  Let $\pi$ be an unitary irreducible representation of a Lie group $G$. $\pi$
defines a moment set $I_\pi$, subset of the dual $\mathfrak g^*$ of the Lie
algebra of $G$. Unfortunately, $I_\pi$ does not characterize $\pi$. However, we
sometimes can find an overgroup $G^+$ for $G$, and associate, to $\pi$, a
representation $\pi^+$ of $G^+$ in such a manner that $I_{\pi^+}$ characterizes
$\pi$, at least for generic representations $\pi$. If this construction is
based on polynomial functions with degree at most 2, we say that $G^+$ is a
quadratic overgroup for $G$. In this paper, we prove the existence of such a
quadratic overgroup for many different classes of $G$.
