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

TL;DR

This paper demonstrates that the Ginzburg-Weinstein diffeomorphism for unitary Lie algebras admits a tropical limit revealing the Gelfand-Zeitlin integrable system, linking symplectic geometry, tropical geometry, and integrable systems.

Contribution

It introduces a tropical limit of the Ginzburg-Weinstein diffeomorphism, connecting it to the Gelfand-Zeitlin system and providing new insights into the structure of Lagrangian tori.

Findings

The tropical limit map's target is a cone times a torus with an integrable system.

Pull-back of action-angle coordinates recovers the Gelfand-Zeitlin system.

Lagrangian tori intersect totally positive matrices for large action coordinates.

Abstract

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 $\mathfrak{u}(n)^*$. 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 $\mathfrak{u}(n)^*$ 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.