Representations of Lie Groups and Supergroups-猫眼课题宝

DOI：

10.4171/owr/2013/13

发表时间：

2013-11

期刊：

Oberwolfach Reports

影响因子：

通讯作者：

中科院分区：

其他

文献类型：

作者：

J. Hilgert;Toshiyuki Kobayashi;K. Neeb;T. Ratiu

研讨会集中讨论了群对象表示理论的最新进展，主要是有限维和无限维光滑流形和超流形。讲座涵盖了广泛的主题，特别强调了基准问题和示例，例如分支、极限行为和对偶。在许多谈话中，与物理的关系起了重要作用。数学学科分类（2000）：22E、32M、46G20、53D、58B、58C。李群和超群的代表研讨会由Joachim Hilgert（帕德博恩）、Toshiyuki Kobayashi（东京）、Karl-Hermann Neeb（埃尔兰根）和Tudor Ratiu（洛桑）组织。从一开始，物理学中的应用就是李群表示研究的主要动机。后来，数论，特别是朗兰兹程序，也成为一股推动力。在20世纪的最后二十年里，人们在酉表示的分类上投入了大量的努力。有大量的信息可供使用，但中心分类问题仍然没有完全解决。目前，这方面的研究主要集中在一些美国研究小组。目前的研究主要集中在基准问题和实例上，如分支、极限行为和对偶对。此外，将表征理论的范围扩展到无限维群和超群，也起到了重要的作用。736 Oberwolfach报告13/2013这些努力的一个共同特点是识别相关类的例子，这些例子足够普遍，很有趣，但同时有足够的限制，允许一般理论。在许多情况下，这些基准例子的选择是由理论物理中的问题指导的。这次讲习班的重点是这些最近的发展。来自许多欧洲国家、加拿大、美国和日本的51名代表参加了会议。会议围绕23个讲座展开，每个讲座50分钟。被选中的演讲者是处于职业生涯各个阶段的研究人员，从非常有前途的年轻博士后到在过去45年里为该领域贡献了关键成果的资深科学家。我们认为这次会议令人兴奋，非常成功。讲座的质量是杰出的，讨论的强度是例外的，即使是奥伯沃尔法赫的标准。这一观察的一个很好的指标是，所有可用的黑板都被讨论组占用了，直到每天晚上会议很晚。更值得注意的是，这些讨论组的组成每天都在变化。特别是，在最近的研究中，无论是有限维、无限维还是超维语境的研究人员都没有停留在自己的圈子里。新的合作已经开始，来自不同大陆的既定研究伙伴有机会亲自讨论进一步的项目。在不讨论太多细节的情况下，让我们提一下一些重要的新事态发展。在无限维李群领域，事物在两个相互作用的层面上运动，一个是酉表示的解析理论，另一个是处理具有群作用的流形上的几何结构（辛的、泊松的等）。基于对特定表示类的新的系统方法，我们已经看到了对振子群、规范群和微分同态群等各种群的精确分类结果（Janssens, Goldin, Zellner）。特别有趣的新方向是将随机分析和量子场论的方法与李论相结合（gordra, Jorgensen, Vershik），而且似乎对于某些无限维李代数类，全局范畴视角可以提供深刻的新见解（Penkov）。在几何方面，对于无限维哈密顿系统的重要类别，偶对的强大方法正在出现（Gay-Balmaz），规范群的不变量理论与奇点理论（Iohara）联系起来，无限维群上微分方程的新的正则性结果已经得到（Glockner）。李超群的解析表示理论（与李超代数的代数表示理论相反，这也是一个蓬勃发展的领域，但不在本次研讨会的范围内）近年来取得了实质性进展，特别是由源自介观物理的调和分析问题推动。通过这一发展，李群表征理论和超群表征理论得到了进一步的发展。在超群表示和Clifford分析之间的相互作用中也可以观察到类似的效果（Alldridge, de Bie, Przebinda, Wurzbacher）。有限维李群的表示理论的主要焦点已经从还未解决的约群的酉对偶的分类问题转移到表示的结构结果，如子群的分支问题，以及约群的最小表示的分析。无限维表示与非紧流形的全局分析的相互作用也得到了积极的研究，这往往给我们带来新的几何见解。有趣的进展包括复流形上可见作用的广义Cartan分解（A. Sasaki）和实球变的广义Cartan分解（B. Krotz），非紧子群分支律的分析，和共形等变微分系统（T.Kubo）。更具体的资料载于本卷后面的摘要中。李群和超群的表示（739）研讨会：李群和超群的表示

The workshop focussed on recent developments in the representation theory of group objects in several categories, mostly finite and infinite dimensional smooth manifolds and supermanifolds. The talks covered a broad range of topics, with a certain emphasis on benchmark problems and examples such as branching, limit behavior, and dual pairs. In many talks the relation to physics played an important role. Mathematics Subject Classification (2000): 22E, 32M, 46G20 53D, 58B, 58C. Introduction by the Organisers The workshop Representations of Lie groups and supergroups was organized by Joachim Hilgert (Paderborn), Toshiyuki Kobayashi (Tokyo), Karl-Hermann Neeb (Erlangen), and Tudor Ratiu (Lausanne). From the very beginning applications in physics were a major motivation for the study of representations of Lie groups. Later also number theory, specifically the Langlands program, became a driving force. A lot of effort has been invested in the classification of unitary representations during the last two decades of the 20th century. There is a huge body of information available, but the central classification problems are still not completely solved. At the moment research in that direction is concentrated with some American research teams. The majority of research nowadays is focused on benchmark problems and examples, such as branching, limit behavior, and dual pairs. Moreover, the extension of the scope of representation theory to infinite dimensional groups on the one hand, and supergroups on the other, plays an important role. A common feature 736 Oberwolfach Report 13/2013 of these efforts is the identification of relevant classes of examples which are general enough to be interesting, but at the same time have enough restrictions to allow a general theory. In many cases the choice of these benchmark examples is guided by problems from theoretical physics. The focus of this workshop was on these recent developments. The meeting was attended by 51 participants from many European countries, Canada, the USA, and Japan. The meeting was organized around a series of 23 lectures each of 50 minutes duration. The set of speakers chosen was a mix of researchers in all stages of their careers, from very promising young post-docs to senior scientists who have been contributing key results to the field over the last 45 years. We feel that the meeting was exciting and highly successful. The quality of the lectures was outstanding and the intensity of discussions was exceptional even for Oberwolfach standards. A good indicator for this observation is the fact that all the available blackboards were occupied by discussion groups until late every evening of the meeting. What is even more remarkable is that the composition of these discussion groups changed every day. In particular, the researchers who have been focussing on either finite dimensional, infinite dimensional, or super contexts in their recent research did not stay among themselves. New collaborations have been started, and established research partners from different continents had the opportunity to discuss further projects in person. Without going too much into detail, let us mention some important new developments. In the area of infinite-dimensional Lie groups things are moving on two mutually interacting levels, one is the analytic theory of unitary representations and the other deals with geometric structures (symplectic, Poisson etc.) on manifolds with group actions. Based on new systematic approaches to specific classes of representations, we have seen precise classification results for various classes of groups such as oscillator groups, gauge groups and diffeomorphism groups (Janssens, Goldin, Zellner). Particulary interesting new directions are concerned with the combination of methods from stochastic analysis and quantum field theory with Lie theory (Gordina, Jorgensen, Vershik) and it also appears that, for certain classes of infinite-dimensional Lie algebras the global categorical perspective can provide deep new insights (Penkov). On the geometric side the powerful method of dual pairs is now emerging for important classes of infinite dimensional Hamiltonian systems (Gay-Balmaz), invariant theory for gauge groups is connected to singularity theory (Iohara) and new regularity results for differential equations on infinite dimensional groups have been obtained (Glockner). The analytic representation theory of Lie supergroups (as opposed to the algebraic representation theory of Lie superalgebras, which is also a thriving field but was not within the scope of this workshop) has made substantial progress in recent years, fueled in particular by questions of harmonic analysis originating from mesoscopic physics. Through this development a rapprochement of the Representations of Lie Groups and Supergroups 737 representation theory of supergroups and traditional representation theory of Lie groups can be observed. A similar effect can be observed for the interplay between representations of supergroups and Clifford analysis (Alldridge, de Bie, Przebinda, Wurzbacher). The main focus of the representation theory of finite dimensional Lie groups has shifted from the classification problem of the unitary dual of reductive groups (which is still unsolved) to structural results of representations such as branching problems to subgroups, and analysis on minimal representations of reductive groups. The interactions of infinite dimensional representations with global analysis on non-compact manifolds have been also actively studied, which often bring us new geometric insights. Interesting progress includes generalized Cartan decompositions for visible actions on complex manifolds (A. Sasaki) and for real spherical varieties (B. Krotz), analysis on branching laws to non-compact subgroups, and conformally equivariant differential systems (T.Kubo). More specific information is contained in the abstracts which follow in this volume. Representations of Lie Groups and Supergroups 739 Workshop: Representations of Lie Groups and Supergroups

Representations of Lie Groups and Supergroups

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