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

在前半期，我考虑了光滑流形M上由diff_0(M)表示的微分同胚群的表示。历史上我们对这个群有很多分析，但我的研究对象是M^∞上的L^2空间上的一个自然表示，它是由M上具有无限质量的光滑测度的一个限制乘积测度ν_E得到的。ν_E在Diff_0(M)的对角作用下是拟不变的，从而得到了Diff_0(M)在L^2空间上的自然表示T。其次，取自然数的有限置换的无限置换群S的么正表示II，取M^∞上具有性质(1)f(xσ)=II(σ)^&lt；-1&gt；f(X)(2)f(X)是平方可和的函数f。设H(Σ)是所有这样的f的空间，我们在这个空间上引入了diff_0(M)的另一种自然表示，类似于上面。把Σ：=(E，II)。然后我们有酉表示(T(G)，H(Σ))，g∈Diff_0(M)，我们已经知道这些表示都是IRRE…在本研究中，我研究了表示T的不可约分支，并阐明它们只不过是上述(T(G)，H(Σ))。在后半期，我将注意力转向DIFF_0(M)的应用。例如，通过无限置换群的分析或Young图的渐近行为理论，通过diff_0(M)的表示来寻找函数方程是很有趣的。另一种是通过diff_0(M)的表示来实现S正则表示的不可约分量的方法。在阅读了前人的文献，特别是E Thoma的文献之后，进一步考虑了正定函数的极端分解，观察了不可约分解与极限分解的区别。众所周知，极端分解包含对应的酉表示的不可约分量，此外还有另一种类似Choqet定理的分解。在这段时间里，我研究了这两个概念是如何相互联系的，并暂时得到了一个结果。但这一结果在具体问题中的应用并不方便，因此我试图将其改进为更有用的形式。我希望它将对分析S.Less的常规代表性有所帮助

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