Combinatorial and Geometric Constructions Associated with the Kostant Cascade

Dmitri I. Panyushev¹

¹Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

Journal of Lie Theory, Tome 33 (2023) no. 2, pp. 497-526

Résumé

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

Détail

Classification : 17B22, 17B20, 17B08, 14L30
Mots-clés : Root system, cascade element, abelian ideal, Frobenius algebra, nilpotent orbit