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
Joint work with Gastón Burrull and Paolo Sentinelli. In this paper we find the characteristic p analogue of the classical Jones-Wenzl idempotent. We hope this to be a building block for the p-canonical basis as sl_2 is a building block for the representation theory of semi-simple Lie algebras.
Joint work with David Plaza. We provide a conjecture (and the proof of the "graded degree part") that links the Hecke category in positive characteristic and a certain "blob category" that we introduce as a quotient of KLR algebras. A part of the latter category can be understood in physical terms when the characteristic is zero. If there was a physical interpretation in the positive characteristic case, it would be majestic, because one could have physical intuition in one of the most difficult problems in representation theory (understanding the p-canonical basis).
joint with Geordie Williamson, Comptes Rendus Mathematique Vol 355 (2017) Issue 8, 853-858. We prove that there are indecomposable Soergel bimodules (in type A) having negative degree endomorphisms. This is quite surprising and proves the existence of a non-perverse parity sheaf in type A.
joint with Ben Elias ; with an appendix by Ben Webster, Trans. Amer. Math. Soc. 369 (2017), 3883-3910. We find the explicit numbers for which "Soergel's conjecture in positive characteristic for Universal Coxeter systems" fail, i.e. we find explicitly the projectors to the indecomposables. These are clearly the most simple Coxeter systems (after the dihedral groups) but even in this case the situation is quite subtle. It would be amazing (for modular representation theory) to find similar results as the ones in this paper, but for Weyl groups or affine Weyl groups.
Joint with Geordie Williamson, Proc. London. Math. Soc. 109 (2014), no. 5, 1264-1280. For any Coxeter system, we establish the existence of analogues of standard and costandard objects in 2-braid groups, thus proving a conjecture that Rouquier stated in the ICM 2006. This result was a key step for the proof by Elias and Williamson of Kazhdan-Lusztig conjectures
New bases of some Hecke algebras via Soergel bimodules , Advances in Math. 228 (2011) 1043-1067. This is a first attempt to find explicitly Soergel indecomposable bimodules for extra-large Coxeter systems. This is very linked with my "Forking path conjecture" (see the paper "Gentle Introduction to Soergel bimodules" above), an extremely strange phenomenon that I would love to understand better.
Chapter 1 is essentially a version of the paper that one could call "Soergel bimodules explained by Soergel" with explanations of the obscure points. Sections 2.4 and 2.5 are original and are not included in any other paper. I give a different (and easier) proof of the fact that Rouquier complexes satisfy the braid relations.