arXiv subjects

Oscar Kivinen

Publications and source records attributed to Oscar Kivinen.

12 recordsLinked to original sources

arXiv2026

Shalika germs for tamely ramified elements in $GL_n$

We prove explicit combinatorial formulas for various germ expansions of orbital integrals of tamely ramified elements in $GL_n(F)$, where $F$ is a nonarchimedean local field. The relevant combinatorics arises from the theory of the elliptic Hall algebra and the representation-theoretic knot superpolynomials defined by Cherednik--Danilenko and Morton--Samuelson. As a byproduct, we give explicit formulas for the weight polynomials of affine Springer fibers in type A and standard orbital integrals of tamely ramified regular semisimple elements. Our formulas subsume most earlier facts about the structure of Shalika germs of $GL_n$ in the literature. As a further corollary, we show that point-counts of compactified Jacobians of locally planar curves are given by non-negative integral polynomials. Our results also provide further evidence for the Oblomkov-Rasmussen-Shende conjecture relating compactified Jacobians and HOMFLY-PT invariants of algebraic knots.

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arXiv2026

On the affine Springer fibers inside the invariant center of the small quantum group

Let $\mathfrak{u}_ζ^\vee$ denote the small quantum group associated with a simple Lie algebra $\mathfrak{g}^\vee$ and a root of unity $ζ$. In [9], a geometric realization of $Z(\mathfrak{u}_ζ^\vee)^{G^\vee}$, the $G^\vee$-invariant part of the center of $\mathfrak{u}_ζ^\vee$, was proposed. We compute the dimension of the geometric subalgebra of the center and in the case where $G=SL_n$, we study a bigraded refinement of the result.

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arXiv2025

A Lie-theoretic generalization of some Hilbert schemes

We define several versions of a class of varieties $X_{\mathfrak{g}}$ attached to a complex reductive Lie algebra $\mathfrak{g}$, generalizing the Hilbert scheme of points on the plane. These include trigonometric and elliptic versions attached to the corresponding groups. We also define the corresponding isospectral varieties $Y_{\mathfrak{g}}$. We prove a Gordon-Stafford localization theorem for $X_{\mathfrak{g}}$ and the corresponding equal-parameter rational Cherednik algebras, relate these varieties to the affine Springer fiber-sheaf correspondence of arXiv:2204.00303, and discuss examples. We conjecture that the torus-fixed points of our varieties are in bijection with two-sided cells in the finite Weyl group and prove this in types $ABC$. We relate these results to known results about Calogero-Moser spaces.

math.AG↗

arXiv2025

The Hilb-vs-Quot Conjecture

Let $R$ be the complete local ring of a complex plane curve germ and $S$ its normalization. We propose a "Hilb-vs-Quot" conjecture relating the virtual weight polynomials of the Hilbert schemes of $R$ to those of the Quot schemes that parametrize $R$-submodules of $S$. By relating the Quot side to a type of compactified Picard scheme, we show that our conjecture generalizes a conjecture of Cherednik's, and that it would relate the perverse filtration on the cohomology of the Picard side to a more elementary filtration. Next, we propose a Quot version of the Oblomkov-Rasmussen-Shende conjecture, relating parabolic refinements of our Quot schemes to Khovanov-Rozansky link homology. It becomes equivalent to the original version under (refined) Hilb-vs-Quot, but can also be strengthened to incorporate polynomial actions and $y$-ification. For germs $y^n = x^d$, where $n$ is either coprime to or divides $d$, we prove the Quot version of ORS through combinatorics. When $n = 3$ and $3 \nmid d$, we deduce Hilb-vs-Quot by an asymptotic argument, and hence, establish the original ORS conjecture for these germs.

math.AG↗

arXiv2025

Discriminants and motivic integration

We study invariants of a plane cuve singularity $(f,0)$ coming from motivic integration on symmetric powers of a formal deformation of $f$. We show that a natural discriminant integral recovers the motivic classes of the principal Hilbert schemes of points on $f$, while the orbifold integral gives the plethystic exponential of the motivic Igusa zeta function of $f$. The latter result also holds in higher dimemsions. Combined with results of Gorsky and Némethi we obtain an interpretation of the discriminant integrals in terms of knot Floer homology, which is reminiscent of the relation between the cohomology of contact loci and fixed point Floer homology proven by de la Bodega and Poza.

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arXiv2025

The affine Springer fiber-sheaf correspondence

Given a semisimple element in the loop Lie algebra of a reductive group, we construct a quasi-coherent sheaf on a partial resolution of the trigonometric commuting variety of the Langlands dual group. The construction uses affine Springer theory and can be thought of as an incarnation of 3d mirror symmetry. For the group $GL_n$, the corresponding partial resolution is $\mathrm{Hilb}^n(\mathbb{C}^\times\times \mathbb{C})$. We also consider a quantization of this construction for homogeneous elements.

math.AG↗

arXiv2022

Generalized affine Springer theory and Hilbert schemes on planar curves

We show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for $GL_n$, as defined by Goresky-Kottwitz-MacPherson. Using a generalization of affine Springer theory for Braverman-Finkelberg-Nakajima's Coulomb branch algebras, we construct a rational Cherednik algebra action on the homology of the Hilbert schemes, and compute it in examples. Along the way, we generalize to the parahoric setting the recent construction of Hilburn-Kamnitzer-Weekes, which may be of independent interest. In the spherical case, we make our computations explicit through a new general localization formula for Coulomb branches. Via results of Hogancamp-Mellit, we also show the rational Cherednik algebra acts on the HOMFLY homologies of torus knots. This work was inspired in part by a construction in three-dimensional $\mathcal{N}=4$ gauge theory.

math.AG↗

arXiv2021

Algebra and geometry of link homology

These notes cover the lectures of the first named author at 2021 IHES Summer School on "Enumerative Geometry, Physics and Representation Theory" with additional details and references. They cover the definition of Khovanov-Rozansky triply graded homology, its basic properties and recent advances, as well as three algebro-geometric models for link homology: braid varieties, Hilbert schemes of singular curves and affine Springer fibers, and Hilbert schemes of points on the plane.

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arXiv2019

Unramified affine Springer fibers and isospectral Hilbert schemes

For any connected reductive group $G$ over $\mathbb{C}$, we revisit Goresky-Kottwitz-MacPherson's description of the torus equivariant Borel-Moore homology of affine Springer fibers $\mathrm{Sp}_γ\subset \mathrm{Gr}_G$, where $γ=at^d$, and $a$ is a regular semisimple element in the Lie algebra of $G$. In the case $G = GL_n$, we relate the equivariant cohomology of $\mathrm{Sp}_γ$ to Haiman's work on the isospectral Hilbert scheme of points on the plane. We also explain the connection to the HOMFLY homology of $(n, dn)$-torus links, and formulate a conjecture describing the homology of the Hilbert scheme of points on the curve $\{x^n=y^{dn}\}$.

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arXiv2018

Quadratic ideals and Rogers-Ramanujan recursions

We give an explicit recursive description of the Hilbert series and Gröbner bases for the family of quadratic ideals defining the jet schemes of a double point. We relate these recursions to the Rogers-Ramanujan identity and prove a conjecture of the second author, Oblomkov and Rasmussen.

math.AG↗

arXiv2017

Homology of Hilbert schemes of reducible locally planar curves

Let $C$ be a complex, reduced, locally planar curve. We extend the results of Rennemo arXiv:1308.4104 to reducible curves by constructing an algebra $A$ acting on $V=\bigoplus_{n\geq 0} H_*(C^{[n]}, \mathbb{Q})$, where $C^{[n]}$ is the Hilbert scheme of $n$ points on $C$. If $m$ is the number of irreducible components of $C$, we realize $A$ as a subalgebra of the Weyl algebra of $\mathbb{A}^{2m}$. We also compute the representation $V$ in the simplest reducible example of a node.

math.AG↗

arXiv2014

Some Computations for Binomial Edge Ideals and Koszul Duality

We make some observations on binomial edge ideals, with the characterization of their Koszulness as motivation. Inspired by results of Ene, Herzog and Hibi, we discuss building Koszul graphs from smaller pieces in a controlled manner. We characterize the Koszul property of cone graphs and compute the dual algebra of the quadratic algebra coming from an arbitrary binomial edge ideal. We compute the first two syzygies of the infinite minimal free resolution of the residue field over the algebra defined by a binomial edge ideal, and observe that they are always linear.

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