Representation theory and measure theory of infinite-dimensional groups and related topics

无限维群的表示论和测度论及相关话题

基本信息

批准号：
16540162
负责人：
SHIMOMURA Hiroaki
金额：
$ 0.64万
依托单位：
Kochi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

项目摘要

In the first half period, I considered the representations of the group of diffeomorphisms denoted by Diff_0(M) on smooth manifolds M.Historically we have many analysis on this group, however my researching object is a natural representation on L^2 space overM^∞ derived from a restricted product measure ν_E of a smooth measure on M with infinite mass. ν_E is quasi-invariant under the diagonal action of Diff_0(M), and hence we have a natural representation T of Diff_0(M) over the L^2 space.Secondly, take a unitary representation II of the infinite permutation group S of the finite permutations of the natural numbers, and take functions f on M^∞ that have properties (1) f(xσ)=II(σ)^<-1>f(x) (2) f(x) is square summable. Let H(Σ) be the space of all such f.We introduce another natural representation of Diff_0(M) on this space, similarly as above. Put Σ:=(E,II). Then we have unitary representations (T(g), H(Σ)), g∈ Diff_0(M), and we have already known that these representations are all irre … More ducible.In the present research, I investigated the irreducible components of the representation T, and clarified that they are nothing but the above (T(g), H(Σ)).In the later half period, I turned my attention to the applications of Diff_0(M). For example, it is interesting to find functional equations through the representations of Diff_0(M) via analysis of the infinite permutation group or the theory of asymptotic behavior of Young diagrams. Another one is an approach to a realization of irreducible component of the regular representation of S through the representations of Diff_0(M).After reading preceding bibliographies, in particular that by E Thoma, and further considering extreme decomposition of positive-definite functions, I observed the difference between irreducible decomposition and extreme one.As is well-known, extreme decomposition contains irreducible ones of the corresponding unitary representation, and besides another decomposition like the Choqet theorem. In this period, I studied how these two concepts connect with each other and obtained a result for the time being. However it is inconvenient for applications of this result to concrete problems, so I am trying to improve it more useful forms. I hope it will be useful in the analysis of the regular representation of S. Less

在前半段，我考虑了平滑歧管上用diff_0（m）表示的一组差异形态的表示，我们对这一组进行了许多分析，但是我的研究对象是从l^2 space overm^∞衍生出的自然表示，这些代表是从限制的产品测量值c上的限制性测量的无限质量质量的限制性测量。 ν_e在diff_0（m）的对角性作用下是准不变的，因此，在l^2空间上，我们具有diff_0（m）的自然表示。第二，以统一的统一置入率s的统一代表II，对天然数量的有限置置置换术的有限置置率，并在属性上功能functions portions portion prounn prounct prounn protion prounct proptions（1）（1）（1）（1） f（xσ）= ii（σ）^<-1> f（x）（2）f（x）是正方形的。令h（σ）为所有此类F的空间。我们在此空间上引入了另一个自然表示，如上所述。 putσ：=（E，II）。然后，我们具有单一的表示（t（g），h（σ）），gd diff_0（m），我们已经知道这些表示形式都不是不可能的……更加可言。例如，有趣的是，通过分析无限置换组或年轻图的不对称行为理论，通过DIFF_0（M）的表示来找到功能方程。另一个是一种方法，可以通过diff_0（m）的代表来实现s定期表示不可证明的组成部分。在阅读之前阅读之前，特别是通过e thoma，并进一步考虑了正常功能的极端分解，我还观察到了不可用的分解和极端的差异。诸如Choqet定理之类的另一个分解。在此期间，我研究了这两个概念如何相互联系，并暂时获得了结果。但是，对于该结果将其用于具体问题是不方便的，因此我试图改善它更有用的形式。我希望它将在分析定期代表的分析中很有用。