The orbit structure of the Gelfand–Zeitlin group on n × n matrices

Mark Colarusso

Vol. 250 (2011), No. 1, 109–138

DOI: 10.2140/pjm.2011.250.109

Abstract

Recently, Kostant and Wallach constructed an action of a simply connected Lie group A ≃ ℂ^n(n−1)∕2 on gl(n) using a completely integrable system derived from the Poisson analogue of the Gelfand–Zeitlin subalgebra of the enveloping algebra. They show that A-orbits of dimension n(n− 1)∕2 form Lagrangian submanifolds of regular adjoint orbits in gl(n) and describe the orbits of A on a certain Zariski open subset of regular semisimple elements. In this paper, we describe all A-orbits of dimension n(n − 1)∕2 and thus all polarizations of regular adjoint orbits obtained using Gelfand–Zeitlin theory.

Keywords

Lie–Poisson structure, integrable system, algebraic group actions, Gelfand–Zeitlin algebra

Mathematical Subject Classification 2000

Primary: 14L30, 14R20, 37J35, 53D17

Milestones

Received: 3 October 2009

Revised: 16 December 2009

Accepted: 17 December 2009

Published: 1 March 2011