Representation Theory of Non-Linear Groups

这个项目关注非线性约化群的表示理论，重点是数论应用。其中一个重点是将角色从线性群体提升到非线性群体。这是基于Flicker， Kazhdan等人的工作，并概括了与Jing-Song Huang在GL（n，R）和Adam在Mp（2n，R）上的共同工作。最终，它可以将非线性群的特征研究简化为更容易理解的线性群，从而将非线性群纳入Langlands程序。在与Steve Rallis和Carey Rader的合作中，Adams还将研究p进域或实域F上SO（2n+1）和Mp（2n，F）之间轨道积分的推测几何匹配。最后Adams计划继续与Dan Barbasch的合作项目，研究r上的对偶对应。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的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，李群不是直接看到的，而是通过它的“表征”出现的，这引起了对李群表征的研究。近年来，更多的奇异的“非线性”李群发挥了越来越重要的作用，尽管它们的表征理论并不像线性情况下那样被很好地理解。本项目的主要目标是将非线性李群表示的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.