Integrable Multi-Hamiltonian Systems from Reduction of an Extended Quasi-Poisson Double of \({\text {U}}(n)\)

We construct a master dynamical system on a \({\text {U}}(n)\) quasi-Poisson manifold, \({\mathcal {M}}_d\), built from the double \({\text {U}}(n) \times {\text {U}}(n)\) and \(d\ge 2\) open balls in \(\mathbb {C}^n\), whose quasi-Poisson structures are obtained from \(T^* \mathbb {R}^n\) by exponentiation. A pencil of quasi-Poisson bivectors \(P_{\underline{z}}\) is defined on \({\mathcal {M}}_d\) that depends on \(d(d-1)/2\) arbitrary real parameters and gives rise to pairwise compatible Poisson brackets on the \({\text {U}}(n)\)-invariant functions. The master system on \({\mathcal {M}}_d\) is a quasi-Poisson analogue of the degenerate integrable system of free motion on the extended cotangent bundle \(T^*\!{\text {U}}(n) \times \mathbb {C}^{n\times d}\). Its commuting Hamiltonians are pullbacks of the class functions on one of the \({\text {U}}(n)\) factors. We prove that the master system descends to a degenerate integrable system on a dense open subset of the smooth component of the quotient space \({\mathcal {M}}_d/{\text {U}}(n)\) associated with the principal orbit type. Any reduced Hamiltonian arising from a class function generates the same flow via any of the compatible Poisson structures stemming from the bivectors \(P_{\underline{z}}\). The restrictions of the reduced system on minimal symplectic leaves parameterized by generic elements of the center of \({\text {U}}(n)\) provide a new real form of the complex, trigonometric spin Ruijsenaars–Schneider model of Krichever and Zabrodin. This generalizes the derivation of the compactified trigonometric RS model found previously in the \(d=1\) case.