Combinatorial and geometric constructions associated with the Kostant cascade

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 $\mathfrak{g}$.

arXiv comment32 pages