The topology of Gelfand–Zeitlin fibers

By definition, a Gelfand–Zeitlin fiber is an affine variety defined by fixing eigenvalues of nested minors, and hence cut out by a system of equations, but this description is complicated and often sheds little light. We give a systematic description of unitary and orthogonal Gelfand–Zeitlin fibers up to diffeomorphism as products of certain Lie double-coset manifolds (biquotients). This resolves a question concerning the diffeomorphism type originating with Guillemin and Sternberg in 1983, the answer to which was previously only known in special cases.

Our answer is given in terms of simple combinatorial rules using the GZ pattern associated to the fiber and our identification is equivariant with respect to the GZ circle actions and related easily understood circle actions on the resulting biquotient.

Key to our result is a new analysis of the bundle tower structure of Gelfand–Zeitlin fibers that describes each stage’s fiber bundle structure completely, whereas previous results merely described each stage’s fibers. This analysis also enables us to describe the local topological structure of Gelfand–Zeitlin fibers (in the spirit of topological normal forms for integrable systems) and to compute their cohomology rings.