  The coadjoint representation of a connected algebraic group $Q$ with Lie
algebra $\mathfrak q$ is a thrilling and fascinating object. Symmetric
invariants of $\mathfrak q$ (= $\mathfrak q$-invariants in the symmetric
algebra $S(\mathfrak q)$) can be considered as a first approximation to the
understanding of the coadjoint action $(Q:\mathfrak q^*)$ and coadjoint orbits.
In this article, we study a class of non-reductive Lie algebras, where the
description of the symmetric invariants is possible and the coadjoint
representation has a number of nice invariant-theoretic properties. If $G$ is a
semisimple group with Lie algebra $\mathfrak g$ and $V$ is $G$-module, then we
define $\mathfrak q$ to be the semi-direct product of $\mathfrak g$ and $V$.
Then we are interested in the case, where the generic isotropy group for the
$G$-action on $V$ is reductive and commutative. It turns out that in this case
symmetric invariants of $\mathfrak q$ can be constructed via certain
$G$-equivariant maps from $\mathfrak g$ to $V$ ("covariants").
