Ginzburg-Weinstein via Gelfand-Zeitlin

Let U(n) be the unitary group, and $u(n{)}^{\ast }$ 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 $(n-1)n\mathrm{/}2$ on $u(n{)}^{\ast }$, with moment map given by the Gelfand-Zeitlin coordinates. A few years later, Flaschka-Ratiu described a similar, `multiplicative’ Gelfand-Zeitlin system for the Poisson Lie group $U(n{)}^{\ast }$. By the Ginzburg-Weinstein theorem, $U(n{)}^{\ast }$ is isomorphic to $u(n{)}^{\ast }$ 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.