FRG: Collaborative Research: Atlas of Lie Groups and Representations

摘要：李群的不可约酉表示集的计算问题是表示理论中尚未解决的主要问题之一。本项目的主要目标是通过数学和计算技术的结合来计算实李群和p进李群的酉对偶。特别是，我们计划开发一套软件包，用于计算李群、可容许表示和酉表示的结构理论。表征理论广泛应用于数学和科学学科。特别重要的是它在现代数论、自同构形式和朗兰兹程序中所起的中心作用。另一方面，表征理论，特别是对一元表征的研究，是一个非常技术性的学科，对非专业人士来说很难。该项目的主要目标是使有关李群和表示的信息为广泛的数学和科学受众所访问。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。这包括用于访问表征理论信息的在线工具。此外，我们正在开发一个软件包来进行结构理论、可容许表示和酉表示的计算。这与用于计算半简单李代数的软件包LiE相当，尽管在相当高的水平上。我们设想这个项目在李群中的作用可以与有限群图集在有限群理论中的作用相媲美。

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.