Topological group representations that are generated by Poisson measures (Q2784529)

Let \(X\) be a manifold, \(G\) the group of diffeomorphisms of \(X\) or the loop group of \(X\) and \(\Sigma^{n},\) \(\Sigma^{\infty}\) the groups formed by all bijective epimorphisms of the set \(\{1,\ldots,n\}\) or of the set of all natural numbers, respectively. To each unitary representation \(q\) of \(\Sigma^{n}\) or \(\Sigma^{\infty}\) in a Hilbert space and each measure \(\mu\) on \(X\) a representation \(U^{q,\mu}\) of \(G\) in a Hilbert space is associated. It is proved that there exist quasi-invariant measures \(\mu\) on \(X\) such that the representations \(U^{q,\mu}\) are irreducible when \(q\) is irreducible. Furthermore a criterion is given that the representations \(U^{\mu}\) and \(U^{\overline{\mu}}\) are equivalent.