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.

arXiv comment22 pages