DAHAs of Type $C^{\vee }{C}_n$ and Character Varieties

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.