DAHAs and character varieties | Integrable System Lunch Seminar

Abstract

Consider the spherical subalgebra of the double affine Hecke algebra of type C^lvee C n. ltdepends on the quantum parameter q and five couplings t=(t 0, t 1, t 2, t 3, t 4). lt is known thatfor g=1 this algebra becomes commutative, so one may ask for its geometric interpretation. Weshow that it is isomorphic to the ring of functions on a certain character variety of a 4-puncturedRiemann sphere. This proves a conjecture of Etingof-Gan-Oblomkov. As a by-product, we establishthat this character variety provides a completed phase space of the classical Koornwinder-vanDiejen particle system, and explicitly integrate its dynamics.This is joint work with Bradley Ryan(Leeds).