The $U(n)$ Gelfand-Zeitlin system as a tropical limit of Ginzburg-Weinstein diffeomorphisms

We show that the Ginzburg-Weinstein diffeomorphism $\mathfrak{u}(n)^* \to U(n)^*$ of Alekseev-Meinrenken admits a scaling tropical limit on an open dense subset of $\mathfrak{u}(n{)}^{\ast }$. The target of the limit map is a product $\mathcal{C}\times T$, where $\mathcal{C}$ is the interior of a cone, $T$ is a torus, and $\mathcal{C}\times T$ carries an integrable system with natural action-angle coordinates. The pull-back of these coordinates to $\mathfrak{u}(n{)}^{\ast }$ recovers the Gelfand-Zeitlin integrable system of Guillemin-Sternberg. As a by-product of our proof, we show that the Lagrangian tori of the Flaschka-Ratiu integrable system on the set of upper triangular matrices meet the set of totally positive matrices for sufficiently large action coordinates.