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

分支定律是指一个群的不可约酉表示在被限制于子群时的不可约分解（例如张量积的分解，物理中的破缺对称性，…）。分支规律的发现是表征理论的主要课题之一。然而，直到90年代中期，除了一些特殊情况外，对酉表示的分支律的研究很少，部分原因是由于无限维的分析困难。本文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，基于几年前发现的新实例，我们提出了离散分支律的表述，并利用微局部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，我们还发现了这些表示理论的新应用。更多的结果是：1)非交换调和分析。构造齐次空间的新的离散级数表示。ii)自同构形式。证明局部黎曼对称空间的模变的消失定理。此外，我们还发现了与共形几何有关的某些情况下的显式分支定律。关于这些主题，我在各种国际会议上做了一个小时的演讲，并在MSJ为春季奖（1999）做了一次全体演讲。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）做过系列讲座。自20世纪80年代末以来，我开始研究伪黎曼齐次流形的紧化clifford - klein形式的存在性问题。近年来，人们通过离散群、遍历理论、辛几何和酉表示理论等不同的方法对这一问题进行了研究，揭示了它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”中提出了一些开放问题，作为2000年世界数学年的一个项目。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