Gelfand–Zeitlin actions and rational maps

Abstract

We observe that the analogue of the Gelfand–Zeitlin action on ${{\mathfrak g \mathfrak l}(n,{\mathbb C})}$, which exists on any symplectic manifold M with an Hamiltonian action of ${GL(n,{\mathbb C})}$, has a natural interpretation as a residual action, after we identify M with a symplectic quotient of ${M\times \prod_{i=1}^n T^\ast GL(i,{\mathbb C})}$. We also show that the Gelfand–Zeitlin actions on ${T^\ast GL(n,{\mathbb C})}$ and on the regular part of ${{\mathfrak g \mathfrak l}(n,{\mathbb C})}$ can be identified with natural Hamiltonian actions on spaces of rational maps into full flag manifolds, while the Gelfand–Zeitlin action on the whole ${{\mathfrak g \mathfrak l}(n,{\mathbb C})}$ corresponds to a natural action on a space of rational maps into the manifold of half-full flags in ${{\mathbb C}^{2n}}$.