Unitary Representations of the Group of Diffeomorphisms via Restricted Product Measures with Infinite Mass II

Abstract

This paper concerns the problem of irreducibly decomposing unitary representations of the group ${\text{Diff}}_0(M)$ of diffeomorphisms with compact support on the smooth manifold $M$. As was shown in [19], these representations are decomposable under a fairly mild condition. In this paper, we consider a specific example of unitary representations $(T,{\text{Diff}}_0(M))$ that has been considered by [4]. $(T,{\text{Diff}}_0(M))$ is already a factor representation of type ${\text{II}}_{\infty }$; in addition, it may be decomposed into irreducible components through the left regular representatation of the group ${\mathfrak{S}}_{\infty }$ of the finite permutations. We describe the concrete realization of these irreducible components. The results obtained herein have some resemblance to the finite-dimensional case of [20] with the exception of the factor representatation. In addition, we will give another proofs of the irreducibility and equivalence that were obtained by [4].