FRG: Collaborative Research: Atlas of Lie Groups and Representations: Unitary Representations

这个项目有两个主要目标。一是解决么正对偶问题：刻画实可约李群的不可约么正表示。主要的工具是计算任何给定群的酉对偶的算法，我们正在“atlas”软件中实现该算法。我们计划使用这些信息来证明关于酉对偶的结果，从Arthur的酉性表示的么正表示开始。第二个主要目标是通过软件、网站、公共研讨会和其他方式，使非专家能够获得关于真实群体的表征理论的信息。Atlas软件在atlas网站上免费提供，并将继续在那里无限期地维护。利用对称性来研究数学和科学问题的想法可以追溯到近200年前傅里叶关于热的工作。在赫尔曼·韦尔、尤金·维格纳和安德烈·韦尔的手中，对称性在量子力学和数论中扮演着核心角色。李群是以挪威数学家索菲斯·李的名字命名的，是对称性背后的数学对象。表象理论研究给定的对称性或李群的所有表现方式。理解所有的“么正”表示(其中的对称操作保持长度不变)是该学科中最重要的悬而未决的问题之一，并且在许多领域都有潜在的应用；例如，它是问题的抽象版本，“什么量子力学系统可以接纳某种对称性？”

This project has two primary goals. The first is to solve the problem of the unitary dual: to describe the irreducible unitary representations of real reductive Lie groups. The primary tool is an algorithm to compute the unitary dual of any given group, which we are implementing inside the "atlas" software. We plan to use this information to prove results about the unitary dual, beginning with the unitarity of Arthur's unipotent representations. The second primary goal is to make information about representation theory of real groups accessible to non-specialists, via the software, a web site, public workshops, and other means. The atlas software is freely available on the atlas web site, and will continue to be maintained there indefinitely.The idea of using symmetry to study problems in mathematics and science dates back to Fourier's work on heat nearly two hundred years ago. In the hands of Hermann Weyl, Eugene Wigner, and Andre Weil, symmetry has come to play a central role in quantum mechanics and in number theory. Lie groups, named after the Norwegian mathematician Sophus Lie, are the mathematical objects underlying symmetry. Representation theory studies all of the ways a given symmetry, or Lie group, can manifest itself. The problem of understanding all "unitary" representations (in which the symmetry operations preserve lengths) is one of the most important unsolved problems in the subject, and has potential applications in many areas; for example, it is an abstract version of the question, "what quantum mechanical systems can admit a certain kind of symmetry?"