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 \(\mathfrak {g}\).