Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory
Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory
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相对迹公式之间的传递算子和 Hankel 变换，II：Rankin-Selberg 理论
DOI：
10.1016/j.aim.2021.108039
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发表时间：
2022
影响因子：
1.7
通讯作者：
Sakellaridis, Yiannis
中科院分区：
文献类型：
--
作者：
Sakellaridis, Yiannis
在“超越内窥镜”计划中重新表述的Langlands泛函猜想，预测了不同群g1， g2的（稳定）迹公式之间的比较，对于它们L群之间的每个态射g1 L→l2 g2。这个猜想可以看作是一个更一般的猜想的特例，这个猜想用球变代替了约化群，用它的推广——相对的迹公式代替了迹公式。本文及其前驱[11]的目的是通过实例证明相对迹公式之间的“传递算子”的存在性，从而推广内窥镜的标量传递因子。这些转移算子具有迹公式比较的所有特性：匹配、Hecke代数的基本引理、（相对）字符的转移。最重要的是，令人惊讶的是，尽管它们包含了非阿贝尔调和分析的泛函关系，但它们似乎具有阿贝尔性质（至少在本文中考虑的低秩示例中是如此）。因此，它们适用于泊松求和公式的应用，以便进行全局比较。此外，我们证明当考虑的空间被它们的“渐近锥”所取代时，这些阿贝尔变换具有一些结构——目前我们无法完全理解它们——作为熟知的算子的变形。在第二篇论文中，我们使用Rankin-Selberg理论来证明Rudnick 1990年论文（将SL 2的稳定迹公式与库兹涅佐夫公式进行比较）和Venkatesh 2002年论文（提供从环面到GL 2的泛函转移的“超越内窥镜”证明）后面的局部转移。事实证明，后者与内镜下转移并不是完全脱节的——事实上，我们通过内镜下转移来证明“因素”。我们还研究了gl2的对称平方l函数的泛函方程，并证明了它在Kuznetsov公式的水平上由显式的“Hankel算子”控制，这也是阿贝尔性质的。一个类似的关于标准l函数的理论以前是由Jacquet开发的（用不同的语言）。
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 G 1 L→ 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 SL 2 with the Kuznetsov formula) and Venkatesh's 2002 thesis (providing a “beyond endoscopy” proof of functorial transfer from tori to 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 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.
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