分支理论是李群表示论中的核心问题之一,一直以来受到很多学者的关注.一个李群的表示在其子群上的限制表示是构造李群表示的重要途径之一.分支理论中的一个自然的问题是,对于一个约化李群G及其约化子群G',是否存在G的无限维不可约酉表示V在G'上的限制是无穷小离散分解的或是可容许的.在本项目中,设G是一个单李群,(G,G')是一个克莱因四元对称对.首先,申请人将研究克莱因对称对的分支规则与对应的对称对的分支规则的关系.其次,对于埃米尔特单李群G,找出所有的克莱因四元对称对(G,G'),使得存在G的无限维不可约酉表示在G'上的限制是无穷小离散分解的.最后,对于一般的单李群G及克莱因对称对(G,G'),申请人将对一些特殊的G的不可约酉表示,研究它们限制在G'上的无穷小离散分解性及可容许性,如离散序列表示,极小表示等.

Branching theory is one of the main topics in representation theory of Lie groups, which is focused by many mathematicians. The restrictions of the representations of Lie groups to the subgroups are important ways to contruct new representations of Lie groups. A natural question in branching theory is, for a reductive Lie group G and its reductive subgroup G', whether there exists an infinite dimensional irreducible unitary representation V of G, which is infinitesimally discretely decomposable or admissible upon restriction to G'. In this project, let G be a simple Lie group, and (G,G') a Klein four symmetric pair. Firstly, the applicant will study the relation between the branching rules for Klein four symmetric pairs and those for the corresponding symmetric pairs. Secondly, for simple Lie groups G of Hermitian type, classify all the Klein four symmetric pairs (G,G') such that there exists an infinite dimensional irreducible unitary representation V of G, which is infinitesimally discretely decomposable upon restriction to G'. Thirdly, for general simple Lie groups G and Klein four symmetric pairs (G,G'), the applicant will study the discrete decomposability and G'-admissiblity of the restrictions of certain classes of irreducibile unitary representations of G to G', for example, discrete series representations, minimal representations, etc.