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The affine Springer fiber-sheaf correspondence

Eugene Gorsky, Oscar Kivinen, Alexei Oblomkov

TL;DR

This work constructs a geometric bridge between affine Springer fibers and the representation theory of trigonometric Cherednik algebras via a partial resolution of the trigonometric commuting variety for the Langlands dual group. By leveraging Coulomb branch technology and ${\mathbb Z}$-algebras, it realizes modules built from sequences of affine Springer fibers as graded modules over a Cherednik-based algebra, encoding homological information of the fibers into quasi-coherent sheaves ${\mathcal F}_{\gamma}^{K_{\gamma}}$ on the partial resolution ${\widetilde{\mathfrak{C}}_{G^{\vee}}}$. The paper also connects to 3d mirror symmetry, knot invariants, and Hilbert schemes, providing explicit descriptions in GL$_n$ via punctual Hilbert schemes and the Procesi bundle. It proves shift isomorphisms for spherical/antispherical subalgebras, identifies the commutative limit with isospectral commuting varieties, and develops a generalized affine Springer theory with a robust geometric ${\mathbb Z}$-algebra framework, culminating in finite-generation conjectures and illustrative examples. Overall, the work offers a unified geometric and algebraic paradigm linking affine Springer theory, Cherednik algebras, and Coulomb branches, with implications for representation theory and low-dimensional topology.

Abstract

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.

The affine Springer fiber-sheaf correspondence

TL;DR

This work constructs a geometric bridge between affine Springer fibers and the representation theory of trigonometric Cherednik algebras via a partial resolution of the trigonometric commuting variety for the Langlands dual group. By leveraging Coulomb branch technology and -algebras, it realizes modules built from sequences of affine Springer fibers as graded modules over a Cherednik-based algebra, encoding homological information of the fibers into quasi-coherent sheaves on the partial resolution . The paper also connects to 3d mirror symmetry, knot invariants, and Hilbert schemes, providing explicit descriptions in GL via punctual Hilbert schemes and the Procesi bundle. It proves shift isomorphisms for spherical/antispherical subalgebras, identifies the commutative limit with isospectral commuting varieties, and develops a generalized affine Springer theory with a robust geometric -algebra framework, culminating in finite-generation conjectures and illustrative examples. Overall, the work offers a unified geometric and algebraic paradigm linking affine Springer theory, Cherednik algebras, and Coulomb branches, with implications for representation theory and low-dimensional topology.

Abstract

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 , the corresponding partial resolution is . We also consider a quantization of this construction for homogeneous elements.
Paper Structure (48 sections, 86 theorems, 246 equations)

This paper contains 48 sections, 86 theorems, 246 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. 3d Mirror Symmetry
  4. Conjectures
  5. Relation to conjectures stemming from knot theory
  6. Outline
  7. Outline of the argument
  8. Outline of the paper
  9. Affine Springer fibers
  10. Definitions
  11. The Springer action
  12. Extended symmetry
  13. The lattice action
  14. Equivariant versions, endoscopy
  15. The commuting variety
  16. ...and 33 more sections

Key Result

Theorem 1.1

Let $\gamma\in \mathfrak{g}(\mathcal{K})$ be a semisimple element and $G_\gamma=C_{G(\mathcal{K})}(\gamma)$ its centralizer in $G(\mathcal{K})$. The group $G_\gamma$ admits a Néron model $J_\gamma/\mathcal{O}$. Then for every subgroup $K_\gamma\subseteq J_\gamma(\mathcal{O})$, there exists a quasi-c Moreover, the sheaf $\mathcal{F}_{\gamma}^{K_{\gamma}}$ is equivariant with respect to the action o

Theorems & Definitions (211)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Proposition \ref{['prop:punctualexample']}
  • Theorem 1.5: Proposition \ref{['prop:procesi']}
  • Remark 1.6
  • Conjecture 1.7
  • Remark 1.8
  • Conjecture 1.9
  • Conjecture 1.10
  • ...and 201 more