Topological Methods in Representation Theory and Automorphic Forms

DMS-0105256 Kari Vilonen首席研究员的工作目标是开发几何技术，用于表示论和几何Langlands程序。从这个观点出发，基本的几何问题就是从微局部的观点来理解反常层。 对旗流形上的微局部反常层有了足够深入的理解，就能得到约化李群的不可约表示的有趣不变量，并有望最终理解幂单表示。 类似的技术也被应用于模块化表示理论。 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。除了微观局部技术的角度惠特克模型的使用。该项目的基本主题是几何和表示理论之间的相互作用。 将各种几何的理解归结为它们的对称群的想法可以追溯到世纪末的费利克斯·克莱因。从这个观点来看，基本对象，理论的“原子”，是给定几何的对称群的不可约表示。事实证明，这些原子产生于特定的几何情况，因此我们可以应用几何方法来分析不可约表示的性质。 这是该提案第一部分的目标。该项目的第二部分涉及朗兰兹的深层结构。这里的几何问题产生于算术的情况，这带来了数论的混合物几何和代表性的理论。

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.