Infinite Dimensional Representations and Related Topics

无限维表示及相关主题

• 批准号: 14540167
• 负责人: SHIMOMURA Hiroaki
• 金额: $ 0.64万
• 依托单位: Kochi University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540167/
• 关键词: Manifold; Diffeomorphism; Unitary Representation; Quasi-Invariant Measure; Direct Product; Infinite Symmetric Group; Dual Pair; Irreducible Decomposition

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

在此期间，得到了以下结果：随后，我考虑了光滑流形M上的非同态群的酉表示。用Diff_o（m）表示具有紧支集的这类同构群.群中包含了原流形M的许多信息，并且与量子动力学有着密切的联系，因此研究Diff_o（M）的表示是一个有趣的问题。事实上，在此之前，许多作者已经研究和构造了许多有趣的酉表示及其线性形式，其中大多数是不可约的，在我的研究中，我通过具有无限质量的光滑测度的限制乘积得到了一系列新的表示，这与现在已经得到的表示本质上是不等价的。更确切地说，设E：={E_n}是M中的可数波莱尔集族（称为μ-酉），它具有以下三个性质。(1)0<μ（E_n）<+∞（2）|1-μ（E_n）|<+∞（3）E_n互不相交。 ...更多信息 我们在M^∞上有一个限制乘积测度ν_E。其次，我们对元素σ只置换有限个自然数的无限对称群取一个不可约酉表示，并考虑M^∞上的可测函数f具有以下性质：（1）f（xσ）= f（σ）^<-1>f（x）（2）f（x）在D_E上平方可和，其中D_E是Borel集，使得由σ在D_E上的作用得到的集合D_Eσ互不相交，它的并集是关于ν_E的完全测度。我们用H（M）表示这样的f的集合，并从Diff_o（M）对H（M）的对角作用导出L^2上的自然表示，其中M =（E，II）.则有一个酉表示（T（g），H（G）），g∈Diff_o（（M））. [1]（T（g），H（g））是不可约的. [2]两个关于A =（E，IT）和A =（E '，II'）的表示是等价的，当且仅当存在一个置换（可能是无限的）a，使得II和II'通过a等价，并且A =（E'，II '）是等价的。|μ（E'_&lt;a（n）&gt;-μ（E_n）|另一个主要结果是证明了Diff_o（（M））的每一个酉表示在一个相当温和的条件下都有一个不可约分解.证明包括分析Mautner关于经典局部紧群的结果和将Shavgulidze测度应用于同构群。少

项目成果

H.Shimomura: "Quasi-invariant measures on the group of diffeomorphisms and smooth vectors of unitary representations"Journal of Functional Analysis. 187. 406-441 (2001)

H.Shimomura：“微分同胚群和酉表示的平滑向量的准不变测度”泛函分析杂志。

H.Shimomura: "Irreducible decompositions of unitary representations of infinite-dimensional groups"Journal of Functional Analysis. (Submitted).

H.Shimomura：“无限维群的单一表示的不可约分解”泛函分析杂志。

H.Shimomura: "Unitary representations of the group of diffeomorphisms via restricted product measures with infinite mass."JPSP-DFG Japan-Germany Joint Seminar, IDHA. (To appear in Proceedings). (2003)

H.Shimomura：“通过无限质量的受限乘积测度微分同胚群的酉表示。”JPSP-DFG 日本-德国联合研讨会，IDHA。

H.Shimomura: "Unitary representations of the group of diffeomorphisms via restricted product measures with infinite mass"Journal of Mathematical Society of Japan. (Submitted).

H.Shimomura：“通过无限质量的受限乘积测度微分同胚群的酉表示”日本数学会杂志。

H.Shimomura: "Unitary representations of the group of diffeomorphisms via restricted product measures with infinite mass."Journal of Mathematical Society of Japan. (Submitted).

H.Shimomura：“通过无限质量的受限乘积测度微分同胚群的酉表示。”日本数学会杂志。