Séparation des représentations par des surgroupes quadratiques

Didier Arnal (1) , Mohamed Selmi (2) , Amel Zergane (1, 2)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 Physique Mathématique

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 $G$.

Mots clés: Application moment; Représentations des groupes de Lie; Surgroupes quadratiques

Domaines: Théorie des représentations [math.RT]

Didier Arnal, Mohamed Selmi, Amel Zergane. Séparation des représentations par des surgroupes quadratiques. Bulletin des Sciences Mathématiques, 2011, 135 (2), pp.141-165. ⟨hal-00394283⟩