Accepted for publication in Journal of the Indian Institute of Science. It is an introduction/survey of representation theory with lots of humor, open questions, and extremely smart insights :)
Joint with Gastón Burrull and David Plaza, to appear in IMRN. We prove the combinatorial invariance conjecture (by G. Lusztig and M. Dyer in the eighties) for the affine A2. This is the first infinite group with non-trivial Kazhdan-Lusztig polynomials where this fascinating conjecture is proved.
Joint with Leo Patimo and David Plaza, Advances in Mathematics 399 (2022). For any affine Weyl group, we introduce the pre-canonical bases, a set of bases of the spherical Hecke algebra that interpolates between the standard basis and the canonical basis. Thus we divide the hard problem of calculating Kazhdan-Lusztig polynomials (or q-analogues of weight multiplicities) into a finite number of much easier problems.
joint with Geordie Williamson, Advances in Mathematics 405 (2022). 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 central result of 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 the Billiards conjecture. Along the way, we introduce the anti-spherical light leaves.
Joint with Leo Patimo, submitted.. We find a surprisingly beautiful basis of the Hom spaces between indecomposable Soergel bimodules for SL(3) (something that we call indecomposable light leaves.
Joint with Geordie Williamson, Journal of Algebra 568 (2021) 181-192. When Soergel's conjecture is satisfied, we produce (finally!) the canonical light leaves, that do not depend on choices. This gives a new approach towards finding a combinatorial interpretation of Kazhdan-Lusztig polynomials.
Joint work with David Plaza. Proc. Lond. Math. Soc. Vol. 121 (2020) Issue3, 656-701. We conjecture (and prove the "graded degree part") an equivalence between the type A affine Hecke category in positive characteristic and a certain blob category that we introduce as a quotient of KLR algebras. This conjecture has been proved recently in an amazing paper by Chris Bowman, Anton Cox, Amit Hazi!! It opens lots of questions...
Joint work with Gastón Burrull and Paolo Sentinelli. Advances in Mathematics 352 (2019) 246-264. In this paper we introduce the p-Jones Wenzl idempotent, a 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.
Sao Paulo Journal of Mathematical Sciences, 13(2) (2019), 499-538. 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. We introduce the Forking path 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 introduce the multicolored Temperley-Lieb 2-category. Using it, we find the explicit numbers for which "Soergel's conjecture in positive characteristic for Universal Coxeter systems" fail.
Advances in Mathematics (2015) 772-807. I introduce the double leaves basis and with it I prove that Lusztig's conjecture reduces to a problem about the light leaves. Using the result in Section 4.3 of this paper Geordie Williamson disproved Lusztig's conjecture! The counterexamples grow exponentially in the Coxeter number. Here is Geordie's paper
Joint with Geordie Williamson, Proc. London. Math. Soc. 109 (2014), no. 5, 1264-1280. We introduce the concept of Δ-exact complexes for any Coxeter system. With that, we establish the existence of analogues of standard and costandard objects in 2-braid groups, thus proving the 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
Advances in Math. 228 (2011) 1043-1067. I introduce a new set of bases for Hecke algebras related to extra-large Coxeter groups, coming from the theory of Soergel bimodules. If my "Forking path conjecture" (see the paper "Gentle Introduction to Soergel bimodules" above) is correct, the same kind of bases would exist for the symmetric group. I believe that they have a deep meaning related to the Hecke category and the p-canonical basis.
J. Pure Appl. Algebra 214 (2010), no. 12, 2265-2278. This was the first time that a "presentation of Soergel bimodules by generators and relations" was attempted. This revolutionary idea (explained to me by Rouquier) was the key of all the impressive subsequent development of the theory.
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.
Journal of Algebra, 320 (2008) 2695-2705. I prove that in Soergel's conjecture it is equivalent to use the "easy" geometric representation or the "difficult" reflection faithful representation used before.
Journal of Algebra 320 (2008) 2675-2694. I introduce the light leaves basis.