\(K\)-orbits on the flag variety and strongly regular nilpotent matrices (Q411331)

\textit{B. Kostant} and \textit{N. Wallach} [Progr. Math. 243, 319--364 (2006; Zbl 1099.14037); ibid. 244, 387--420 (2006; Zbl 1099.14038)] 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, the first author [Pac. J. Math. 250, No. 1, 109--138 (2011; Zbl 1218.14039)] studied the nilfiber, which consists of strongly regular elements such that each \(i \times i\) submatrix in the upper left corner is nilpotent. In the present paper, it is proven that every Borel subalgebra contains strongly regular elements and that the Borel subalgebras containing elements of the nilfiber is determined by the theory of \({K_{i} = \mathrm{GL}(i - 1,\mathbb{C}) \times \mathrm{GL}(1,\mathbb{C})}\)-orbits on the flag variety for \({\mathfrak{gl}(i,\mathbb{C})}\) for \(2 \leq i \leq n + 1\). As a consequence, a more precise description of the nilfiber is obtained. 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 \(i \leq n\).