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DAHAs of Type $C^\vee C_n$ and Character Varieties

Oleg Chalykh, Bradley Ryan

Abstract

This paper studies the spherical subalgebra of the double affine Hecke algebra of type $C^\vee C_n$ and relates it, at the classical level $q = 1$, to a certain character variety of the four-punctured Riemann sphere. This establishes a conjecture from math.QA/0504089. As a by-product, we find a completed phase space for the trigonometric van Diejen system, explicitly integrate its dynamics and explain how it can be obtained via Hamiltonian reduction.

DAHAs of Type $C^\vee C_n$ and Character Varieties

Abstract

This paper studies the spherical subalgebra of the double affine Hecke algebra of type and relates it, at the classical level , to a certain character variety of the four-punctured Riemann sphere. This establishes a conjecture from math.QA/0504089. As a by-product, we find a completed phase space for the trigonometric van Diejen system, explicitly integrate its dynamics and explain how it can be obtained via Hamiltonian reduction.

Paper Structure

This paper contains 24 sections, 39 theorems, 123 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The DAHA
  3. The DAHA
  4. Duality
  5. The Basic Representation
  6. Character Varieties
  7. General Properties
  8. Calogero--Moser Spaces of Type $C^\vee C_n$
  9. Alternative Form
  10. Representations of Quivers and Multiplicative Quiver Varieties
  11. From Representations of the DAHA to Character Varieties
  12. The Etingof-Gan-Oblomkov Map
  13. From DAHA to Matrices
  14. The EGO Map on a Chart
  15. Coordinates on the Character Variety
  16. ...and 9 more sections

Key Result

Theorem 1.1

For $W = S_{n}$ and $c \neq 0$, we have an isomorphism between the spherical subalgebra and the algebra of regular functions on the Calogero-Moser space. Equivalently, $\mathop{\mathrm{Spec}}\nolimits(e\mathsf{H}_{0,c}e)\cong \mathcal{C}_n$ as affine algebraic varieties.

Figures (5)

  • Figure 1: The star-shaped quiver $Q$.
  • Figure 2: The quiver $Q^\infty$ corresponding to the Calogero--Moser space $\mathcal{M}_n$.
  • Figure 3: The quivers associated with Problem \ref{['prob: DS problem for X']}.
  • Figure 4: Positive roots supported at the central node.
  • Figure 5: The graph corresponding to the four-punctured sphere.

Theorems & Definitions (81)

  • Theorem 1.1: cf. etingofginzburg
  • Theorem 1.2: etingofginzburg
  • Theorem 1.3: oblomkovJun04
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.4
  • proof
  • Theorem 2.5: sahi, Duality Isomorphism
  • Definition 2.6
  • ...and 71 more