Mathematical Sciences: Sheaves on Witt Schemes and Trace Formula with Application to Representation Theory

数学科学：维特方案和迹公式及其在表示论中的应用

• 批准号: 9700458
• 负责人: Alexander Polishchuk
• 金额: $ 8.25万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9700458&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Sheaves; Witt; Schemes

Polishchuk 9700458 This research is concerned with generalizing the approach of character sheaves from the case of groups over finite fields to groups over p-adic fields. The work involves two parts. The first part is to devise a notion of sheaves on ind-schemes mimicking that of locally-constant functions on p-adic varieties. The second part is to generalize the Lefschetz-Verdier trace formula and its application to the construction of the discrete series representation of a reductive group over a finite field based on the gluing of perverse sheaves. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.

本文研究了将特征束的方法从有限域上群的情况推广到p进域上群的情况。这项工作包括两部分。第一部分是设计一个在独立格式上的束的概念，模仿p进变量上的局部常数函数。第二部分推广了Lefschetz-Verdier迹公式，并将其应用于构造有限域上基于反常束胶合的约化群的离散级数表示。这是代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。