• Charges via the affine Grassmannian
    We give a new construction of Lascoux-Schützenberger’s charge statistic in type A motivated by the geometry of the affine Grassmannian. Using the geometric Satake equivalence, we construct the charge by understanding how hyperbolic localization changes on a family of cocharacters in the affine Grassmannian. Although this framework works for any reductive group, in type A there are two crucial features that make this procedure behave well: the atomic decomposition of crystal, which we revise here, and a technical property of twisted Bruhat graphs.
  • Pre-canonical Bases on affine Hecke algebras (joint with Nicolás Libedinsky and David Plaza)
    We introduce some new bases of the spherical Hecke algebras of affine Weyl groups, called the pre-canonical bases, which interpolate between the standard and KL bases. We conjecture that in type A they have remarkable positive properties, namely that the $(i+1)$-th pre-canonical basis has positive coefficients in the $i$th basis. This would reduce the problem of finding KL polynomials in a sequence of easier problems. We also compute the pre-canonical bases explicitly up to rank 4.
  • On the Affine Hecke Category (joint with Nicolás Libedinsky)
    We study the Hecke category for the affine Weyl group $\tilde{A}_2$ in characteristic 0. We determine the Kazhdan-Lusztig basis and give a recursive formula to compute the indempotent corresponding to every indecomposable object. Then we produce a basis of the morphisms between indecomposable objects, that we call "indecomposable light leaves." As a corollary, we also obtain a combinatorial objects counting KL polynomials
  • On the Induction of p-Cells (joint with Thorge Jensen)
    We generalize to p-cells (i.e. cells constructed using the p-canonical basis) a result of Geck, called parabolic induction of cells. In terms of cell modules, this translates to the fact that cell modulex of a right p-cell in a standard parabolic subgroup decompose as a direct sum of cell modules after induction to the bigger Coxeter group. A crucial point in the proof is a new positivity result of the p-canonical basis with respect of the hybrid p-canonical basis, which we introduce here.
  • Bases of the Intersection Cohomology of Grassmannian Schubert Varieties
    There are several explicit combinatorial formula that describe Kazhdan-Lusztig polynomials for Grasmannians, one of which, described by Shigechi and Zinn-Justin, involves counting certain Dyck partitions. We ''lift'' this combinatorial formula to the intersection cohomology of Schubert varieties in Grassmannians and as a consequence obtain many distinguished bases of the intersection cohomology which extend the classical Schubert basis of the ordinary cohomology. These bases are an useful tool when studying the intersection cohomology of Schubert varieties as a module: for example, we can write in these bases a generalization of the Pieri's formula
  • Singular Rouquier Complexes
    This is a revised version of Chapter 4 of my PhD thesis. Rouquier complexes are an important tool for studying Soergel bimodules and the Hecke category. Here we generalise the construction of Rouquier complexes to the setting of singular Soergel bimodules. We show that they retain many of the properties of ordinary Rouquier complexes: they are ∆ -split, they satisfy a vanishing formula and, when Soergel's conjecture holds they are perverse. As an application, we use singular Rouquier complexes to establish Hodge theory for singular Soergel bimodules.
  • A combinatorial formula for the coefficient of q of Kazhdan-Lusztig polynomials
    to appear in: IMRN
    We study the coefficient of q in Kazhdan-Lusztig polynomials. Using moment graphs, for finite groups of type ADE we prove that this coefficient can be computed via a formula which only depends on the poset structure of the Bruhat interval.
  • The Néron-Severi Lie Algebra of a Soergel module
    Transform. Groups 23 (2018), no. 4
    Following Looijenga and Lunts, we introduce for any Soergel module a semisimple Lie algebra generated by all the Lefschetz operators and their adjoints. This algebra, called the Néron-Severi Lie algebra, can be used to provide a simple proof of the well-known fact that a Schubert variety is rationally smooth if and only if its Betti numbers satisfy Poincaré duality. We also determine a large set of elements, for finite Coxeter group, where the Néron-Severi Lie algebra coincides with the full Lie algebra of endomorphism preserving the intersection form.
  • The Hard Lefschetz Theorem in positive characteristic for the Flag Varieties
    Int. Math. Res. Not. IMRN 2018, no. 18
    We show that the Hard Lefschetz theorem holds for the flag variety of some reductive group in characteristic p, if p is larger than the number of positive roots. The converse also holds, a part from 3 exceptional cases. One can also deduce an algebraic proof of the Hard Lefschetz theorem in the already known characteristic 0 case.

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