Moment sets and the unitary dual of a nilpotent Lie group (Q2718663)

Let \(G\) be a simply connected nilpotent Lie group with Lie algebra \(\mathfrak g\) and unitary dual \(\widehat G\). The moment map for \(\pi\in \widehat G\) sends smooth vectors in the representation space of \(\pi\) to \(\mathfrak g^*\) and the closure of its image is called the moment set. \textit{N. Wildberger} has proved in [Invent. Math. 98, 281-292 (1989; Zbl 0684.22005)], that the moment set for \(\pi_l\) coincides with the closure \(\overline{Conv({\mathcal O}_l)}\) of the convex hull of the corresponding coadjoint orbit \(\Omega_l\) \((l\in\mathfrak g^*)\). The authors say that \(\widehat G\) is moment separable when the moment sets differ for any pair of distinct irreducible unitary representations. By Pukanszky's parametrization of the coadjoint orbits of \(G\), there is a layering \(\mathfrak g^*=\bigcup_{e\in \mathcal E} \Omega_e\) of \(\mathfrak g^*\) by semi-algebraic \(G\)-invariant sets, a total order \(\prec\) on \(\mathcal E\) and for every \(e\in \mathcal E\) a polynomial function \(P_e\) on \(\mathfrak g^*\), such that \(\Omega_e=\{ l\in\mathfrak g^*: P_e(l)\neq 0, P_{e'}(l)=0,\) for \(e'\succ e\}\). The authors consider groups \(G\) which satisfy the following condition: for every \(e\in\mathcal E\), for every \(l\in \Omega _e\), \(\overline{\text{Conv}({\mathcal O}_l)}\subset \bigcup_{e'\succeq e}\Omega _{e'}\). In this paper sufficient and necessary conditions for moment separability for groups with this property are provided.