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

分支律是指当一个群被限制到一个子群时，一个群的一个不可约么正表示的不可约分解(例如，张量积的分解，物理上的对称性破坏，…)。寻找分支规律是表象理论的主要课题之一。然而，除了90年代中期以前的一些特殊情况外，关于么正表示的分支规律的研究很少，部分原因是无限维1的分析困难。在此期间，我们的主要结果是建立了“半单李群的无限维表示的离散分支律”的基本理论。也就是说，在几年前发现的新例子的基础上，我们提出了一个离散分支律的提法，并利用微局部分析和代数表示理论证明了分支律是离散可分解的一个判据。此外，我们还发现了这些表示理论Res…的新应用由此引出以下问题：i)非对易调和分析。构造齐次空间的新的离散级数表示。ii)自同构形式。证明了局部黎曼对称空间的模簇的一个零化定理，并发现了与共形几何有关的某些环境下的显式分枝规律.在这些主题上，我在各种国际会议上作了一小时的演讲，并在MSJ上作了一次全体演讲(1999年).此外，我还在欧洲学校(2000)、哈佛大学(2001)和捷克共和国冬季学校(2002)2作了一系列演讲。从20世纪80年代末开始，我开始研究伪黎曼齐次流形的紧致CliffordKlein形式的存在性问题。近年来，人们用离散群、遍历理论、辛几何和么正表示等方法对这一问题进行了研究，揭示了这一问题与数学其他分支之间的相互作用。我在这方面写了一篇说明性的综述，并作为世界数学年2000年的一个项目在《数学无限，2001及以后》中提出了一些开放问题。较少

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