The Langlands functoriality conjecture, as reformulated in the “beyond endoscopy” program, predicts comparisons between the (stable) trace formulas of different groups $G_1,G_2$ for every morphism ${}^{L}G_1{\to }^L{G}_2$ between their L-groups. This conjecture can be seen as a special case of a more general conjecture, which replaces reductive groups by spherical varieties and the trace formula by its generalization, the relative trace formula.

The goal of this article and its precursor [11] is to demonstrate, by example, the existence of “transfer operators” between relative trace formulas, which generalize the scalar transfer factors of endoscopy. These transfer operators have all properties that one could expect from a trace formula comparison: matching, fundamental lemma for the Hecke algebra, transfer of (relative) characters. Most importantly, and quite surprisingly, they appear to be of abelian nature (at least, in the low-rank examples considered in this paper), even though they encompass functoriality relations of non-abelian harmonic analysis. Thus, they are amenable to application of the Poisson summation formula in order to perform the global comparison. Moreover, we show that these abelian transforms have some structure — which presently escapes our understanding in its entirety — as deformations of well-understood operators when the spaces under consideration are replaced by their “asymptotic cones”.

In this second paper we use Rankin–Selberg theory to prove the local transfer behind Rudnick's 1990 thesis (comparing the stable trace formula for ${\mathrm{SL}}_2$ with the Kuznetsov formula) and Venkatesh's 2002 thesis (providing a “beyond endoscopy” proof of functorial transfer from tori to ${\mathrm{GL}}_2$). As it turns out, the latter is not completely disjoint from endoscopic transfer — in fact, our proof “factors” through endoscopic transfer. We also study the functional equation of the symmetric-square L-function for ${\mathrm{GL}}_2$, and show that it is governed by an explicit “Hankel operator” at the level of the Kuznetsov formula, which is also of abelian nature. A similar theory for the standard L-function was previously developed (in a different language) by Jacquet.

在“超越内窥镜”程序中重新表述的朗兰兹函数猜想预测了不同组的（稳定）迹线公式之间的比较 $G_1,G_2$ 对于每一个态射 ${}^{升}G_1{\to }^{升}{G}_2$在它们的L组之间。这个猜想可以看作是一个更一般的猜想的特例，它用球形簇代替了还原基团，用它的广义相对迹公式代替了迹公式。

本文及其前身 [11] 的目的是通过实例证明相对迹公式之间“传递算子”的存在，该公式概括了内窥镜的标量传递因子。这些传递算子具有可以从迹公式比较中期望的所有属性：匹配、Hecke 代数的基本引理、（相对）字符的传递。最重要且非常令人惊讶的是，它们似乎具有阿贝尔性质（至少在本文中考虑的低阶示例中），即使它们包含非阿贝尔调和分析的函子关系。因此，它们适合应用泊松求和公式以执行全局比较。而且，

在第二篇论文中，我们使用 Rankin-Selberg 理论来证明 Rudnick 1990 年论文背后的局部转移（比较稳定迹公式 ${\mathrm{SL}}_2$ 使用 Kuznetsov 公式）和 Venkatesh 2002 年的论文（提供了一个“超越内窥镜”的证明从 tori 到 ${\mathrm{GL}}_2$）。事实证明，后者与内窥镜转移并非完全脱节——事实上，我们通过内窥镜转移的证明“因素”。我们还研究了对称平方L函数的函数方程${\mathrm{GL}}_2$，并证明它在库兹涅佐夫公式层次上由一个显式的“汉克尔算子”控制，这也是阿贝尔性质的。Jacquet 先前（以不同的语言）开发了标准L函数的类似理论。