Topics in Automorphic Forms

傅立叶分析意味着周期性的函数-例如，依赖于时间的量在固定时间后重复其先前值-可以实现为三角函数的总和。 具有更复杂、非交换、周期性的函数是现代数论的核心。 朗兰兹的一个基本观点预测了这些函数与算术相关的对称性之间的联系，即来自多项式方程的根。 这一建议涉及周期函数的一个新的幌子，当功能的定义不是一个组，但在有限覆盖的一组。这以一种新的方式融合了算术和分析。 此外，所要研究的技术可能与数学物理中的构造有关，本项目的主要研究对象是覆盖群上的自守形式。 主要研究者将首先集中在爱森斯坦系列，这是通过平均过程获得的。 自守形式的许多标准工具并没有延续（例如惠特克泛函通常不是唯一的），但一个令人惊讶的图片混合表示理论（规范基础，Mirkovi\'c-Vilonen循环）和数论正在出现。 首席研究员将调查这些和他们的数论应用。他还将研究metaplectic爱森斯坦系列，“高θ系列的残留物。“要理解这些物体的幂幺轨道，发展不同群上的这些系列之间的关系，并在全局和局部构造中使用它们，还有很多工作要做。第三个项目涉及独特的泛函和岩堀Hecke代数。 它提供了独特的泛函的新的建设的潜力。

Fourier analysis implies that functions that are periodic -- for example, quantities depending on time that repeat their previous value a fixed amount of time later -- can be realized as sums of trigonometric functions. Functions with more complicated, non-commutative, periodicities are at the heart of modern number theory. A fundamental vision of Langlands predicts connections between such functions and symmetries related to arithmetic, that is, coming from roots of polynomial equations. This proposal is concerned with periodic functions in a new guise, when the functions are defined not on a group but on a finite cover of a group. This blends arithmetic and analysis in a new way. Moreover, the techniques to be studied may have connections to constructions in mathematical physics.The main objects of study in this project are automorphic forms on covering groups. The principal investigator will focus first on Eisenstein series, which are obtained by an averaging process. Many of the standard tools from automorphic forms do not carry over (e.g. Whittaker functionals are typically not unique), but a surprising picture blending representation theory (canonical bases, Mirkovi\'c-Vilonen cycles) and number theory is emerging. The principal investigator will investigate these and and their number-theoretic applications. He also will study the residues of metaplectic Eisenstein series, "higher theta series." There is much to be done to understand the unipotent orbits attached to these objects, to develop relations between such series on different groups, and to use them both globally and in local constructions. A third project concerns unique functionals and Iwahori Hecke algebras. It offers the potential for new constructions of unique functionals.