Multiple Dirichlet series, Whittaker functions, and the cohomology of arithmetic groups

本研究涉及数论与表象理论的相互作用。数论研究的是整数的性质，是数学中最古老的分支。表征理论是对对称性的系统研究，通过发展简单的数学对象来编码基本的不可约的对称性。该提案的主要目的是在“朗兰兹哲学”的精神下探索这两个学科之间的关系，该哲学预测了数论和表征理论之间的深层联系。今天，这些学科所研究的问题和现象成为当代数学研究的驱动力。此外，各个领域本身也在编码和数据传输、化学、物理和理论计算机科学等不同领域贡献了许多应用。本课题将研究的问题集中在算术群和多重狄利克雷级数上，这些对象与自同构形式密切相关。第一部分讨论了算术群的上同调及其相关领域的问题，如局部对称空间的几何以及将上同调与算术几何和伽罗瓦表示联系起来的猜想。本部分的中心目标是计算研究各种算术群上同调中的扭转及其与算术的关系。第二部分研究Weyl群上的多重Dirichletseries。这些是几个复变量的无穷级数，满足一组混合所有变量的泛函方程。本部分的主题强调了这些级数与组合学、表示论、自同构形式和数学物理之间的联系。

This research deals with the interactions between number theory andrepresentation theory. Number theory is the study of the propertiesof the whole numbers, and is the oldest branch of mathematics.Representation theory is the systematic study of symmetry, through thedevelopment of simple mathematical objects that encode the fundamentalirreducible pieces of symmetry. A principal aim of the proposal is toexplore relationships between these two subjects in the spirit of the"Langlands philosophy," which predicts deep connections between numbertheory and representation theory. Today the questions and phenomenaaddressed by these subjects serve as driving forces in much ofcontemporary mathematics research. Moreover, the individual areasthemselves have contributed many applications in such diverse areas ascoding and data transmission, chemistry, physics, and theoreticalcomputer science.The problems that will be investigated in this project focus onarithmetic groups and multiple Dirichlet series, objects that areintimately related to automorphic forms. The first part addressestopics related to cohomology of arithmetic groups and allied areas,such as the geometry of locally symmetric spaces and the conjectureslinking cohomology to arithmetic geometry and Galois representations.A central goal of this part is the computational investigation oftorsion in the cohomology of various arithmetic groups and itsrelation to arithmetic. The second part studies multiple Dirichletseries attached to Weyl groups. These are infinite series in severalcomplex variables that satisfy a group of functional equationsintermixing all the variables. The topics in this part emphasizeconnections between these series and combinatorics, representationtheory, automorphic forms, and mathematical physics.