Gelfand-Zeitlin actions and rational maps

Details Gelfand-Zeitlin actions and rational maps Gelfand-Zeitlin actions and rational maps

2006-12-13

arxiv.org/abs/math/0612365v2

Mathematics

Symplectic Geometry

24 pages; v.2 clarifies a confusing statement in the introduction

Scientific paper

We observe that the analogue of the Gelfand-Zeitlin action on gl(n,C), which exists on any symplectic manifold M with an Hamiltonian action of GL(n,C), has a natural interpretation as a residual action, after we identify M with a symplectic quotient of the product of M with T*GL(n-1,C)x...xT*GL(1,C). We also show that the Gelfand-Zeitlin actions on T*GL(n,C) and on the regular part of gl(n,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 gl(n,C) corresponds to a natural action on a space of rational maps into the manifold of half-full flags in C^{2n}.

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