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

TL;DR

This paper explores the connection between the spherical subalgebra of the double affine Hecke algebra of type C^vee C_n and character varieties, confirming a conjecture and providing explicit integration of the trigonometric van Diejen system.

Contribution

It establishes a link between DAHA of type C^vee C_n and character varieties, confirming a prior conjecture and analyzing the phase space of the van Diejen system.

Findings

Confirmed the conjecture relating DAHA and character varieties.

Explicitly integrated the dynamics of the trigonometric van Diejen system.

Provided a Hamiltonian reduction perspective for the phase space.

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.