Semi-direct products of Lie algebras and covariants

Let Q be a connected algebraic group with Lie algebra q. Symmetric invariants of q, i.e., the Q-invariants in the symmetric algebra S(q) of q, is a first approximation to the understanding of the coadjoint action (Q:q⁎) and coadjoint Q-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 g and V is G-module, then we define q to be the semi-direct product of 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 q can be constructed via certain G-equivariant maps from g to V (“covariants”).