We deal with reductions of integrable `master systems' living on the `classical doubles' of any semisimple, connected and simply connected compact Lie group G. The doubles in question are the cotangent bundle, the Heisenberg double and the internally fused quasi-Poisson double, each of which carries two natural integrable systems. In the cotangent bundle case, one of the integrable systems is generated by the class functions of G and the other one by the invariant functions of its Lie algebra. The reduction is defined by taking quotient by the cotangent lift of the conjugation action of G on itself, and this naturally generalizes to the other two doubles. The quotient space of the internally fused double represents the moduli space of flat principal G-connections on the torus with a hole. We explain that degenerate integrability of the master systems is inherited on the smooth component of the Poisson quotient corresponding to the principal orbit type for the pertinent G-action, and present explicit formulas for the reduced Poisson structure and equations of motion in terms of dynamical r-matrices after further restriction to a dense open subset.
Lecture 1. The integrable master systems on the classical doubles and the definition of their reductions. The warm up case of the cotangent bundle.
Lecture 2. Degenerate integrability on the Poisson quotient of the Heisenberg double corresponding to the principal orbit type and the interpretation of the reduced systems as Ruijsenaars--Schneider (alias relativistic Calogero--Moser) type many--body systems extended by `spin' degrees of freedom.
Lecture 3. The case of the quasi-Poisson double. Specific examples on small symplectic leaves for G=SU(n):
compact counterparts of the trigonometric Ruijsenaars--Schneider system.