Abstract For a semisimple Lie group G satisfying the equal rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In this paper, we study some of the branching laws for these when restricted to a subgroup H of the same type by combining the classical results with the recent work of T. Kobayashi. We analyze aspects of having differential operators being symmetry-breaking operators; in particular, we prove in the so-called admissible case that every symmetry breaking (H-map) operator is a differential operator. We prove discrete decomposability under Harish-Chandra's condition of cusp form on the reproducing kernels. Our techniques are based on realizing discrete series representations as kernels of elliptic invariant differential operators.

中文翻译：

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半单李群和再生核的分支问题

摘要 对于满足等秩条件的半单李群G，最基本的酉不可约表示族是Harish-Chandra 发现的离散级数。在本文中，我们通过将经典结果与 T. Kobayashi 的最新工作相结合，研究了当限制在同一类型的子群 H 时这些分支的一些分支定律。我们分析了微分算子是对称破坏算子的方面；特别是，我们在所谓的可容许情况下证明了每个对称破坏（H-map）算子都是一个微分算子。我们证明了在 Harish-Chandra 的再生核上的尖峰形式条件下的离散可分解性。我们的技术基于将离散序列表示实现为椭圆不变微分算子的内核。