Coadjoint orbitopes

TL;DR

This paper investigates the geometric and face structure of coadjoint orbitopes, revealing that all faces are exposed and can be characterized via momentum maps and root data, with connections to complex geometry.

Contribution

It provides a complete description of the face structure of coadjoint orbitopes, showing all faces are exposed and classifiable through momentum maps and root data.

Findings

All faces of coadjoint orbitopes are exposed.

Faces are determined by the momentum polytope and root data.

Extreme sets of faces correspond to closed orbits of parabolic subgroups.

Abstract

We study coadjoint orbitopes, i.e. convex hulls of coadjoint orbits of a compact Lie group. We show that all the faces of such an orbitope are exposed. The face structure is studied by means of the momentum map and it is shown that every face is again a coadjoint orbitope. Up to conjugation the faces are completely determined by the momentum polytope and can be described in a simple way in terms of root data. Finally we consider the complex geometry of the coadjoint orbit and we prove that the submanifolds of the orbit that are extreme sets of a face are exactly the closed orbits of parabolic subgroups.