Reduction of a bi-Hamiltonian hierarchy on $T^*\mathrm{U}(n)$ to spin Ruijsenaars--Sutherland models

We first exhibit two compatible Poisson structures on the cotangent bundle of the unitary group $\mathrm{U}(n)$ in such a way that the invariant functions of the $\mathfrak{u}(n)^*$-valued momenta generate a bi-Hamiltonian hierarchy. One of the Poisson structures is the canonical one and the other one arises from embedding the Heisenberg double of the Poisson-Lie group $\mathrm{U}(n)$ into $T^*\mathrm{U}(n)$, and subsequently extending the embedded Poisson structure to the full cotangent bundle. We then apply Poisson reduction to the bi-Hamiltonian hierarchy on $T^*\mathrm{U}(n)$ using the conjugation action of $\mathrm{U}(n)$, for which the ring of invariant functions is closed under both Poisson brackets. We demonstrate that the reduced hierarchy belongs to the overlap of well-known trigonometric spin Sutherland and spin Ruijsenaars--Schneider type integrable many-body models, which receive a bi-Hamiltonian interpretation via our treatment.