arXiv subjects

Eric Vasserot

Publications and source records attributed to Eric Vasserot.

At least 19 recordsLinked to original sources

arXiv2026

Hecke operators on symplectic surfaces and $\chi$-independence

We prove Toda's chi-independence conjecture for the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces, relative to the Chow variety. We also identify the BPS Lie algebra associated with one-dimensional Mukai vectors with the subspace of tautological classes, giving an extension of Markman's tautological generation theorem from primitive to arbitrary Mukai vectors. The main structure input is a bialgebra structure on the cohomological Hall algebra of coherent sheaves on a quasi-projective symplectic variety S. The coproduct is obtained, by dimensional reduction, from a factorization coproduct for 3d cohomological Hall algebras, and gives rise to a global BPS Lie algebra attached to the stack of coherent sheaves on S. The link between this structure and the applications to chi-independence and tautological generation is provided by Hecke operators on BPS cohomology, which modify one-dimensional sheaves by zero-dimensional quotients. To make this construction work, we prove that there is an identification between the affinized BPS cohomology of the semistable locus and the primitive part of the coproduct on the entire moduli stack

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arXiv2026

Cohomological Hall algebras of one-dimensional sheaves on surfaces and Yangians

This paper provides the first algebraic characterization of an algebra of cohomological Hecke operators associated with modifications of coherent sheaves on a smooth surface $X$ along a fixed proper curve $Z \subset X$ (possibly singular and reducible), establishing a direct connection with Yangians. It is based on the theory of equivariant nilpotent cohomological Hall algebras $\mathbf{HA}^T_{X,Z}$, developed by the same authors. More precisely, let $X$ be a resolution of a Kleinian singularity (for example, $X = T^\ast\mathbb{P}^1$) and let $Z$ be the exceptional divisor. One of the main results of this paper is an explicit isomorphism $\mathbf{HA}^T_{X,Z} \simeq \mathbb{Y}^+_\infty$, where $\mathbb{Y}^+_\infty$ is a completed, nonstandard, positive half of the affine Yangian $\mathbb{Y}(\mathfrak{g})$ of the corresponding affine ADE Lie algebra $\mathfrak{g}$. Furthermore, the generators of $\mathbf{HA}^T_{X,Z}$--given by fundamental classes of substacks of zero-dimensional sheaves and of pushforwards of line bundles on $Z$--are expressed explicitly in terms of Yangian generators. Our main tools, which may be of independent interest, are: (i) a `continuity' theorem describing the behavior of cohomological Hall algebras of objects in the heart of $t$-structures $\tau_n$ when the sequence $(\tau_n)_n$ converges, in an appropriate sense, to a fixed $t$-structure $\tau_\infty$; (ii) the definition of a multi-parameter Yangian $\mathbb{Y}_Q$ for an arbitrary quiver $Q$, given by generators and relations; (iii) a theorem relating the algebraic action of the braid group $B_Q$ on the Yangian $\mathbb{Y}_Q$ to the action of $B_Q$ on the equivariant 2-dimensional cohomological Hall algebra $\mathbf{HA}^T_Q$ of $Q$, where the latter can be described in terms of derived reflection functors of the bounded derived category of modules over the preprojective algebra of $Q$.

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arXiv2025

On the geometric Satake equivalence for Kac-Moody groups

This article establishes a geometric Satake equivalence for affine Kac-Moody groups as an equivalence of abelian semisimple categories over algebraically closed fields. We define a well-behaved category of equivariant sheaves on the double affine grassmannian \Gr_{G}, seen as a infty-stack, that we equip with a t-structure. We obtain an Braden's hyperbolic localization theorem for such a stack and prove that the constant term functor is t-exact using dimension estimates for affine MV-cycles. We then deduce the sought-for equivalence and prove that the IC-complexes match with the irreducible highest weight representations of the Langlands dual group G^{\vee}.

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arXiv2025

Representations of shifted affine quantum groups and Coulomb branches

We compare the integral category O of shifted affine quantum groups of symmetric and non symmetric types. To do so we compute the K-theoretic analog of the Coulomb branches with symmetrizers introduced by Nakajima and Weekes. This yields an equivalence of the category O with a module category over a new type of quiver Hecke algebras. At the decategorified level, this establishes a connection between the Grothendieck group of O and a finite-dimensional module over a simple Lie algebra of unfolded symmetric type. We compute this module in certain cases and give a combinatorial rule for its crystal.

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arXiv2025

Nilpotent cohomological Hall algebras of surfaces

This paper develops a framework for systematically studying cohomological "Hecke operators" associated with modifications of coherent sheaves on a smooth surface $X$ along a fixed proper curve $Z \subset X$ (possibly singular and reducible), using the theory of cohomological Hall algebras. More precisely, we construct a moduli stack of coherent sheaves $\mathbf{Coh}(\widehat{X}_Z)$ on $X$ with set-theoretic support $Z$ and we prove that its reduced is an Artin stack locally of finite type. This provides a vast generalization of the global nilpotent cone. Subsequently, we develop the needed background to define the (motivic, $T$-equivariant) cohomological Hall algebra $\mathbf{HA}^{T}_{X,Z}$ of the moduli stack of coherent sheaves on $X$ with set-theoretic support on $Z$, in the setting of a general motivic formalism $\mathbf{D}$ in the sense of Khan. The algebra $\mathbf{HA}^{\mathbf{D}, A}_{X,Z}$ is functorial with respect to closed immersions $Z' \subset Z$ and transformations of the motivic formalism $\mathbf{D}$, and only depends on the formal neighborhood $\widehat{X}_Z$ of $Z$ in $X$. In the companion paper arXiv:2603.03386, we use the nilpotent COHA $\mathbf{HA}^{T}_{X,Z}$ to answer a question previously raised in arXiv:2004.13685 about the precise relationship between the COHA of a minimal resolution of a Kleinian singularity and the corresponding preprojective COHA.

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arXiv2023

Cohomological Hall algebras of quivers and Yangians

We construct an isomorphism between the preprojective cohomological Hall algebra of an arbitrary quiver and a positive half of the corresponding Maulik-Okounkov Yangian, which intertwines the respective actions on the cohomology of the Nakajima quiver varieties. We use this to prove a conjecture of Okounkov relating the character of the Maulik-Okounkov Lie algebra to Kac polynomials.

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arXiv2023

Coherent sheaves on surfaces, COHAs and deformed $W_{1+\infty}$-algebras

We compute the cohomological Hall algebra of zero-dimensional sheaves on an arbitrary smooth quasi-projective surface $S$ with pure cohomology, deriving an explicit presentation by generators and relations. When $S$ has trivial canonical bundle, this COHA is isomorphic to the enveloping algebra of deformed trigonometric $W_{1+\infty}$-algebra associated to the ring $H^*(S,\mathbb{Q})$. We also define a double of this COHA, show that it acts on the homology of various moduli stacks of sheaves on $S$ and explicitly describe this action on the products of tautological classes. Examples include Hilbert schemes of points on surfaces, the moduli stack of Higgs bundles on a smooth projective curve and the moduli stack of $1$-dimensional sheaves on a $K3$ surface in an ample class. The double COHA is shown to contain Nakajima's Heisenberg algebra, as well as a copy of the Virasoro algebra.

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arXiv2023

Non symmetric quantum loop groups and K-theory

We realize the quantum loop groups and shifted quantum loop groups of arbitrary types, possibly non symmetric, using critical K-theory. This generalizes the Nakajima construction of symmetric quantum loop groups via quiver varieties to non symmetric types. We also give a new geometric construction of some simple modules of both quantum loop groups and shifted quantum loop groups.

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arXiv2023

Quantum loop groups and critical convolution algebras

We realize geometrically a family of simple modules of (shifted) quantum loop groups including Kirillov-Reshetikhin and prefundamental representations. To do this, we introduce a new family of algebras attached to quivers with potentials, using critical K-theory and critical Borel-Moore homology, which generalizes the convolution algebras attached to quivers defined by Nakajima.

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arXiv2022

A geometric realization of the center of the small quantum group

We propose a new geometric model for the center of the small quantum group using the cohomology of certain affine Springer fibers. More precisely, we establish an isomorphism between the equivariant cohomology of affine Spaltenstein fibers for a split element and the center of the deformed graded modules for the small quantum group. We also obtain an embedding from the invariant part of the nonequivariant cohomology under the action of the extended affine Weyl group to the invariant part of the center of the small quantum group under Langlands dual group action, which we conjecture to be an isomorphism. Finally, we give a dimension formula for the invariants on the cohomology side, thus providing a lower bound for the dimension of the center.

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arXiv2020

K-theoretic Hall algebras, quantum groups and super quantum groups

We first prove that the K-theoretic Hall algebra of a preprojective algebra of affine type is isomorphic to the positive half of a quantum toroidal quantum group. An essential step consists to deform the K-theoretic Hall algebra so that the deformation is torsion free over some polynomial subalgebra. Next, we compare super toroidal quantum groups of type A with K-theoretic Hall algebras of quivers with potential, which are defined using the Grothendieck groups of categories of singularities of some Landau-Ginzburg models.

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arXiv2019

Coherent categorification of quantum loop algebras : the $SL(2)$ case

We construct an equivalence of graded Abelian categories from a category of representations of the quiver-Hecke algebra of type $A_1^{(1)}$ to the category of equivariant perverse coherent sheaves on the nilpotent cone of type $A$. We prove that this equivalence is weakly monoidal. This gives a representation-theoretic categorification of the preprojective K-theoretic Hall algebra considered by Schiffmann-Vasserot. Using this categorification, we compare the monoidal categorification of the quantum open unipotent cells of type $A_1^{(1)}$ given by Kang-Kashiwara-Kim-Oh-Park in terms of quiver-Hecke algebras with the one given by Cautis-Williams in terms of equivariant perverse coherent sheaves on the affine Grassmannians.

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arXiv2019

The cohomological Hall algebra of a surface and factorization cohomology

For a smooth quasi-projective surface S over complex numbers we consider the Borel-Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative algebra by a version of the Hall multiplication. This multiplication involves data (virtual pullbacks) governing the derived moduli stack, i.e., the perfect obstruction theory naturally existing on the non-derived stack. By restricting to sheaves with support of given dimension, we obtain several types of Hecke operators. In particular, we study R(S), the Hecke algebra of 0-dimensional sheaves. For the flat case S=A^2, we show that R(S) is an enveloping algebra and identify it, as a vector space, with the symmetric algebra of an explicit graded vector space. For a general S we find the graded dimension of R(S), using the techniques of factorization cohomology.

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arXiv2017

On cohomological Hall algebras of quivers : generators

We study the cohomological Hall algebra Y of a lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties. We prove that Y is pure and we compute its Poincare polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov. As a corollary, we prove a variant of Okounkov's conjecture, which is a generalization of the Kac conjecture relating the constant term of Kac polynomials to root multiplicities of Kac-Moody algebras.

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arXiv2017

On cohomological Hall algebras of quivers : Yangians

We consider the cohomological Hall algebra Y of a Lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and its actions on the cohomology of quiver varieties. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov, and we construct an embedding of Y in the Yangian, intertwining the respective actions of both algebras on the cohomology of quiver varieties.

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arXiv2016

Categorical actions on unipotent representations of finite classical groups

We review the categorical representation of a Kac-Moody algebra on unipotent representations of finite unitary groups in non-defining characteristic given by the authors. Then, we extend this construction to finite reductive groups of types B or C, in non-defining characteristic. We show that the decategorified representation is isomorphic to a direct sum of level 2 Fock spaces. We deduce that the Harish-Chandra branching graph coincides with the crystal graph of these Fock spaces. We also obtain derived equivalences between blocks, yielding Broue's abelian defect group conjecture for unipotent l-blocks at linear primes.

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arXiv2015

Categorical actions on unipotent representations I. Finite unitary groups

Using Harish-Chandra induction and restriction, we construct a categorical action of a Kac-Moody algebra on the category of unipotent representations of finite unitary groups in non-defining characteristic. We show that the decategorified representation is naturally isomorphic to a direct sum of level 2 Fock spaces. From our construction we deduce that the Harish-Chandra branching graph coincide with the crystal graph of these Fock spaces, solving a recent conjecture of Gerber-Hiss-Jacon. We also obtain derived equivalences between blocks, yielding Broué's abelian defect groups conjecture for unipotent $\ell$-blocks at linear primes $\ell$.

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arXiv2013

Koszul duality of affine Kac-Moody algebras and cyclotomic rational DAHA

We give a proof of the parabolic/singular Koszul duality for the category O of affine Kac-Moody algebras. The main new tool is a relation between moment graphs and finite codimensional affine Schubert varieties. We apply this duality to q-Schur algebras and to cyclotomic rational double affine Hecke algebras.

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