Let U(n) be the unitary group, and u(n)¤ the dual of its Lie algebra, equipped with the Kirillov Poisson structure. In their 1983
paper, Guillemin-Sternberg introduced a densely defined Hamiltonian action of a torus of dimension (n−1)n/2 on u(n)¤, with moment map given by the Gelfand-Zeitlin coordinates. A few years later, Flaschka-Ratiu described a similar, ‘multiplicative’ GelfandZeitlin system for the Poisson Lie group U(n)¤.
By the Ginzburg-Weinstein theorem, U(n)¤ is isomorphic to u(n)¤ as a Poisson manifold. Flaschka-Ratiu conjectured that
one can choose the Ginzburg-Weinstein diffeomorphism in such a way that it intertwines the linear and nonlinear Gelfand-Zeitlin
systems. Our main result gives a proof of this conjecture, and produces a canonical Ginzburg-Weinstein diffeomorphism.