Branching laws for 1-parameter families of representations of Lie groups and their asymptotic behavior

李群表示的 1 参数族的分支定律及其渐近行为

• 批准号: 121466542
• 负责人: Professor Dr. Joachim Hilgert
• 金额: --
• 依托单位: Fachgebiet Lie-Theorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2009
• 资助国家: 德国
• 起止时间: 2008-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/121466542?language=en
• 关键词: Branching; laws; parameter; families; representations

The determination of branching laws, i.e., the decomposition of representations of a group G into irreducible ones upon restriction to a subgroup H is a fundamental problem of representation theory. In physical applications it describes the breaking of symmetries for quantum mechanical systems. Many branching laws are known in principle in the form of complicated combinatorial or topological formulas expressing multiplicities as alternating sums. Exceptions to this rule are results of Littelmann and Knutson, which so far are available only for commutative H. If a representation can be realized by holomorphic sections (as in the case of compact Lie groups via the Borel-Weil Theorem) it admits a reproducing kernel which carries the entire information of the representation. Thus decompositions of this kernel into suitable pieces invariant under the action of H yield decompositions of the restricted representations. In specific examples such decompositions have been obtained by Taylor expansions transversal to an H-orbit, an idea going back to S. Martens in the 1970s. We propose a systematic study of reproducing kernels associated with representations of compact Lie groups G with the goal to describe cancelation free branching laws and their asymptotic behavior.

分支律的确定，即，将群G的表示分解为限制在子群H上的不可约表示是表示论的一个基本问题。在物理应用中，它描述了量子力学系统的对称性破缺。许多分支定律原则上以复杂的组合或拓扑公式的形式被知道，这些公式将多重性表示为交替和。Littelmann和Knutson的结果与这一规则相类似，但到目前为止，这些结果只适用于交换H。如果一个表示可以通过全纯部分来实现（就像通过博雷尔-韦尔定理的紧李群的情况一样），那么它就允许一个携带该表示的全部信息的再生核。因此，这个内核分解成适当的片不变H的作用下产生分解的限制表示。在特定的例子中，这样的分解已经通过横向于H轨道的泰勒展开获得，这是一个可以追溯到S的想法。1970年代的马滕斯我们提出了一个系统的研究再生核与表示的紧李群G的目标是描述取消自由分支法律和他们的渐近行为。