Quasi-invariant Measures on the Group of Diffeomorphisms and Smooth Vectors of Unitary Representations
Quasi-invariant Measures on the Group of Diffeomorphisms and Smooth Vectors of Unitary Representations
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酉表示的微分同胚群和光滑向量的拟不变测度
DOI：
10.1006/jfan.2001.3807
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
Hiroaki Shimomura
中科院分区：
文献类型：
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作者：
Hiroaki Shimomura
设M为光滑流形，Diff 0 (M)为M上具有紧支持的所有光滑微分同态的群。本文主要研究了差分同态群上某些拟不变测度的存在性，以及对于Diff* 0 (M)中单位元的连通分量Diff* 0 (M)的给定酉表示U， C∞-向量的密集性。首先推广了shavgullidze关于微分同构群上拟不变测度的一些结果。然后我们证明了以下结果：假设M是紧的，并且U具有这样的性质：对于某有限k，作用连续扩展到与恒等式同伦的ck个微同态群Diff* k (M)。那么U有一个C∞向量的密集集合。我们也将定理推广到非紧M。
Abstract 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* 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* 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 non-compact M .