Geometric methods in local and global representation theory

局部和全局表示论中的几何方法

• 批准号: 1261660
• 负责人: Zhiwei Yun
• 金额: $ 4.35万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1261660&HistoricalAwards=false
• 关键词: Geometric; methods; local; global; representation

The proposal aims to apply methods from algebraic geometry to study representations of algebraic groups. The PI and his collaborators will further develop his theory of global Springer representations, which involves the geometry of Hitchin integrable systems, and apply it to representations of p-adic groups. The proposal also proposes to use geometric methods of Laumon, Ngo, etc. to prove orbital integral identities (Fundamental Lemmas) in relative trace formulae and to construct explicit automorphic forms (sheaves) for function fields and their Hecke eigenvalues (local systems), especially those with wild ramifications. e.g., Kloosterman sheaves for general reductive groups. Finally the proposed project also plans to study Koszul duality patterns for sheaves on generalized flag varieties.This project naturally sits at the intersection of algebraic geometry, representation theory, number theory, and mathematical physics. In these subjects as well as in understanding our physical world, symmetry is a central theme. Group theory is a uniform way to study such symmetries, and representation theory is trying to classify the actions of groups on vector spaces. As often happens, the most interesting representations come from geometric objects with symmetries, which in turn appear in physics. This is why geometric methods are so powerful in solving representation-theoretic problems.Through this project, the PI hopes to shed light on hard problems in representation theory and number theory, and to discover more symmetries that appear in geometry.

该提案旨在应用代数几何的方法来研究代数群的表示。PI和他的合作者将进一步发展他的全局Springer表示理论，其中涉及希钦可积系统的几何，并将其应用于p-adic群的表示。该提案还建议使用Laumon，Ngo等的几何方法来证明相对迹公式中的轨道积分恒等式（基本引理），并为函数场及其Hecke特征值（局部系统）构造显式自守形式（层），特别是那些具有野生分支的。例如，在一个实施例中，一般约化群的Kloosterman层。最后，拟议的项目还计划研究广义旗簇上层的Koszul对偶模式。这个项目自然位于代数几何，表示论，数论和数学物理的交叉点。在这些学科中，以及在理解我们的物理世界中，对称性是一个中心主题。群论是研究这种对称性的统一方法，而表示论试图对向量空间上的群的作用进行分类。正如经常发生的那样，最有趣的表示来自具有对称性的几何对象，这些对象反过来又出现在物理学中。这就是为什么几何方法在解决表示论问题上如此强大的原因。通过这个项目，PI希望阐明表示论和数论中的难题，并发现更多出现在几何中的对称性。