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

分支律的确定，即群G的表示在约束于子群H时分解为不可约的表示，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对称性的破缺。许多分支定律在原则上以复杂的组合或拓扑公式的形式已知，这些公式将多重性表示为交替和。这个规则的例外是Littelmann和Knutson的结果，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截面实现（如通过Borel-Weil定理的紧李群的情况），它就允许一个复制核，它携带了表示的全部信息。因此，在H作用下，将该核分解为合适的不变量块，得到了限制表示的分解。在一些具体的例子中，这种分解是通过横向的h轨道泰勒展开得到的，这个想法可以追溯到20世纪70年代的马丁斯。我们提出了一个系统的研究与紧李群G表示相关的再现核，目的是描述消去自由分支律及其渐近行为。

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.