## On the affine Hecke category

Joint with Leo Patimo. We find a surprisingly beautiful basis of the Hom spaces between indecomposable Soergel bimodules for SL(3) (something that we call "indecomposable light leaves").

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# Papers and preprints

## Highlighted Papers

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On the affine Hecke category

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Gentle introduction to Soergel bimodules I: The basics

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Light leaves and Lusztig’s conjecture

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Pre-canonical bases on affine Hecke algebras

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Kazhdan-Lusztig polynomials and subexpressions

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p-Jones Wenzl idempotent

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Blob algebra approach to modular representation theory

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A non-perverse Soergel bimodule in type A

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The Anti-Spherical Category

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Indecomposable Soergel bimodules for Universal Coxeter groups

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Standard objects in 2-braid groups

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New bases of some Hecke algebras via Soergel bimodules

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Presentation of right-angled Soergel categories by generators and relations

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My Ph.D thesis

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Équivalences entre conjectures de Soergel

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Sur la catégorie des bimodules de Soergel

Joint with Leo Patimo. We find a surprisingly beautiful basis of the Hom spaces between indecomposable Soergel bimodules for SL(3) (something that we call "indecomposable light leaves").

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July 1, 2020

** 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.

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September 1, 2017

**Advances in Mathematics **(2015) 772-807.
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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January 1, 2015

Joint with Leo Patimo and David Plaza. 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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July 10, 2020

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.

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May 2, 2020

Joint work with Gastón Burrull and Paolo Sentinelli. . **Advances in Mathematics ** 352 (2019) 246-264. 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.

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February 16, 2019

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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July 24, 2018

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.

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January 1, 2016

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.

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November 11, 2017

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.

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June 1, 2017

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 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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January 1, 2014

**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.

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June 29, 2011

**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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January 27, 2009

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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November 17, 2008

**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.

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July 30, 2008

**Journal of Algebra** 320 (2008) 2675-2694.
I introduce the light leaves basis.

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July 14, 2008