arXiv subjects
Publications and source records attributed to Amel Zergane.
We present a one-to-one correspondence between equivalence classes of unitary irreducible representations and coadjoint orbits for a class of pro-Lie groups including all connected locally compact nilpotent groups and arbitrary infinite direct products of nilpotent Lie groups.
We develop a semigroup approach to representation theory for pro-Lie groups satisfying suitable amenability conditions. As an application of our approach, we establish a one-to-one correspondence between equivalence classes of unitary irreducible representations and coadjoint orbits for a class of pro-Lie groups including all connected locally compact nilpotent groups and arbitrary infinite direct products of nilpotent Lie groups. The usual $C^*$-algebraic approach to group representation theory positivey breaks down for infinite direct products of non-compact locally compact groups, hence the description of their unitary duals in terms of coadjoint orbits is particularly important whenever it is available, being the only description known so far.
For the semi simple and deployed Lie algebra $\mathfrak g=\mathfrak{sl}(n, \R)$, we give an explicit construction of an overalgebra $\mathfrak g^+=\mathfrak g\rtimes V$ of $\mathfrak g$, where $V$ is a finite dimensional vector space. In such a setup, we prove the existence of a map $Φ$ from the dual $\mathfrak g^\star$ of $\mathfrak g$ into the dual $(\mathfrak g^+)^\star$ of $\mathfrak g^+$ such that the coadjoint orbits of $Φ(ξ)$, for generic $ξ$ in $\mathfrak g^\star$, have a distinct closed convex hulls. Therefore, these closed convex hulls separate 'almost' the generic coadjoint orbits of $G$.
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 $G$.