on the combinatorial invariance conjecture, which is a fascinating conjecture concerning Bruhat graphs (certain directed graphs built out of Coxeter groups), and Kazhdan-Lusztig polynomials.
Some mathematical maturity will be assumed. If you want to get started, the book “Introduction to Soergel bimodules” by Elias, Makisumi, Thiel and Williamson is a good place to start.