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

分支律指的是群的不可约酉表示在限制于子群时的不可约分解（例如张量积的分解，物理学中的对称破缺，.）。寻找分支律是表示论的主要课题之一。然而，除了一些特殊情况，直到90年代中期，很少有人研究酉表示的分支律，部分原因是无限维带来的分析困难。我们在这一时期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。也就是说，基于我们几年前发现的新的例子，我们提出了一个制定的离散分支法律，并证明了一个标准的分支法律是离散可分解的微局部分析和代数表示理论。此外，我们还发现了这些表示理论的新应用 关于我们 得到了以下几个问题：（1）非对易调和分析。构造齐性空间的新的离散级数表示。ii）自守形式。证明了局部黎曼对称空间的模簇的消失定理。此外，我们发现了与共形几何有关的某些设置中的显式分支律。关于这些主题，我在各种国际会议上发表了一个小时的演讲，并在MSJ的春季奖（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）发表了一系列演讲。自20世纪80年代末以来，我开始研究伪黎曼齐次流形的紧致CliffordKlein型的存在性问题。近年来，人们用离散群、遍历理论、辛几何和酉表示理论等方法研究了这一问题，揭示了它与其他数学分支的相互作用，我在2000年世界数学年项目《Mathematics Unlimited，2001 and beyond》中对这一问题作了简要的综述，并提出了一些开放问题。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