Ruijsenaars wavefunctions as modular group matrix coefficients

We give a description of the Halln\"as--Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element $S\in SL(2,\mathbb{Z})$ acting on the Hilbert space of $GL(2)$ quantum Teichm\"uller theory on the punctured torus. The $GL(2)$ Macdonald polynomials are then obtained as special values of the analytic continuation of these matrix coefficients. The main tool used in the proof is the cluster structure on the moduli space of framed $GL(2)$-local systems on the punctured torus, and an $SL(2,\mathbb{Z})$-equivariant embedding of the $GL(2)$ spherical DAHA into the quantized coordinate ring of the corresponding cluster Poisson variety.