The Frobenius semiradical, generic stabilisers, and Poisson centre for nilradicals

Abstract

Let $\mathfrak g$ be a complex simple Lie algebra and $\mathfrak n$ the nilradical of a parabolic subalgebra of $\mathfrak g$. We consider some properties of the coadjoint representation of $\mathfrak n$ and related algebras of invariants. This includes (i) the problem of existence of generic stabilisers, (ii) a description of the Frobenius semiradical of $\mathfrak n$ and the Poisson centre $Z(\mathfrak n)$ of the symmetric algebra $S(\mathfrak n)$, (iii) the structure of $S(\mathfrak n)$ as $Z(\mathfrak n)$-module, and (iv) the description of square integrable (= quasi-reductive) nilradicals. Our main technical tools are the Kostant cascade in the set of positive roots of $\mathfrak g$ and the notion of optimisation of $\mathfrak n$.