Shin and Templier studied families of automorphic representations with local restrictions: roughly, Archimedean components contained in a fixed $L$-packet of discrete series and non-Archimedean components ramified only up to a fixed level. They computed limiting statistics of local components as either the weight of the $L$-packet or level went to infinity. We extend their weight-aspect results to families where the Archimedean component is restricted to a single discrete-series representation instead of an entire $L$-packet.

We do this by using a so-called “hyperendoscopy” version of the stable trace formula of Ferarri. The main technical difficulties are first, defining a version of hyperendoscopy that works for groups without simply connected derived subgroup and second, bounding the values of transfers of unramified functions. We also present an extension to noncuspidal groups of Arthur’s simple trace formula since it does not seem to appear elsewhere in the literature.

Shin 和 Templier 研究了具有局部限制的自守表示系列：粗略地说，阿基米德分量包含在一个固定的$\mathrm{大号}$- 离散系列和非阿基米德组件的数据包仅分支到固定级别。他们计算了局部组件的限制统计数据，即$\mathrm{大号}$-数据包或级别达到无穷大。我们将它们的权重方面结果扩展到阿基米德分量仅限于单个离散系列表示而不是整个系列的系列$\mathrm{大号}$-包。

我们通过使用 Ferarri 稳定微量公式的所谓“内窥镜检查”版本来做到这一点。主要的技术困难是首先，定义一个适用于没有简单连接派生子组的组的超内窥镜检查版本，其次，限制未分支函数的传递值。我们还提出了对亚瑟简单迹公式的非尖峰组的扩展，因为它似乎没有出现在文献的其他地方。