The Gelfand-Zeitlin system is a completely integrable system on the Poisson manifold \(\mathfrak{u}(n)^*\), which has an elegant and elementary description due to Guillemin and Sternberg. It is an open problem to construct analogous integrable systems on \(\mathfrak{k}^*\), for other compact Lie algebras \(\mathfrak{k}\). I will present progress towards such a construction.

The moment map image of the Gelfand-Zeitlin system is a polyhedral cone, and integral points of this cone parametrize canonical bases for irreducible representations of \(\mathfrak{u}(n)\). Our approach is to consider analogous polyhedral cones, described by Berenstein, Kazhdan, and Zelevinsky, which parametrize canonical bases for irreducible representations of \(\mathfrak{k}\) of other type.

This is based on joint works with Alekseev, Berenstein, Lane, and Li.