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

TL;DR

This paper explores the structure of strongly regular elements in the Lie algebra gl(n+1,c), characterizing Borel subalgebras containing nilpotent elements and linking orbit theory to discrete series representations.

Contribution

It proves every Borel subalgebra contains strongly regular elements and classifies those with nilpotent elements using orbit theory, refining the understanding of the nilfibre.

Findings

Every Borel subalgebra contains strongly regular elements.

Classification of Borel subalgebras with nilpotent elements via $K_i$-orbits.

Connection between orbits and discrete series representations.

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 $i\times i$ 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 $K_{i}=GL(i-1,\C) \times GL(1,\C)$-orbits on the flag variety for $\fgl(i,\C)$ for $2\leq i\leq n+1$. 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 $i \le n$.