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

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 $\mathfrak{n}$.