Representation Theory of Non-Linear Groups

抽象的亚当斯 这个项目关注非线性约化群的表示理论，重点是数论应用。一个重点将是从线性到非线性组的字符提升。这建立在Flicker、Kazhdan等人的工作基础上，并推广了与Jing-Song Huang在GL（n，R）上的联合工作，以及Adam在Mp（2n，R）上的工作。最后，它可以减少非线性群的特征的研究，线性群，这是更好地理解，从而把非线性群的朗兰兹计划。 在与Steve Rallis和Carey Rader的联合工作中，亚当斯还将研究SO（2n+1）和Mp（2n，F）在p-adic或真实的域F上的轨道积分的几何匹配。最后，亚当斯计划继续他与丹·巴巴施的合作项目，研究R上的对偶对对应。 李群（以挪威数学家Sophus Lie命名）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。它们在任何发现对称性的地方都起着重要的作用，例如在粒子加速器的设计中。一般来说，李群 李群是直接的，但通过它的“表示”出现，这引起了李群表示的研究，近年来，更奇特的“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不像线性情况下那样被理解。 该项目的主要目标是将非线性李群的表示研究提升到线性李群的水平。

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.