Semi-direct products of Lie algebras and covariants

Abstract

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").