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Theory of branching laws of unitary representations of reductive Lie groups and geometric realization of representations

还原李群酉表示的分支定律理论及表示的几何实现

基本信息

批准号：
11440018
负责人：
KOBAYASHI Toshiyuki
金额：
$ 3.39万
依托单位：
KYOTO UNIVERSITY (2001)The University of Tokyo (1999-2000)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

The branching law means the irreducible decomposition of an irreducible unitary representation of a group when restricted to a subgroup (e.g. decomposition of tensor products, breaking symmetry in physics,…). It is one of principal subjects in representation theory to find branching laws. Nevertheless, very little has been studied on branching laws of unitary representations, except for some special cases until mid-90s, partly because of analytic difficulties arising from infinite dimensions.1. Our main results during this period are to establish a basic theory of "discrete branching laws of infinite dimensional representations of semisimple Lie groups. Namely, based on new examples that we had found some years ago, we proposed a formulation of discrete branching laws, and proved a criterion for branching laws to be discretely decomposable by using both micro-local analysis and algebraic representation theory. Furthermore, we found new applications of these representation theoretic res … More ults to the following problems :i) Non-commutative harmonic analysis. To construct new discrete series representations for homogeneous spaces.ii) Automorphic forms. To prove a vanishing theorem of modular varieties for locally Riemannian symmetric spaces.Moreover, we found explicitly branching laws in certain settings in connection with conformal geometry.On these topics, I gave one-hour lectures in various international conferences, and a plenary lecture at MSJ for the Spring Prize (1999). Also, I gave series of lectures at European School (2000), at Harvard University (2001), and the Winter School at Czech Republic (2002)2. Since the late 1980s, I have initiated the study of the existence problem of compact CliffordKlein forms of pseudo-Riemannian homogeneous manifolds. Recently, this problem has been studied by different methods such as discrete groups, ergodic theory, symplectic geometry and unitary representation theory, revealing the interactions with other branches of mathematics.I wrote an expository survey on this area and posed some open problems in "Mathematics Unlimited, 2001 and beyond" as a project of the World Mathematical Year 2000. Less

分支定律意味着当限制为子群时，群的不可约酉表示的不可约分解（例如张量积的分解、物理学中的对称性破缺……）。寻找分支规律是表示论的主要课题之一。然而，直到90年代中期，除了一些特殊情况外，对酉表示的分支定律的研究还很少，部分原因是无限维引起的分析困难。1．这一时期我们的主要成果是建立了“半单李群无限维表示的离散分支律”的基本理论。即根据我们几年前发现的新例子，提出了离散分支律的表述，并利用微局域分析和代数表示理论证明了分支律可离散分解的判据。此外，我们还发现了这些新的应用。表示理论导致以下问题：i) 非交换调和分析。为齐次空间构造新的离散级数表示。ii) 自同构形式。为了证明局部黎曼对称空间的模簇消失定理。此外，我们在与共形几何相关的某些设置中发现了明确的分支定律。在这些主题上，我花了一个小时在各种国际会议上发表演讲，并在 MSJ 获得春季奖（1999 年）的全体演讲。此外，我还在欧洲学校（2000 年）、哈佛大学（2001 年）和捷克共和国冬季学校（2002 年）2 进行了一系列讲座。自20世纪80年代末以来，我开始研究伪黎曼齐次流形的紧CliffordKlein形式的存在问题。最近，这个问题已经通过离散群、遍历理论、辛几何和酉表示论等不同方法进行了研究，揭示了与其他数学分支的相互作用。我写了一篇关于这一领域的说明性调查，并在“数学无限，2001年及以后”作为2000年世界数学年的项目中提出了一些开放性问题。