2011/11/11 by Mark Colarusso, Colarusso, Mark, Sam Evens +1
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1111.2868
33 pages
arxiv created 2011/11/11 · openalex publication_date 2011/11/11 · arxiv updated 2011/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this expository paper, we provide an overview of the Gelfand-Zeiltin integrable system on the Lie algebra of n× n complex matrices \fgl(n,\C) introduced by Kostant and Wallach in 2006. We discuss results concerning the geometry of the set of strongly regular elements, which consists of the points where Gelfand-Zeitlin flow is Lagrangian. We use the theory of Kn=GL(n-1,\C)× GL(1,\C)-orbits on the flag variety Bn of GL(n,\C) to describe the strongly regular elements in the nilfiber of the moment map of the system. We give an overview of the general theory of orbits of a symmetric subgroup of a reductive algebraic group acting on its flag variety, and illustrate how the general theory can be applied to understand the specific example of Kn and GL(n,\C).