Quasi-invariant Measures on the Group of Diffeomorphisms and Smooth Vectors of Unitary Representations

Let M be a smooth manifold and Diff0(M) the group of all smooth diffeomorphisms on M with compact support. Our main subject in this paper concerns the existence of certain quasi-invariant measures on groups of diffeomorphisms, and the denseness of C∞-vectors for a given unitary representation U of Diff*0(M), the connected component of the identity in Diff0(M). We first generalize some results of Shavgulidze on quasi-invariant measures on diffeomorphism groups. Then we prove the following result: Suppose that M is compact and U has the property that the action extends continuously to Diff*k(M), the group of Ck diffeomorphisms which are homotopic to the identity, for some finite k. Then U has a dense set of C∞-vectors. We also give an extension of our theorem to non-compact M.