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的值--可以实现为三角函数的和。具有更复杂的非对易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和与算术有关的对称性之间的联系，即来自多项式方程的根。当周期函数不是定义在群上而是定义在群的有限覆盖上时，这一建议涉及一种新的伪装的周期函数。这以一种新的方式融合了算术和分析。此外，所要研究的技巧可能与数学物理中的构造有关。本课题的主要研究对象是覆盖群上的自同构形式。首席研究员将首先关注爱森斯坦级数，它是通过平均过程获得的。许多来自自同构形式的标准工具并不是唯一的(例如，惠特克泛函通常不是唯一的)，但一种令人惊讶的图画混合表示理论(正则基、米尔科维奇c-维洛宁圈)和数论正在出现。首席研究员将研究这些和及其数论应用.他还将研究变质艾森斯坦系列的残留物，即“更高的θ系列”。要理解这些物体上的幂等轨道，发展不同群上这些级数之间的关系，以及在全局和局部结构中使用它们，还有很多工作要做。第三个项目涉及唯一泛函和Iwahori Hecke代数。它为构造独特的泛函提供了新的可能性。

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.