Pre-canonical bases on affine Hecke algebras

Joint with Leo Patimo and David Plaza. Submitted to Advances in Mathematics.  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.

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p-Jones Wenzl idempotent

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.

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Blob algebra approach to modular representation theory

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...

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The Anti-Spherical Category

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 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.

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Gentle introduction to Soergel bimodules I: The basics

 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.

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Light leaves and Lusztig’s conjecture

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  

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Standard objects in 2-braid groups

Joint with Geordie WilliamsonProc. 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

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New bases of some Hecke algebras via Soergel bimodules 

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.

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Presentation of right-angled Soergel categories by generators and relations

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.

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My Ph.D thesis

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.

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