Representation Theory and Complex Analysis
Representation Theory and Complex Analysis
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表示论与复分析
DOI：
10.1007/978-3-540-76892-0
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
M. Picardello
中科院分区：
文献类型：
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作者：
M. Cowling;E. Frenkel;M. Kashiwara;A. Valette;D. Vogan;N. Wallach;E. C. Tarabusi;A. D'agnolo;M. Picardello
本书收录了 2004 年 7 月 10 日至 17 日在威尼斯举行的 CIME 会议“表示论与复分析”期间所做的六个系列讲座的笔记。我们感谢威尼斯国际大学在美丽的圣塞尔沃洛岛举办的盛情款待。我们组织这次会议的目的是向观众展示有关该主题的广泛的最新成果，从具有分析风格的主题到更多代数或几何导向的主题，同时不忽略与其他领域（例如量子计算）的相互作用。两篇论文概括介绍了半单李群及其酉表示的分析思想和性质。迈克尔·考林 (Michael Cowling) 全面展示了半单群和对称空间上表示论与调和分析之间的各种相互作用。在这种情况下会出现意想不到的现象，例如昆泽-斯坦因性质，它揭示了这些群和群行为与经典服从群（阿贝尔群的扩展）之间的巨大差异。此类结果与单一表示的系数消失密切相关。作为补充，阿兰·瓦莱特回顾了顺从性的概念，并研究了它与半单群酉表示系数消失以及遍历行为的关系。他应用这些想法来展示半简单群及其格表示的另一个令人惊讶的特性，即马古利斯的超刚性。三篇论文详细讨论了半简单群表示的硬分析。理想情况下，这种分析可以分为实群或复群的表示，或局部域上的代数群的表示。 Masaki Kashiwara 对现实世界和复杂世界之间的相互作用进行了深入的阐述，他的论文研究了实半简单李群的表示论和与相应的复杂代数群相关的标志流形的（微局域）几何之间的关系。这些结果，其中相当一部分是与 W. Schmid 的联合工作，几年前公布，并发表在
This volume collects the notes of six series of lectures given on the occasion of the CIME session Representation Theory and Complex Analysis held in Venice on July 10–17, 2004. We thank Venice International University for its hospitality at the beautiful venue of San Servolo island. Our aim in organizing this meeting was to present the audience with a wide spectrum of recent results on the subject of the title, ranging from topics with an analytical flavor, to more algebraic or geometric oriented ones, without neglecting interactions with other domains, such as quantum computing.Two papers present a general introduction to ideas and properties of analysis on semi-simple Lie groups and their unitary representations. Michael Cowling presents a panorama of various interactions between representation theory and harmonic analysis on semisimple groups and symmetric spaces. Unexpected phenomena occur in this context, as for instance the Kunze–Stein property, that reveal a dramatic difference between these groups and group actions and the classical amenable group (an extension of abelian groups). Results of this type are strongly related to the vanishing of coefficients of unitary representations. Complementarily, Alain Valette recalls the notion of amenability and investigates its relations with vanishing of coefficients of unitary representations of semi-simple groups and with ergodic actions. He applies these ideas to show another surprising property of representations of semi-simple groups and their lattices, namely Margulis’ super-rigidity. Three papers deal in full detail with the hard analysis of semisimple group representations. Ideally, this analysis could be split into representations of real groups or complex groups, or of algebraic groups over local fields. A deep account of the interaction between the real and complex world is given by Masaki Kashiwara, whose paper studies the relations between the representation theory of real semisimple Lie groups and the (microlocal) geometry of the flag manifolds associated with the corresponding complex algebraic groups. These results, a considerable part of which are joint work with W. Schmid, were announced some years ago, and are published here in