Commutative polarisations and the Kostant cascade

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.