Infinite Dimensional Representation, Measure Theory and Related Topics

在这两年的时间里，我致力于研究群的么正表示。光滑流形M上具有紧支集diff_0(M)或其子群的微分同胚当M是紧的时，这类群是无限维李群。因此，有可能用李代数方法来分析这些表示。在这些考虑下，我得到了如下关于线性一的么正表示的可约性的结果。1由一个对应于通常李群上的Campbell-Hausdorff公式的公式来保证线性性。(在我们的例子中，这个公式来自于对一些自治微分方程解的性态的估计)2.由李代数的映象通过指数映射生成的子群的特征。对于上面的问题，我已经看到继续我们的理论是没有问题的，例如在diff_0(M)的情况下，子群在中立元的连通分支中稠密。最后，对于C^*-向量的丰富存在性问题，我现在继续讨论它，当然是在无穷维表示上，并将上述结果应用于1-上循环的diff_0(M)，得到了一些基本结果。特别地，上循环形式与M上的几何结构有密切的联系。

Between these two years I contained to study on unitary representations of the group of. diffeomorphisms with compact support Diff_0(M) or of its subgroups on smooth manifolds M.It is known that these groups are infinite dimensional Lie groups, whenever M is compact. Hence there is possibility to analyze these representations with the Lie algebraic method. Under these considerations I have obtained the following results for reducibility our unitary representations to the linear one.1 The linearlity is assured by a formula which corresponds to the Campbell-Hausdorff formula on the usual Lie group. (In our case, the formula comes from an evaluation for the behavior of solutions of some autonomus differential equations)2. A chracterization of the subgroup generated by the image of Lie algebra by the exponential mapping.For the above problem I have seen that it is no problem to proceed our theories, for example in the case of Diff_0(M), the subgroup is dense in the connected component of the neutral element.3. Lastly, for the problem of rich existence of C^*-vectors I am continuing to discuss it now, of course on infinite dimensional representations.Moreover I applied the above results to 1-cocycles in terms of Diff_0(M) and obtained some fundamental results. In particular the cocycle form has a close connection with the geometrical structure on M.