Abstract

Gelfand–Zeitlin systems are a well-known family of examples in symplectic geometry, singular Lagrangian torus fibrations whose total spaces are coadjoint orbits of an action of a unitary or special orthogonal group and whose base spaces are certain convex polytopes. They are easily defined in terms of matrices and their truncations, but do not fit into the familiar framework of integrable systems with nondegenerate singularities, and hence are studied as a sort of edge case.

It is known that the fibers of these systems are determined as iterated pullbacks by the combinatorics of joint eigenvalues of systems of truncated matrices, but the resulting expressions can be rather inexplicit. We provide a new interpretation of Gelfand–Zeitlin fibers as balanced products of Lie groups (or biquotients), and pursue these viewpoints to a determination of their cohomology rings and low-dimensional homotopy groups which can be read transparently off of the combinatorics.

This all represents joint work with Jeremy Lane.