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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.

Let G be a connected and simply connected solvable Lie group. The moment map for π in G ˆ , unitary dual of G, sends smooth vectors in the representation space of π to g ⁎ , dual space of g . The closure of the image of the moment map for π is called its moment set, denoted by I π . Generally, the moment set I π , π ∈ G ˆ does not characterize π, even for generic representations. However, we say that G ˆ is moment separable when the moment sets differ for any pair of distinct irreducible unitary representations. In the case of an exponential solvable Lie group G, D. Arnal and M. Selmi exhibited an accurate construction of an overgroup G + , containing G as a subgroup and an injective map Φ from G ˆ into G + ˆ in such a manner that Φ ( G ˆ ) is moment separable and I Φ ( π ) characterizes π, π ∈ G ˆ . In this work, we provide the existence of a quadratic overgroup for the diamond Lie group, which is the semi-direct product of R n with ( 2 n + 1 ) -dimensional Heisenberg group for some n ⩾ 1 .