On the duality between the hyperbolic Sutherland and the rational Ruijs']jsenaars-Schneider models

We consider two families of commuting Hamiltonians on the cotangent bundle of the group GL(n, C), and show that upon an appropriate single symplectic reduction they descend to the spectral invariants of the hyperbolic Sutherland and of the rational Ruijsenaars-Schneider Lax matrices, respectively. The duality symplectomorphism between these two integrable models that was constructed by Ruijsenaars using direct methods can then be interpreted geometrically simply as a gauge transformation connecting two cross sections of the orbits of the reduction group.