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

✍ Scribed by Hiroaki Shimomura

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 242 KB
• Volume: 187
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2001.3807

✦ Synopsis

Let M be a smooth manifold and Diff 0 (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 g 0 (M), the connected component of the identity in Diff 0 (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 g k (M), the group of C k 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 noncompact M.