Commutative polarisations and the Kostant cascade

TL;DR

This paper classifies certain parabolic subalgebras of complex simple Lie algebras based on the commutativity of their nilradicals' polarisations, using the Kostant cascade and related invariant theory.

Contribution

It provides a classification of parabolic subalgebras with commutative polarisation nilradicals in terms of the Kostant cascade, introducing the notions of optimal nilradicals and abelian ideals.

Findings

Classification of parabolic subalgebras with commutative polarisation nilradicals

Connection to the Kostant cascade and abelian ideals

Invariant-theoretic implications of commutative polarisation

Abstract

Let $\mathfrak g$ be a complex simple Lie algebra. We classify the parabolic subalgebras $\mathfrak p$ of $\mathfrak g$ such that the nilradical of $\mathfrak p$ has a commutative polarisation. The answer is given in terms of the Kostant cascade. It requires also the notion of an optimal nilradical and some properties of abelian ideals in a Borel subalgebra of $\mathfrak g$. Some invariant-theoretic consequences of the existence of a commutative polarisation are also discussed.