Infinite Dimensional Representations and Related Topics

在此期间，我考虑了光滑流形m上的微分同态群的酉表示，用Diff_o(m)表示这些微分同态的紧支持群。这个群有很多原始流形M的信息，并且它与量子动力学有密切的联系，因此研究Diff_o(M)的表示是很有趣的。事实上，在此之前，许多作者已经研究并构造了许多有趣的酉表示及其线性版本，其中大多数都是不可约的。在我的研究中，我通过无限质量的光滑测度的受限积得到了一系列新的表示，这与现在得到的本质上是不等价的。更确切地说，设E:={E_n}是M中的Borel集合的一个可数族（称为μ-一元），它具有以下三个性质。(1) 0 <μ(E_n) < +∞(2)Σ| 1 -μ(E_n) | < +∞(3)E_n是相互分离的。利用μ-单位E，首先得到M^∞上的受限积测度ν_E。其次，我们取元素σ只置换有限个自然数的无限对称群的不可约酉表示Π，并考虑M^∞上的可测函数f具有以下性质：(1)f(xσ)=Π（σ）^<-1>f(x) (2)f(x)在D_E上是平方可和的，其中D_E是一个Borel集合，使得由σ作用于D_E得到的D_Eσ是互不相交的集合，并且它的并集是关于ν_E的满测度。让我们用H（Σ）表示这样一个f的集合，并推导出由Diff_o((M)对H（Σ）， whereΣ=（E，II）的对角线作用得到的L^2上的自然表示。然后我们有一个酉表示(T（g)，H（Σ））， g∈Diff_o（(M)）。主要结果如下：[1](T（g)，H（Σ））是不可约的。[2]两个表示onΣ=（E，IT）和Σ=（E‘，II’）是等价的，当且仅当存在一个排列（可以是无限的），其中II和II‘通过a是等价的，并且Σ|μ(E’_<a(n)>-μ(E_n) |<+∞。另一个主要结果是证明了在相当温和的条件下，Diff_o（(M)）的每一个酉表示都具有不可约分解。该证明包括分析经典局部紧群上的Mautner结果和在微分同构群上应用shavgullidze测度。少

During this period, the following results were obtained :Subsequently I have considered unitary representations of the group of diffeomorphisms on smooth manifolds M. I denote a group of these diffeomorphisms with compact support by Diff_o(m). The group has many informations of the original manifold M, and it has close connections with Quantum Dynamics, so it is interesting to investigate representations of Diff_o(M). In fact, previously, various authors have studied and constructed many interesting unitary representations and their linear versions, most of them were irreducible.In my research, I have obtained a series of new representations via restricted product of smooth measures with infinite mass, which is essentially inequivalent from what have been obtained now. More exactly, let E:={E_n} be a countable family(which is called μ-unital) of Borel sets in M that has the following three properties.(1)0<μ(E_n)<+∞(2)Σ|1-μ(E_n)|<+∞(3)E_n are mutually disjoint.Using μ-unital E, firstly … More we have a restricted product measure ν_E on M^∞. Secondly, we take an irreducible unitary representations Π of the infinite symmetric group whose element σ permutes only a finite number of natural numbers, and consider measurable functions f on M^∞ that have the properties ;(1)f(xσ)=Π(σ)^<-1>f(x) (2)f(x) is square summable on D_E, where D_E is a Borel set such that the sets D_Eσ obtained from the action of σ on D_E are mutually disjoint, and its union is full measure with respect to ν_E. Let us denote a set of such f by H(Σ), and induce natural representation on L^2 obtained from the diagonal action of Diff_o((M) on H(Σ), whereΣ=(E,II). Then we have a unitary representation (T(g),H(Σ)), g∈Diff_o((M). The main results on the representation is as follows :[1](T(g),H(Σ)) is irreducible.[2]Two representations onΣ=(E,IT) and Σ=(E',II') are equivalent, if and only if there exists a permutation(may be an infinite one) a auch that II and II' are equivalent through a, and Σ|μ(E'_<a(n)>-μ(E_n) |<+∞.One more main result is to show that every unitary representations of Diff_o((M) has an irreducible decomposition under a fairly mild condition. The proof consists of analyzing Mautner's result on classical locally compact groups and applying the Shavgulidze measure on the diffeomorphism groups. Less