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

在前半部分，我考虑了表示为Diff_0（M）的群在光滑流形M上的表示。历史上我们对这个群有很多分析，但我的研究对象是M ^∞上L^2空间上的一个自然表示，它是由M上具有无穷质量的光滑测度的一个限制乘积测度ν_E导出的。其次，取自然数的有限置换的无限置换群S的酉表示II，并取M^∞上的函数f，使其具有以下性质：（1）f（xσ）=II（σ）^<-1>f（x）（2）f（x）是平方可和的。设H（M）是所有这样的f的空间，我们在这个空间上引入Diff_0（M）的另一个自然表示，与上面类似。将E：=（E，II）。则我们有酉表示（T（g），H（G）），g∈ Diff_0（M），并且我们已经知道这些表示都是非正则的 关于我们 本文研究了表示T的不可约分支，阐明了它们就是表示T的不可约分支（T（g），H（g）），在后半部分，我们把注意力转向Diff_0（M）的应用。例如，通过对无穷置换群的分析或杨氏图的渐近性态理论，通过Diff_0（M）的表示来寻找函数方程是很有趣的。另一个是通过Diff_0（M）的表示来实现S的正则表示的不可约分量的方法.通过阅读前人的文献，特别是E. Thoma的文献，并进一步考虑正定函数的极值分解，观察到不可约分解与极值分解的区别.极端分解包含相应的酉表示的不可约的，除了另一个分解，如Choqet定理。在此期间，我研究了这两个概念是如何相互联系的，并暂时取得了一个结果。然而，这是不方便的应用，这一结果的具体问题，所以我试图改善它更有用的形式。我希望这将有助于分析S的正则表示。少

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