Universal overgroup
It is well known that to any unitary representation \(\pi\) of a Lie group \(G\) on a finite-dimensional complex vector space \(V\) one can associate the moment map of the corresponding Hamiltonian action on the complex projective space \(PV\). This concept is extended for infinite-dimensional representations \(\pi\) on a Hilbert space \((H,(\cdot|\cdot))\) in \textit{N. J. Wildberger} [Invent. Math. 98, No.~2, 281--292 (1989; Zbl 0684.22005)]. Here, instead of \(H\) one should consider a subspace \(V\) of smooth vectors for a representation \(\pi\). The range \(l_\pi\) of the moment map does not characterize the representation \(\pi\). Following \textit{D. Arnal} and \textit{M. Selmi} [Bull. Sci. Math. 132, No.~1, 54--69 (2008; Zbl 1138.22006)], a way to separate irreducible unitary representations \(\pi\) for a Lie group \(G\) by moment sets is to use an overgroup \(\widetilde G\) and extensions of each representation \(\pi\) to a representation \(\widetilde {\pi}\) of \(\widetilde G\). In this paper the construction of a universal overgroup \(\widetilde G\) is proposed. It is an infinite-dimensional Fréchet-Lie group and extend each \(\pi\) to a Hamiltonian action \(\widetilde{\pi}\) of \(\tilde G\), such that the moment set of \(\widetilde{\pi}\) characterizes \(\pi\). The authors use the following interesting geometric construction. They define a natural Hamiltonian action \(\lambda\) of a Fréchet-Lie group \(G \times V\) (semidirect product with a real abelian group \(V\)) on \(V\) and prove that the moment set \(\ell\) of \(\lambda\) is the graph of the map \(\theta: V\to \text{Lie}(G)^*\), \(\theta(v)(x)= \text{Im}(\pi'(x)v|v)\). The projection of the moment set \(\ell\) on the dual \(\text{Lie}(G)^*\) is the positive cone supported by the the usual moment set associated to the representation \(\pi\). Here \(\pi'\) is the associated representation of the Lie algebra \(\text{Lie}(G)\).
scientific article; zbMATH DE number 5836605