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

TL;DR

This paper constructs an overalgebra extension of rak{sl}(n, \u211d) to distinguish generic coadjoint orbits via convex hulls, aiding their separation in representation theory.

Contribution

It provides an explicit construction of an overalgebra and a map that separates generic coadjoint orbits by their convex hulls, a novel approach in Lie algebra theory.

Findings

Existence of a map  from rak{g}^* to (rak{g}^+)^* with orbit separation properties.

Construction of an overalgebra rak{g}^+ = rak{g} times V for rak{sl}(n, \u211d).

Convex hulls of coadjoint orbits are distinct for generic elements.

Abstract

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 $\Phi$ 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 $G$.