Let G be a group and let H be a subgroup of G. The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G, determine the occurrence of an irreducible representation σ of H in the restriction of π to H. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation σ of H, find an irreducible representation π of G such that σ occurs in the restriction of π to H. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan–Gross–Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in []. The method may be applied to other classical groups as well.

中文翻译：

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自构表示和全局Vogan数据包的对等分支问题

令G为组，令H为G的子组。经典的分支规则（或对称破坏）要求：对于G的不可约表示π ，确定在π到H的约束下H的不可约表示σ的出现。这个经典分支问题的倒数分支问题是：对于H的不可约表示σ ，找到G的不可约表示π，使得σ限制在π到H的范围内。对于经典群的自守形态表示，分支问题已通过著名的全局Gan–Gross–Prasad猜想解决。在本文中，我们使用[]中开发的扭曲自同构下降方法研究特殊正交群自同构表示的倒数分支问题。该方法也可以应用于其他经典组。