Séparation des représentations par des surgroupes quadratiques

Abstract

Let $π$ be an unitary irreducible representation of a Lie group $G$. $π$ defines a moment set $I_π$, subset of the dual $\mathfrak g^*$ of the Lie algebra of $G$. Unfortunately, $I_π$ does not characterize $π$. However, we sometimes can find an overgroup $G^+$ for $G$, and associate, to $π$, a representation $π^+$ of $G^+$ in such a manner that $I_{π^+}$ characterizes $π$, at least for generic representations $π$. 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 $G$.