Combinatorial and geometric constructions associated with the Kostant cascade

TL;DR

This paper explores combinatorial and geometric structures related to the Kostant cascade in complex simple Lie algebras, including constructions of a cascade element, abelian ideals, nilpotent orbits, and involutions, with applications to Lie theory.

Contribution

It introduces new constructions and properties associated with the Kostant cascade, enhancing understanding of Lie algebra structures and their geometric and combinatorial aspects.

Findings

Defined the cascade element $x_{\ ext{K}}$ in the Cartan subalgebra.

Analyzed properties of abelian ideals associated with the cascade.

Explored applications to nilpotent orbits and involutions in Lie algebras.

Abstract

Let $\mathfrak g$ be a complex simple Lie algebra and $\mathfrak b=\mathfrak t\oplus\mathfrak u^+$ a fixed Borel subalgebra. Let $\Delta^+$ be the set of positive roots associated with $\mathfrak u^+$ and $\mathcal K\subset\Delta^+$ the Kostant cascade. We elaborate on some constructions related to $\mathcal K$ and applications of $\mathcal K$. This includes the cascade element $x_{\mathcal K}$ in the Cartan subalgebra $\mathfrak t$ and properties of certain objects naturally associated with $\mathcal K$: an abelian ideal of $\mathfrak b$, a nilpotent $G$-orbit in $\mathfrak g$, and an involution of $\mathfrak g$.