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

This paper concerns the problem of irreducibly decomposing unitary representations of the group Diff0(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, Diff0(M)) that has been considered by [4]. (T, Diff0(M)) is already a factor representation of type II∞; in addition, it may be decomposed into irreducible components through the left regular representation of the group S∞ of finite permutations. We describe concrete realization of these irreducible components. The results obtained herein bear some resemblance to the finite-dimensional case of [20] with the exception of the factor representation. In addition in the Appendix, we will give another proof of the irreducibility and equivalence that were obtained in [4]. 2010 Mathematics Subject Classification: Primary 58D05; Secondary 22E65.