Local normal forms for multiplicity free $U(n)$ actions on coadjoint orbits

Jeremy Lane

Vol. 309 (2020), No. 2, 401–419

DOI: 10.2140/pjm.2020.309.401

Abstract

Actions of $U\left(n\right)$ on $U\left(n+1\right)$ coadjoint orbits via embeddings of $U\left(n\right)$ into $U\left(n+1\right)$ are an important family of examples of multiplicity free spaces. They are related to Gelfand–Zeitlin completely integrable systems and multiplicity free branching rules in representation theory. This paper computes the Hamiltonian local normal forms of all such actions, at arbitrary points, in arbitrary $U\left(n+1\right)$ coadjoint orbits. The results are described using combinatorics of interlacing patterns; gadgets that describe the associated Kirwan polytopes.

Keywords

coadjoint orbits, multiplicity free spaces, local normal form, Gelfand–Zeitlin, integrable systems

Mathematical Subject Classification 2010

Primary: 53D20

Secondary: 14M27, 37J35

Milestones

Received: 21 February 2020

Revised: 1 October 2020

Accepted: 8 October 2020

Published: 14 January 2021