Representation and measure theory of infinite dimensional moues and its applications

无限维运动的表示与测度理论及其应用

• 批准号: 18540184
• 负责人: SHIMOMURA Hiroaki
• 金额: $ 0.74万
• 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: 2006
• 资助国家: 日本
• 起止时间: 2006 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540184/
• 关键词: infinite dimensional group; representation theory; measure theory; diffeomorphism; irreducible representation; positive-deftnite-function; extreme decomposition; 位相群; G-N-S構成法; ユニタリ表現; 既約分解; 無限対称群

The main subjects of this period concerns with positive-definite functions on topological groups :The first one is an extension of the G-N-S construction method for the positive-definite functions on the usual locally compact groups to that for infinite dimensional groups. We showed that the extension is also possible, if we have a quasi-invariant probability measure whose admissible shifts form a denge subgroup. Such a measure plays a key role in place of the Haar measures on the locally compact groups. An interesting and important example of the infinite dimensional groups which admit such measures is the group of diffeomorphisms on smooth manifolds. The detailed arguments are already published in Math Z.The second one is the following problem :When and only when do extreme decompositions of positive-definite functions coincide with irreducible decompositions of unitary representations through the G-N-S construction on the usual locally compact groups ?Up to the present time, we already have obtained a necessary and sufficient condition for the above problem. However its description is not so clear that we are looking for another more concrete condition than that one.Of course, the problem is not always affirmative. We give the negative interesting examples whose origin is due to Thoma :In 1964, Thoma had a complete classification of the characters which is by definition normal traces on factors of type H. Then after about 30 years of this work, Obata had a disintegration of these characters.Hence, we have a problem whether the disintegration corresponds to irreducible decomposition or not.We will write here a partial answer for this problem briefly : It depends on the Thoma parameter which characterize his description ; It is negative if the parameter takes some values. Whether it is always negative or not is left for us, as well as the study of more specific equivalent conditions for the main problem.

这一时期的主要主题涉及拓扑组的正定功能：第一个是G-N-S构建方法的扩展，用于在通常的局部紧凑型组上对无限尺寸群体的正常组的阳性函数的扩展。我们表明，如果我们有一个准不变的概率度量，其可允许的偏移形成了ANEGE亚组，则也可以进行扩展。这种度量在当地紧凑型组上的HAAR措施中起着关键作用。接收这种度量的无限尺寸群体的一个有趣而重要的例子是平滑歧管上的一组差异性。详细的论点已经在数学z中发表。第二个问题是以下问题：何时和何时以及何时何时出现正常函数的极端分解与通常在本地紧凑的组上的G-N-S构建的不可约合的分解，直到当前的时间，我们已经获得了上述问题的足够条件，并且已经为上述问题提供了足够的条件。但是，它的描述并不清楚，我们正在寻找比那个更具体的条件。当然，问题并不总是肯定的。我们给出了负面的有趣示例，其起源是由于Thoma的原因：1964年，Thoma对角色进行了完整的分类，从定义上讲，这是H的正常痕迹。然后，在这项工作的大约30年之后，Obata对这些字符的崩解有崩溃。 ;如果参数采用一些值，则为负。无论它总是负面的，我们都留下了负面影响，以及针对主要问题的更具体的等效条件的研究。

项目成果

Irreducible decomposition of unitary representations of infinite-dimensional groups.

无限维群的单一表示的不可约分解。

• 发表时间: 2006
• 期刊: Math. Z. 251
• 作者: Kazushi;Yoshitomi;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Fumio Hiroshima;Fumio Hiroshima;Nagahata Yukio;Shigeki Aida;會田茂樹;会田 茂樹;会田 茂樹;会田 茂樹;会田 茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;会田 茂樹;會田茂樹;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;會田茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;H. Shimomura;H. Shimomura;H. Shimomura;H. Shimomura;H. Shimomura;H.Shimomura
• 通讯作者: H.Shimomura

Proceedings of JSPS-DFG Japan Germany joint Seminar, Infinite Dimensional Harmonic Analysis IV, World Scientific

JSPS-DFG日德联合研讨会论文集，无限维谐波分析IV，世界科学

• 作者: Kazushi;Yoshitomi;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Fumio Hiroshima;Fumio Hiroshima;Nagahata Yukio;Shigeki Aida;會田茂樹;会田 茂樹;会田 茂樹;会田 茂樹;会田 茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;会田 茂樹;會田茂樹;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;會田茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;H. Shimomura;H. Shimomura;H. Shimomura;H. Shimomura;H. Shimomura;H.Shimomura;H.Shimomura;H. Shimomura
• 通讯作者: H. Shimomura

Unitary representations and quasi-invariant measures on infinite dimensiomal groups

无限维群的酉表示和拟不变测度

• 发表时间: 2007
• 作者: Kazushi;Yoshitomi;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Fumio Hiroshima;Fumio Hiroshima;Nagahata Yukio;Shigeki Aida;會田茂樹;会田 茂樹;会田 茂樹;会田 茂樹;会田 茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;会田 茂樹;會田茂樹;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;會田茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;H. Shimomura;H. Shimomura;H. Shimomura;H. Shimomura;H. Shimomura;H.Shimomura;H.Shimomura;H. Shimomura;H. Shimomura;H. Shimomura;H. Shimomura;H. Shimomura;H. Shimomura;H. Shimomura
• 通讯作者: H. Shimomura

Irreducible decompositions of unitary representations and extreme decompositions of positive-definite functions

酉表示的不可约分解和正定函数的极端分解

• 发表时间: 2008
• 期刊: Proceedings of JSPS-DFG Japan Germany joint Seminar, Infinite Dimensional Harmonic Analysis IV, World Scientific.(掲載確定) 未定
• 作者: Kazushi;Yoshitomi;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Fumio Hiroshima;Fumio Hiroshima;Nagahata Yukio;Shigeki Aida;會田茂樹;会田 茂樹;会田 茂樹;会田 茂樹;会田 茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;会田 茂樹;會田茂樹;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;會田茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;H. Shimomura
• 通讯作者: H. Shimomura

Irreducible decompositions of unitary representations and extreme decompositions of positive-definite functions on groups

酉表示的不可约分解和群上正定函数的极端分解

• 发表时间: 2008
• 期刊: Proceedings of JSPS-DFG Japan-Germany joint seminar, Infinite Dimensional Harmonic Analysis IV, World Scientific (掲載確定)
• 作者: Kazushi;Yoshitomi;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Fumio Hiroshima;Fumio Hiroshima;Nagahata Yukio;Shigeki Aida;會田茂樹;会田 茂樹;会田 茂樹;会田 茂樹;会田 茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;会田 茂樹;會田茂樹;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;Shigeki Aida;會田茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;會田茂樹;H. Shimomura;H. Shimomura
• 通讯作者: H. Shimomura