Ginzburg-Weinstein via Gelfand-Zeitlin

Abstract

Let U(n) be the unitary group, and u(n)* the dual of its Lie al- gebra, equipped with the Kirillov Poisson structure. In their 1983 paper, Guillemin-Sternberg introduced a densely defined Hamil- tonian action of a torus of dimension (n−1)n/2 on u(n)*, with mo- ment 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)*. 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.