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 \({{\mathbb C}^{2n}}\).