Topological Methods in Representation Theory and Automorphic Forms

主要研究者的工作目标是开发几何技术在表示理论和几何朗兰兹程序中的应用。从这个角度来看，基本的几何问题是从微局部的角度来理解扭曲的麦束。对标志流形上的微局部反常束的足够深入的理解应该会导致可约李群的不可约表示的有趣的不变量，并希望最终能够理解无能表示。类似的技术也被应用于模表示理论。最后一个项目的目标是深入了解几何朗兰兹猜想的各种形式。除了微局部技术外，还使用了惠特克模型的观点。该项目的基本主题是几何和表征理论之间的相互作用。将对各种几何的理解简化为它们的对称群的想法可以追溯到19世纪末的菲利克斯·克莱因。从这个观点来看，基本对象，即理论中的“原子”，是给定几何对称群的不可约表示。事实证明，这些原子产生于一种特殊的几何情况，因此我们可以应用几何方法来分析不可约表示的性质。这是提案第一部分的目标。项目的第二部分涉及朗兰兹的深刻猜想。这里所讨论的几何是从算术情况中产生的，这使得数论成为几何和表示理论的混合体。

DMS-0105256Kari VilonenThe goal of the the work of the principal investigator is to develop geometric techniques for applications in representation theory and the geometric Langlands program. From this point of view the basic geometric problem is to understand perverse sheaves from the microlocal point of view. A sufficiently deep understanding of microlocal perverse sheaves on flag manifolds should lead to interesting invariants of irreducible representations of reductive Lie groups and, hopefully, in the end, to an understanding of unipotent representations. Similar techniques are also being applied to modular representation theory. The goal of the last project is to gain insight in the geometric Langlands conjecture in its various forms. In addition to microlocal techniques the point of view of Whittaker models is used.The fundamental theme of the project is the interplay between geometry and representation theory. The idea of reducing the understanding of various geometries to their groups of symmetries goes back to Felix Klein at the end of the nineteenth century. From this point of view the basic objects, the "atoms" of the theory, are the irreducible representations of the group of symmetries of a given geometry. It turns out that these atoms arise from a particular geometric situation and hence we can apply geometric methods to analyze the properties of the irreducible representations. This is the goal of the first part of the the proposal. The second part of the project deals with the deep conjectures of Langlands. Here the geometry in question arises from an arithmetic situation and this brings number theory in the mixture of geometry and representation theory.