Representation Theory of Reductive Groups over Local Fields

局部域上的还原群表示论

• 批准号: 1100943
• 负责人: Ju-Lee Kim
• 金额: $ 13.98万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1100943&HistoricalAwards=false
• 关键词: Representation; Theory; Reductive; Groups; over

The proposed research is on representation theory of reductive groups over local fields, i.e. groups like the group of invertible matrices over fields like the field of real or p-adic numbers. The proposed problems have direct applications in the theory of automorphic forms and in the Langlands program. They also have analytic and geometric aspects. The problems are divided into three topics: Harmonic analysis, representation theory of groupoids and the integrability theorem. Here are brief descriptions of those topics: The PI views harmonic analysis on spherical spaces as a generalization of representation theory. His aim is to transfer certain fundamental results from representation theory to the realm of harmonic analysis on spherical spaces.The notion of a groupoid is an interesting generalization of the notion of group. The PI proposes to re-build the representation theory of p-adic groups for the case of groupoids. The integrability theorem is a theorem from the theory of D-modules which has powerful applications in the theory of invariant distributions on real manifolds, which in turn is an important ingredient of representation theory. The theory of D-modules, however, is not applicable to the p-adic case. Based on a previous partial result, the PI proposes to provide an analog of the integrability theorem for the p-adic case.The propose project is about representation theory and harmonic analysis. A model example of the problems in this project can be the Fourier series. The Fourier series is a decomposition of a function on the circle as a sum of imaginary exponent (which are closely related to the trigonometric functions sine and cosine). These exponents change in a very simple way when you rotate the circle. The problems that the PI studies are, in a sense, a generalization of this construction for higher dimensional cases. In general, group theory can be viewed as the study of symmetries of mathematical objects, representation theory - as the study of symmetries of vector spaces, and harmonic analysis - as the study of spaces of functions over geometric objects that possess symmetries. The geometric objects that are studied are real and p-adic manifolds. Real manifolds are geometric objects that locally look like a line, a plain, a three dimensional space (like the one we live in) or a higher dimensional space. p-adic manifolds are certain analogues of real manifolds. Representation theory of real groups and harmonic analysis on real spaces have various applications in geometry, analysis and subsequently in physics, signal processing, image processing and biology. Both the p-adic and the real case have many important applications in number theory. More specifically, in the theory of automorphic forms and in the Langlands program.

拟研究的是局部域上的约化群的表示理论，即像真实的或p-adic数域上的可逆矩阵群这样的群。提出的问题有直接的应用理论的自守形式和朗兰兹计划。它们也有分析和几何方面。问题分为三个主题：调和分析，群胚的表示理论和可积性定理。以下是这些主题的简要描述：PI将球面空间上的调和分析视为表示论的推广。他的目的是转移某些基本成果从代表理论领域的调和分析的球形spaces.The概念的广群是一个有趣的推广概念的团体。PI建议重新建立群胚情况下的p-adic群的表示理论。可积性定理是D-模理论中的一个定理，它在真实的流形上的不变分布理论中有着强大的应用，而不变分布理论又是表示论的一个重要组成部分。然而，D-模的理论不适用于p-adic的情况。基于先前的部分结果，PI提出了一个类似的p-adic情形的可积性定理，这个计划是关于表示理论和调和分析的。在这个项目中的问题的一个模型例子可以是傅立叶级数。傅里叶级数是一个函数在圆上的分解，作为虚指数的和（这与三角函数正弦和余弦密切相关）。当你旋转圆时，这些指数会以一种非常简单的方式变化。PI研究的问题，在某种意义上说，是这种构造在高维情况下的推广。一般来说，群论可以被视为研究数学对象的对称性，表示论-研究向量空间的对称性，调和分析-研究具有对称性的几何对象上的函数空间。所研究的几何对象是真实的和p-adic流形。真实的流形是局部看起来像直线、平面、三维空间（就像我们生活的空间）或高维空间的几何对象。p-adic流形是真实的流形的某些类似物。真实的群的表示论和真实的空间上的调和分析在几何学、分析学以及随后的物理学、信号处理、图像处理和生物学中有着广泛的应用。p-adic和真实的情形在数论中都有许多重要的应用。更具体地说，在自守形式理论和朗兰兹纲领中。