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 $\Gamma_x$ of configurations (that is, of locally finite subsets) in $X$. These representations are constructed from a quasi-invariant measure $\mu$ on $\Gamma_x$. In particular, necessary and sufficient conditions are established for the representations to be irreducible. In the case of the Poisson measure $\mu$ a description is given of the corresponding representation ring.