On the generalized moment separability theorem for type 1 solvable Lie groups

Let G be a type 1 connected and simply connected solvable Lie group. The generalized moment map for π in {\widehat{G}} , the unitary dual of G, sends smooth vectors of the representation space of π to {{\mathcal{U}(\mathfrak{g})}^{*}} , the dual vector space of {\mathcal{U}(\mathfrak{g})} . The convex hull of the image of the generalized moment map for π is called its generalized moment set, denoted by {J(\pi)} . We say that {\widehat{G}} is generalized moment separable when the generalized moment sets differ for any pair of distinct irreducible unitary representations. Our main result in this paper provides a second proof of the generalized moment separability theorem for G.