The topology of Gelfand-Zeitlin fibers

TL;DR

This paper provides a comprehensive topological analysis of Gelfand-Zeitlin fibers on unitary and orthogonal coadjoint orbits, including explicit descriptions, homotopy groups, and cohomology rings, unifying and extending previous results.

Contribution

It introduces a unifying framework for the topology of Gelfand-Zeitlin fibers, including explicit fiber descriptions, a local normal form, and new computations of homotopy and cohomology.

Findings

Explicit diffeomorphism types of fibers in many cases

First computation of homotopy groups for orthogonal fibers

Cohomology rings of fibers computed for the first time

Abstract

We prove several new results about the topology of fibers of Gelfand--Zeitlin systems on unitary and orthogonal coadjoint orbits, at the same time finding a unifying framework recovering and shedding light on essentially all known results. We find completely explicit descriptions of the diffeomorphism type of the fiber in many instances a direct factor decomposition of the fiber, and a torus factor corresponding to the action given by the Thimm trick. The new description also gives us a weak local normal form for a coadjoint orbit, which we use to define a topological toric degeneration, new in the orthogonal case. We also compute the first three homotopy groups (new in the orthogonal case) and cohomology rings of a fiber (new in both cases). All these descriptions can be read in a straightforward manner from the combinatorics of the associated Gelfand--Zeitlin pattern.