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. PDF The Anti-Spherical Category
This paper is the first of a series of introductory papers on the fascinating world of Soergel bimodules. It is combinatorial in nature and should be accessible to a broad audience. PDF Gentle introduction to Soergel bimodules
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 PDF Light leaves and Lusztig's conjecture
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. PDF Indecomposable Soergel bimodules for Universal Coxeter groups
Standard objects in 2-braid groups, joint with Geordie Williamson, Proc. London. Math. Soc. 109 (2014), no. 5, 1264-1280. Standard objects in 2-braid 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 PDF Standard objects in 2-braid groups
My Ph.D thesis, Paris 7 University, November 2008. Chapter 1 is a essentially a version of this paper (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.
Sur la catégorie des bimodules de Soergel, Journal of Algebra 320 (2008) 2675-2694. I introduce the light leaves basis, a basis of the Hom space between Soergel bimodules. They have been useful in the disproof of Lusztig and James conjectures, in the algebraic proof of KL conjecture, in the proof of the positivity of the coefficients of KL polynomials (and its parabolic version), in the algebraic proof of Jantzen conjecture, and in many other contexts. It has also been a key ingredient to the new approach to modular representation theory by Williamson and Riche. PDF Sur la catégorie des bimodules de Soergel