Representations of Quivers and Cherednik Algebras

We study the spherical subalgebra of the double affine Hecke algebra of type $C^\vee C_n$ and relate it, at the classical level, to a certain character variety of the Riemann sphere with four punctures that we call the Calogero-Moser space. This establishes a conjecture from Etingof-Gan-Oblomkov (2006). As a by-product, we construct a completed phase space for the trigonometric van Diejen system and explicitly integrate the dynamics. We conclude by suggesting how one could quantise the main isomorphism, and discussing some preliminary work that aims to reconcile the Poisson bracket on the Calogero-Moser space with the bracket coming via its interpretation as the moduli space of flat connections on a punctured Riemann surface by Fock-Rosly (1999).