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

Abstract

In two 2006 papers, Kostant and Wallach constructed a complexified Gelfand–Zeitlin integrable system for the Lie algebra $${\mathfrak{gl}(n + 1,\mathbb{C})}$$ and introduced the strongly regular elements, which are the points where the Gelfand–Zeitlin flow is Lagrangian. Later Colarusso studied the nilfiber, 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 nilfiber by using the theory of $${K_{i} = GL(i - 1,\mathbb{C}) \times GL(1,\mathbb{C})}$$ -orbits on the flag variety for $${\mathfrak{gl}(i,\mathbb{C})}$$ for 2 ≤ i ≤ n + 1. As a consequence, we obtain a more precise description of the nilfiber. The K i -orbits contributing to the nilfiber are closely related to holomorphic and anti-holomorphic discrete series for the real Lie groups U(i, 1), with i ≤ n.