内窥镜（endoscopy）计划的近乎完成引发了关于下一步研究方向的疑问。本文从宏观视角梳理了超越内窥镜计划的各类研究思路，重点阐明它们之间的关联，并强调局部与整体层面的关系。这些思路的核心是R. P. Langlands在2000年的一次讲座中提出的构想，旨在从群G的稳定迹公式中提取出与L群态射${^LH} \to {^LG}$对应的猜想函子性提升像中的绝大部分自守表示。随着函子性问题被拓展到球簇及相关空间的“相对”情形，这类比较背后的部分结构已开始显现。在一个看似无关的方向上，Braverman–Kazhdan同样在2000年前后发起的一项计划——将Godement–Jacquet的函数方程证明推广到任意L函数——近年来重新受到关注。本文也综述了这一方向的思路与进展，并讨论了两个研究计划之间的关联。
英文摘要：
The near-completion of the program of endoscopy poses the question of what lies next. This article takes a broad view of ideas beyond the program of endoscopy, highlighting the connections among them, and emphasizing the relationship between local and global aspects. Central among those ideas is the one proposed in a 2000 lecture of R. P. Langlands, aiming to extract from the stable trace formula of a group G the bulk of those automorphic representations in the image of the conjectural functorial lift corresponding to a morphism of L-groups ${^LH} \to {^LG}$. With the extension of the problem of functoriality to the "relative" setting of spherical varieties and related spaces, some structure behind such comparisons has started to reveal itself. In a seemingly unrelated direction, a program initiated by Braverman--Kazhdan, also around 2000, to generalize the Godement--Jacquet proof of the functional equation to arbitrary L-functions, has received renewed attention in recent years. We survey ideas and developments in this direction, as well, and discuss the relationship between the two programs.
Beyond Endoscopy for $\mathrm{GL}(3, \mathbb{Q})$: Isolation of the residual spectrum via Poisson Summation
Yanqi Lake Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
We discuss generalizations of the Langlands program, from reductive groups to the local and automorphic spectra of spherical varieties, and to more general representations arising as "quantizations" of suitable Hamiltonian spaces. To a spherical $G$-variety $X$, one associates a dual group ${^LG_X}$ and an $L$-value (encoded in a representation of ${^LG_X}$), which conjecturally describe the local and automorphic spectra of the variety. This sets up a problem of functoriality, for any morphism ${^LG_X}\to {^LG_Y}$ of dual groups. We review, and generalize, Langlands' "beyond endoscopy" approac…
