joint with Geordie Williamson, We prove that (sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients for ANY Coxeter system and ANY choice of a parabolic subgroup, thus generalizing to the parabolic setting the result The Hodge theory of Soergel bimodules by Elias and Williamson. We also prove a monotonicity conjecture of Brenti. The new techniques were used by Williamson and Lusztig to calculate many new elements of the p-canonical basis and thus make this conjecture.
Advances in Mathematics (2015) 772-807. I prove that Lusztig's conjecture reduces to a problem about the light leaves (as defined in my first paper). Using the result in Section 4.3 of this paper (that he discovered independently) Geordie Williamson disproved Lusztig's conjecture! The counterexamples grow exponentially in the Coxeter number. Here is Geordie's paper