Geometric methods in local and global representation theory

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施普林格表示理论，这涉及到Hitchin可积系统的几何，并将其应用于p进群的表示。提出利用Laumon、Ngo等几何方法证明相对迹公式中的轨道积分恒等式（基本引理），构造函数场及其Hecke特征值（局部系统）的显式自同构形式（束），特别是具有野分支的函数场及其Hecke特征值。例如，一般约化群的Kloosterman轴。最后，本项目还计划研究广义旗种上的束的Koszul对偶模式。这个项目自然地处于代数几何、表示理论、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，而表示理论试图对群在向量空间上的行为进行分类。正如经常发生的那样，最有趣的表征来自具有对称性的几何物体，而这些物体又出现在物理学中。这就是几何方法在解决表示理论问题时如此强大的原因。通过这个项目，PI希望能够阐明表示理论和数论中的难题，并发现几何中出现的更多对称性。

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.