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

在前半部分，我考虑了光滑流形M上用Diff_0(M)表示的微分同态群的表示。历史上我们有很多关于这一群的分析，但是我的研究对象是在M^∞上的L^2空间上的自然表示，它是由M上具有无限质量的光滑测度的受限积测度ν_E推导出来的。ν_E在Diff_0(M)的对角线作用下是拟不变的，因此我们得到了L^2空间上Diff_0(M)的自然表示T。其次，取自然数的有限排列的无限排列群S的酉表示II，并取M^∞上具有以下性质的函数f(1) f(xσ)=II(σ)^<-1>f(x) (2) f(x)是平方可和的。设H（Σ）是所有这些f的空间。我们在这个空间上引入Diff_0(M)的另一种自然表示，与上面类似。Σ:= (E, II)。那么我们就有了酉表示(T（g)， H（Σ））， g∈Diff_0(M)，并且我们已经知道这些表示都是更具有延展性的。在本研究中，我研究了表征T的不可约成分，并澄清它们只不过是上述(T（g)， H（Σ））。在后半段，我把注意力转向了Diff_0(M)的应用。例如，通过分析无穷置换群或杨图的渐近行为理论，通过Diff_0(M)的表示来找到泛函方程是很有趣的。另一种方法是通过Diff_0(M)的表示来实现S的正则表示的不可约分量。在阅读了前面的参考文献，特别是E Thoma的文献，并进一步考虑正定函数的极值分解后，我发现了不可约分解与极值分解的区别。众所周知，极值分解包含了相应的幺正表示的不可约分解，此外还有另一种分解，如Choqet定理。在这段时间里，我研究了这两个概念是如何相互联系的，并获得了一个暂时的结果。但是这个结果不方便应用到具体的问题中，所以我试图将它改进成更有用的形式。希望对S. Less正则表示的分析有所帮助

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