Geometric and Microlocal Study of Automorphic Periods

在谐波分析中，人们把空间上的函数表示为不同频率波的叠加。在数论和朗兰兹规划中，人们对某些齐次空间上的函数感兴趣，其中的“波”是拉普拉斯算子和赫克算子的特殊特征函数，称为自同构形式。自同构形式中最神秘和最重要的不变量是l函数，它是黎曼ζ函数的一大类推广。一个重要的，但不是很好理解的原理是，它们的叠加通常代表一个可以独立描述的函数，根据所谓的球形变化，产生一个称为周期分布的分布。这个分布的谱分解的振幅是l函数的特殊值。该项目将使用量子化的思想（其根源在于数学物理）来研究周期分布和l函数之间的猜想联系。PI还计划召开年度会议，就与该提案相关的主题培训学生和博士后。根据阿贝尔奖获得者罗伯特·朗兰兹自60年代以来发展的远见计划，l函数应该被理解为自同构表示的不变量；这些是“算术流形”的“特征频率”，或者说是（约化的）李群G的表示，以及它的Hecke算子代数，这些Hecke算子表现为商L\G上的函数，其中L是一个算术格。在这种情况下，l函数的精确化身是通过某些称为“周期”的分布来实现的，PI和其他人近年来已经研究并组织成一个连贯的理论。本奖项旨在利用辛几何的思想来研究这些时期。在其他目标中，该项目将研究：(1)周期和l函数之间的对偶性作为群及其对偶群的哈密顿空间之间的对偶性（建立在最近与Ben-Zvi和Venkatesh的工作基础上）；(2)将Waldspurger的相对示踪公式与Knop研究的矩图几何相结合，得到球变的局部谱；(3)泛函的“传递算子”，在不同群和空间的相对轨迹公式之间，体现了朗兰兹“超越内窥镜”的精神。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In harmonic analysis, one represents functions on a space as a superposition of waves with varying frequencies. In number theory and the Langlands program, one is interested in functions on certain homogeneous spaces, where the "waves" are special eigenfunctions of Laplacians and Hecke operators, called automorphic forms. Some of the most mysterious and important invariants of the automorphic forms are the L-functions, a vast class of generalizations of the Riemann zeta function. A significant, but not well-understood, principle is that their superposition often represents a function that can be described independently, in terms of what are known as spherical varieties that give rise to a distribution called the period distribution. The amplitudes of the spectral decomposition of this distribution turn out to be special values of L-functions. The project will investigate conjectural connections between period distributions and L-functions using ideas of quantization (whose roots lie in mathematical physics). The PI also plans yearly meetings to train students and postdocs on the topics related to this proposal. According to the visionary program developed since the '60s by Abel Prize recipient Robert P. Langlands, L-functions should be understood as invariants of automorphic representations; those are the "eigenfrequencies" of "arithmetic manifolds", or else the representations of a (reductive) Lie group G, and of its algebra of Hecke operators, which appear as functions on a quotient L\G, where L is an arithmetic lattice. The precise incarnation of L-functions in this setting is by means of certain distributions called "periods", which the PI and others have studied and organized into a coherent theory in recent years. The present award aims to utilize ideas of symplectic geometry in the study of these periods. Among other goals, this project will study: (1) the duality between periods and L-functions as a duality between Hamiltonian spaces for the group and its dual group (building up on recent work with Ben-Zvi and Venkatesh); (2) the local spectrum of spherical varieties by combining the relative trace formula of Waldspurger with the geometry of the moment map studied by Knop; (3) the "transfer operators" of functoriality, in the spirit of Langlands' "beyond endoscopy", between the relative trace formulas of different groups and spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.