Notes sur la notion d'invariant caract\'eristique

Let $G$ be a Lie group acting on a vector space $V$. Given a set of $G$-invariants, one can ask the question : does this set of invariants characterize the group $G$ ? We recall here some known results, ask questions and state some conjectures for different choices of invariants : polynomial functions, orbits, distributions, and different types of groups.