Singular fibers of the Gelfand--Cetlin system on $\mathfrak{u}(n)^*$

TL;DR

This paper characterizes all singular fibers of the Gelfand--Cetlin system on unitary group orbits, revealing their structure as smooth isotropic submanifolds, often homogeneous spaces or Lie groups, with a combinatorial dimension formula.

Contribution

It provides a detailed geometric and combinatorial analysis of singular fibers, including their classification, structure, and explicit descriptions, using novel methods involving Lie groupoids and complex ellipsoids.

Findings

Singular fibers are smooth isotropic submanifolds diffeomorphic to quotients of compact Lie groups.

Many singular fibers are homogeneous spaces or diffeomorphic to compact Lie groups.

A combinatorial formula for the dimensions of all singular fibers is established.

Abstract

In this paper, we show that every singular fiber of the Gelfand--Cetlin system on coadjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a $2$-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibers can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibers, and give a detailed description of these singular fibers in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibers are degenerate for the Gelfand--Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids, and also general ideas coming from the theory of singularities of integrable Hamiltonian systems.