# Unramified affine Springer fibers and isospectral Hilbert schemes

@article{Kivinen2018UnramifiedAS,
title={Unramified affine Springer fibers and isospectral Hilbert schemes},
author={Oscar Kivinen},
journal={arXiv: Algebraic Geometry},
year={2018}
}
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 $\mathrm{Sp}_\gamma\subset \mathrm{Gr}_G$, where $\gamma=at^d$, 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 $\mathrm{Sp}_\gamma$ to Haiman's work on the isospectral Hilbert scheme of points on the plane. We also explain the… Expand
6 Citations
Affine Springer Fibers, Procesi bundles, and Cherednik algebras
• Mathematics
• 2021
Let g be a semisimple Lie algebra, t its Cartan subalgebra and W the Weyl group. The goal of this paper is to prove an isomorphism between suitable completions of the equivariant Borel-Moore homologyExpand
Hilbert schemes on plane curve singularities are generalized affine Springer fibers
• Mathematics, Physics
• 2020
In this paper, we show that Hilbert schemes of planar curve singularities can be interpreted as generalized affine Springer fibers for $GL_n$. This leads to a construction of a rational CherednikExpand
Fix points and components of equivalued affine Springer fibers
For G a semisimple algebraic group, we revisit the description of the components of the affne Springer fiber given by ts, with s a regular semisimple element. We then compute the fixed points of eachExpand
A G ] 2 3 A ug 2 02 1 Algebra and geometry of link homology Lecture notes from the IHES 2021 Summer School
3 Khovanov-Rozansky homology: definitions and computations 6 3.1 Soergel bimodules and Rouquier complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Khovanov-Rozansky homology . . .Expand
A combinatorial formula for the nabla operator
• Mathematics
• 2020
We present an LLT-type formula for a general power of the nabla operator applied to the Cauchy product for the modified Macdonald polynomials, and use it to deduce a new proof of the generalizedExpand
Generalized affine Springer theory and Hilbert schemes on planar curves
• Mathematics
• 2020
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 byExpand

#### References

SHOWING 1-10 OF 77 REFERENCES
Geometric representations of graded and rational Cherednik algebras
• Mathematics
• 2014
We provide geometric constructions of modules over the graded Cherednik algebra $\mathfrak{H}^{gr}_\nu$ and the rational Cherednik algebra $\mathfrak{H}^{rat}_\nu$ attached to a simple algebraicExpand
Equivariant homology and K-theory of affine Grassmannians and Toda lattices
• Mathematics
• Compositio Mathematica
• 2005
For an almost simple complex algebraic group G with affine Grassmannian $\text{Gr}_G=G(\mathbb{C}(({\rm t})))/G(\mathbb{C}[[{\rm t}]])$, we consider the equivariant homology $H^{G(\mathbb{C}[[{\rmExpand Homology of Hilbert schemes of reducible locally planar curves Let$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]},Expand
The cohomology ring of certain compactified Jacobians
• Mathematics
• 2017
We provide an explicit presentation of the equivariant cohomology ring of the compactified Jacobian $J_{q/p}$ of the rational curve $C_{q/p}$ with planar equation $x^{q}=y^{p}$ for $(p,q)=1$. We alsoExpand
Affine Schubert calculus and double coinvariants
• Mathematics
• 2018
We first define an action of the double coinvariant algebra $DR_n$ on the homology of the affine flag variety $\widetilde{Fl}_n$ in type $A$, and use affine Schubert calculus to prove that itExpand
Rational Cherednik algebras and Hilbert schemes
• Mathematics
• 2004
Let $H_c$ be the rational Cherednik algebra of type $A_{n-1}$ with spherical subalgebra $U_c = e H_c e$. Then $U_c$ is filtered by order of differential operators, with associated graded ring Expand
Hilbert schemes and $y$-ification of Khovanov-Rozansky homology
• Mathematics
• 2017
Author(s): Gorsky, Eugene; Hogancamp, Matthew | Abstract: We define a deformation of the triply graded Khovanov-Rozansky homology of a link $L$ depending on a choice of parameters $y_c$ for eachExpand
Hilbert schemes, polygraphs and the Macdonald positivity conjecture
We study the isospectral Hilbert scheme X_n, defined as the reduced fiber product of C^2n with the Hilbert scheme H_n of points in the plane, over the symmetric power S^n C^2. We prove that X_n isExpand
Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ gauge theories, II
Consider the $3$-dimensional $\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G_c$ and its quaternionic representation $\mathbf M$. Physicists study its Coulomb branch,Expand
Science Fiction and Macdonald's Polynomials
• Mathematics
• 1998
This work studies the remarkable relationships that hold among certain m-tuples of the Garsia-Haiman modules ${\bf M}_\mu$ and corresponding elements of the Macdonald basis. We recall that \${\bfExpand