Vol. 309, No. 2, 2020

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

Jeremy Lane

Vol. 309 (2020), No. 2, 401–419
Abstract

Actions of U(n) on U(n + 1) coadjoint orbits via embeddings of U(n) into U(n + 1) 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(n + 1) 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
Authors
Jeremy Lane
Department of Mathematics
McMaster University
Hamilton, ON
Canada