A program for branching problems in the representation theory of real reductive groups
A program for branching problems in the representation theory of real reductive groups
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DOI：
10.1007/978-3-319-23443-4_10
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发表时间：
2015-09
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影响因子：
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通讯作者：
Toshiyuki Kobayashi
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文献类型：
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作者：
Toshiyuki Kobayashi
我们希望理解群的不可约表示在被限制到子群时是如何表现的（分支问题）。我们主要关注的是还原李群的表示，它涉及代数和解析方法。我们将分支问题分为三个阶段：(A)限制的抽象特征；(B)分支定律（限制的不可约分解）；(C)几何模型上对称破缺算子的构造。我们可以期待对阶段B和C的分支问题进行简单而详细的研究，这些分支问题在阶段a中被优先认为是“好的”，相反，在阶段C中，新的结果和方法可能会为分支问题（包括阶段a）开辟另一个富有成效的方向。本文的目的是提供关于主题的新视角，解释基于一些最新进展的方法，并提出一些猜想和开放性问题。
We wish to understand how irreducible representations of a groupGbehave when restricted to a subgroupG′ (thebranching problem). Our primary concern is with representations of reductive Lie groups, which involve both algebraic and analytic approaches. We divide branching problems into three stages: (A) abstract features of the restriction; (B) branching laws (irreducible decompositions of the restriction); and (C) construction of symmetry breaking operators on geometric models. We could expect a simple and detailed study of branching problems in Stages B and C in the settings that area prioriknown to be “nice” in Stage A, and conversely, new results and methods in Stage C that might open another fruitful direction of branching problems including Stage A. The aim of this article is to give new perspectives on the subjects, to explain the methods based on some recent progress, and to raise some conjectures and open questions.