Weak quadratic overgroups for a class of Lie groups

Abstract

Let \(G\) be a connected and simply connected Lie group with Lie algebra \(\mathfrak g \). We say that a subset \(X\) in the set \(\mathfrak g ^\star / G\) of coadjoint orbits is convex hull separable when the convex hulls differ for any pair of distinct coadjoint orbits in \(X\). In this paper, we define a class of solvable Lie groups, and we give an explicit construction of an overgroup \(G^+\) and a quadratic map \(\varphi \) sending each generic orbit in \(\mathfrak g ^\star \) to a \(G^+\)-orbit in \(\mathfrak{g ^+}^\star \), in such a manner that the set \(\varphi (\mathfrak g ^\star _{gen}){/ G^+}\) is convex hull separable. We then call \(G^+\) a weak quadratic overgroup for \(G\). Thanks to this construction, we prove that any nilpotent Lie group, with dimension at most 7 admits such a weak quadratic overgroup. Finally, we produce different examples of solvable Lie groups, having weak quadratic overgroups, but which are not in our class of Lie groups and for which usual constructions fail to hold.