FRG: Collaborative Research: Atlas of Lie Groups and Representations

FRG：协作研究：李群和表示图集

• 批准号: 0554278
• 负责人: Jeffrey Adams
• 金额: $ 81.35万
• 依托单位: American Institute of Mathematics
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0554278&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Atlas; Lie

AbstactAdamsThe problem of computing the set of irreducible unitaryrepresentations of a Lie group is one of the main unsolved problems inrepresentation theory. The primary goal of this project is to computethe unitary dual of real and p-adic Lie groups, by a combination ofmathematical and computational techniques. In particular we plan todevelop a set of software packages for computing structure theory ofLie groups, admissible representations, and unitary representations.Representation theory has applications to a broad spectrum ofmathematical and scientific disciplines. Of particular significance isthe central role it plays in modern number theory, automorphic formsand the Langlands program. On the other hand representation theory, inparticular the study of unitary representations, is a very technicalsubject, and difficult for non-specialists. A primary goal of thisproject is to make information about Lie groups and representationaccessible to a wide mathematical and scientific audience. Everythingwe do is being documented and made available through our web site,www.liegroups.org. This includes on-line tools for accessinginformation about representation theory. In addition we are developinga software package to do computations in structure theory, admissiblerepresentations, and unitary representations. This is comparable tothe software package LiE for computing with semisimple Lie algebras,although at a considerably higher level. We envision this projectplaying a role in Lie groups comparable to the one the Atlas of FiniteGroups plays in finite group theory.

李群不可约酉表示集的计算问题是表示论中尚未解决的主要问题之一。这个项目的主要目标是计算酉对偶的真实的和p-adic李群，通过数学和计算技术相结合。特别是，我们计划开发一套软件包计算结构理论的李群，可接受的陈述，和unitary representations. Representationtheory已应用到广泛的数学和科学学科。特别重要的是它在现代数论、自守形式和朗兰兹纲领中所起的中心作用。另一方面，表示论，特别是酉表示的研究，是一个非常技术性的课题，对于非专业人员来说是困难的。这个项目的主要目标是使有关李群和表示的信息可以被广泛的数学和科学观众所接受。我们所做的一切都被记录下来，并通过我们的网站www.liegroups.org提供。这包括在线工具，用于访问有关表征理论的信息。此外，我们正在开发一个软件包做计算的结构理论，admissiblerepresentations，和unitary representation。这是可比的软件包李E计算与半单李代数，虽然在相当高的水平。我们设想这个项目在李群中发挥的作用与有限群论中的阿特拉斯群相当。