Theory of branching laws of unitary representations and non-commutative harmonic analysis by transformation groups of geometric structures

分支律是当群表示限于一个子群时的不可约分解(例如，张量积的分解、物理中的对称性破坏等)。分支规律的分析是表象理论的主要课题之一。然而，直到20世纪90年代中期，由于分析无限维的困难，对约化群的酉表示的分支律的研究很少。这一时期我们的主要结果是：1.发表了酉表示的分支律的基本理论，重点是离散可分解情形及其在表示理论本身、非对易调和分析和局部对称空间拓扑中的应用。2.利用Yamabe算子构造伪黎曼流形的共形群的规范表示。特别地，我们使用这种机器来研究in…中的最小表示更确定的正交群。此外，我们还找到了某些超双曲型方程解的(共形)守恒律。当分支律仅限于等距群时，分支律在我们的分析中起着重要的作用(与B.Orsted联合)。3.借助于保轨ANI-全纯映射，给出了复流形的双全纯变换群的无重数定理的统一方法。特别地，我们使用这种机器来研究一般线性群的无重张量积表示。4.在1980年代末，我开始研究伪黎曼齐次空间的不连续群。在A.Borel的纪念卷中，我发表了一篇关于这一领域的综述文章，并给出了半单对称空间及其切对称空间(与T.Yoshino联合)上紧不连续群存在的新判据。5.在这些主题上，我在各种国际会议上做了一小时的特邀演讲，包括ICM 2002、2004泛非会议(全体演讲)和2005亚洲会议。较少

The branching law is the irreducible decomposition of a group representation when restricted to a subgroup (e.g. decomposition of tensor products, breaking symmetry in physics,...). Analysis of branching laws is one of principal subjects in representation theory. Nevertheless, very little has been studied on branching laws of unitary representations of reductive groups, except for some special cases until mid 1990s, partly because of analytic difficulties arising from infinite dimensions.Our main results during this period are :1.To publish the basic theory of branching laws of unitary representations with emphasis on discrete decomposable cases and its applications to representation theory itself, non-commutative harmonic analysis, and topology of locally symmetric spaces.2.To construct the canonical representations of the conformal groups of pseudo-Riemannian manifolds by means of the Yamabe operator. In particular, we use this machinery for the study of minimal representations of in … More definite orthogonal groups. Besides, we find (conformal) conservation laws of solutions to certain ultra-hyperbolic equations. Branching laws when restricted to isometry groups play an important role in our analysis (joint with B.Orsted).3.To make a unified approach to multiplicity-free theorems of the biholomorphic transformation groups of complex manifolds by means of orbit-preserving anthi-holomorphic maps. In particular, we use this machinery for the study of multiplicity-free tensor product representations of general linear groups.4.In the late 1980, I initiated the study of discontinuous groups for pseudo-Riemannian homogeneous spaces. In the memorial volume of A.Borel, I published a survey article on this area, and also gave new criteria for the existence of co-compact discontinuous groups for semisimple symmetric spaces and their tangential symmetric spaces (joint with T.Yoshino). Besides, the deformation spaces of discontinuous groups are investigated.5.On these topics, I gave one-hour invited lectures in various international conferences, including ICM 2002, Pan-African Congress 2004 (plenary address), and Asian Congress 2005. Less