K-orbits on the flag variety and strongly regular nilpotent matrices

2011/05/09 by Mark Colarusso, Colarusso, Mark, Sam Evens +1
Mathematics · #20G20 #53D17 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Symplectic Geometry (math.SG) #math.RT #math.SG #msc:20G20 #msc:53D17

paper · pdf · doi:10.48550/arxiv.1105.1726

19 pages

arxiv created 2011/05/09 · openalex publication_date 2011/05/09 · arxiv updated 2011/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In two 2006 papers, Kostant and Wallach constructed a complexified Gelfand-Zeitlin integrable system for the Lie algebra \fgl(n+1,\C) and introduced the strongly regular elements, which are the points where the Gelfand-Zeitlin flow is Lagrangian. Later Colarusso studied the nilfibre, which consists of strongly regular elements such that each i× i submatrix in the upper left corner is nilpotent. In this paper, we prove that every Borel subalgebra contains strongly regular elements and determine the Borel subalgebras containing elements of the nilfibre by using the theory of K_i=GL(i-1,\C) × GL(1,\C)-orbits on the flag variety for \fgl(i,\C) for 2≤ i≤ n+1. As a consequence, we obtain a more precise description of the nilfibre. The K_i-orbits contributing to the nilfibre are closely related to holomorphic and anti-holomorphic discrete series for the real Lie groups U(i,1), with i ≤ n.

Related

• The Gelfand-Zeitlin integrable system and K-orbits on the flag variety
• General Linear and Symplectic Nilpotent Orbit Varieties
• The Gelfand-Zeitlin integrable system and its action on generic elements of gl(n) and so(n)
• The Complex Orthogonal Gelfand-Zeitlin System
• The orbit structure of the Gelfand-Zeitlin group on n x n matrices