Overalgebras and separation of generic coadjoint orbits of $SL(n,\mathbb{R})$

Abstract

For the semi simple and deployed Lie algebra $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{R})$, we give an explicit construction of an overalgebra ${\mathfrak{g}}^{+}=\mathfrak{g}⋊V$ of $\mathfrak{g}$, where $V$ is a finite dimensional vector space. In such a setup, we prove the existence of a map $\Phi$ from the dual ${\mathfrak{g}}^{\star }$ of $\mathfrak{g}$ into the dual $({\mathfrak{g}}^{+}{)}^{\star }$ of ${\mathfrak{g}}^{+}$ such that the coadjoint orbits of $\Phi (\xi )$, for generic $\xi$ in ${\mathfrak{g}}^{\star }$, have a distinct closed convex hulls. Therefore, these closed convex hulls separate 'almost' the generic coadjoint orbits of $G$.