Representation Theory of Non-Linear Groups

摘要亚当斯这个项目涉及非线性还原群的表示理论，重点是数论的应用。其中一个重点将是将字符从线性组提升到非线性组。本文在Flicker、Kazhdan等人工作的基础上，推广了与黄静松关于GL(n，R)的工作，以及Adam关于MP(2n，R)的工作。最终，它可以将对非线性群的特征的研究简化为更好地理解的线性群，从而将非线性群纳入朗兰兹计划。在与Steve Rallis和Carey Rader的合作中，亚当斯还将研究p-进或实场F上SO(2n+1)和MP(2n，F)之间的轨道积分的猜想几何匹配。最后，亚当斯计划继续他与Dan Barbasch关于R.Lie群(以挪威数学家Sophus Lie命名)上的对偶对应的联合项目。R.Lie群是数学和科学中最普遍的对象之一。李群是研究对称性的数学框架。它们在发现对称性的任何地方都扮演着重要的角色，例如在粒子加速器的设计中。通常情况下，李群不是直接可见的，而是通过它的“表示”出现的，这引发了对李群表示的研究。近年来，更奇异的“非线性”李群发挥了越来越重要的作用，尽管它们的表示理论没有在线性情况下理解得那么好。这个项目的主要目标是将非线性李群的表示的研究提高到线性李群的水平。

Abstract Adams This project concerns representation theory of non-linear reductive groups, with an emphasis on number-theoretic applications. One focus will be on lifting of characters from linear to non-linear groups. This builds on work of Flicker, Kazhdan and others, and generalizes joint work with Jing-Song Huang on GL(n,R), and Adam's work on Mp(2n,R). Ultimately it may reduce the study of characters of non-linear groups to linear groups, which are more well understood, and thereby bring non-linear groups into the Langlands program. In joint work with Steve Rallis and Carey Rader, Adams will also look at a conjectural geometric matching of orbital integrals between SO(2n+1) and Mp(2n,F) over a p-adic or real field F. Finally Adams plan to continue his joint project with Dan Barbasch on the dual pair correspondence over R. Lie groups (named after the Norwegian mathematician Sophus Lie) are among the most ubiquitous objects in mathematics and science. Lie groups are the mathematical framework for the study of symmetry. They play an important role wherever symmetry is found, for example in the design of particle accelerators. Typically a Lie group is not seen directly, but appears via a "representation" of it, which gives rise to the study of representations of Lie groups, In recent years more exotic "non-linear" Lie groups have played an increasingly important role, although their representation theory is not as well understood as in the linear case. The main goal of this project is to bring the study of representations of non-linear Lie groups up to the level of linear Lie groups.