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Quantization of the minimal nilpotent orbits and the quantum Hikita conjecture

Xiaojun Chen, Weiqiang He, Sirui Yu

Abstract

We show that the specialized quantum D-module of the equivariant quantum cohomology ring of the minimal resolution of an ADE singularity is isomorphic to the D-module of graded traces on the minimal nilpotent orbit in the Lie algebra of the same type. This generalizes a recent result of Shlykov [Hikita conjecture for the minimal nilpotent orbit, to appear in Proc. AMS, https://doi.org/10.1090/proc/15281] and hence verifies in this case the quantum version of Hikita's conjecture, proposed by Kamnitzer, McBreen and Proudfoot [The quantum Hikita conjecture, Advances in Mathematics 390 (2021) 107947]. We also show analogous isomorphisms for singularities of BCFG type.

Quantization of the minimal nilpotent orbits and the quantum Hikita conjecture

Abstract

We show that the specialized quantum D-module of the equivariant quantum cohomology ring of the minimal resolution of an ADE singularity is isomorphic to the D-module of graded traces on the minimal nilpotent orbit in the Lie algebra of the same type. This generalizes a recent result of Shlykov [Hikita conjecture for the minimal nilpotent orbit, to appear in Proc. AMS, https://doi.org/10.1090/proc/15281] and hence verifies in this case the quantum version of Hikita's conjecture, proposed by Kamnitzer, McBreen and Proudfoot [The quantum Hikita conjecture, Advances in Mathematics 390 (2021) 107947]. We also show analogous isomorphisms for singularities of BCFG type.
Paper Structure (29 sections, 61 theorems, 175 equations)

This paper contains 29 sections, 61 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Equivariant quantum cohomology of ADE resolutions
  3. Kleinian singularities
  4. The $\mathbb C^\times$-equivariant quantum cohomology
  5. The $(\mathbb C^\times)^2$-equivariant quantum cohomology of $A_n$ resolutions
  6. Equivariant cohomology
  7. Equivariant quantum cohomology
  8. Quantization of the minimal nilpotent orbits
  9. The coordinate ring of minimal orbits
  10. Quantization of the minimal nilpotent orbits
  11. Joseph's quantization of the minimal nilpotent orbits
  12. Garfinkle's construction of the Joseph ideals
  13. Joseph ideal for type A Lie algebras
  14. Joseph ideal for type D and E Lie algebras
  15. The $B$-algebras
  16. ...and 14 more sections

Key Result

Theorem 1.2

Let $\mathfrak g$ be a complex semisimple Lie algebra of ADE type, and let $\overline{\mathcal{O}}_{min}$ be the closure of the minimal nilpotent orbit in $\mathfrak g$. Let $\widetilde{{\mathbb C^2}/\Gamma}$ be the minimal resolution of the singularity of the same type. Then the quantum Hikita conj where $\mathrm{QH}^\bullet(-)$ is the specialized quantum D-module, and $Q(\mathscr A(-))$ is the D

Theorems & Definitions (123)

  • Conjecture 1.1: Hikita Hi
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: Slodowy slice S
  • Theorem 2.2: Brieskorn Br and Slodowy S
  • Theorem 2.3: BG
  • Proposition 2.4
  • Definition 2.5
  • ...and 113 more