S\'eparation des repr\'esentations par des surgroupes quadratiques

TL;DR

This paper introduces the concept of quadratic overgroups for Lie groups, demonstrating their existence for various classes, which helps uniquely characterize certain representations via associated moment sets.

Contribution

The paper proves the existence of quadratic overgroups for many classes of Lie groups, enabling better representation characterization through moment sets.

Findings

Existence of quadratic overgroups for various Lie groups

Quadratic overgroups help characterize representations uniquely

Construction based on polynomial functions of degree at most 2

Abstract

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 $G$.