Representations of the Group of Diffeomorphisms

This article contains a survey of results on representations of the diffeomorphism group of a noncompact manifold X associated with the space ΓX of configurations (that is, of locally finite subsets) in X. These representations are constructed from a quasi-invariant measure μ on ΓX. In particular, necessary and sufficient conditions are established for the representations to be irreducible. In the case of the Poisson measure μ a description is given of the corresponding representation ring.