Tamely Ramified Langlands Correspondence

本文试图利用从驯服Weil-Deligne群到G的对偶群的同态，对分裂约p进群G的平方可积的驯服分枝表示及其内形进行分类。这个项目是这类群的局部Langlands猜想的一部分。这些表示被划分为有限的集合，称为“l包”。该方法是有限约化群的Lusztig Jordan分解的p进模拟。从函子的观点来看，驯服的l包是拟分裂群及其内部形式的幂偶l包的双目标提升。l包的性质，如形式度和惠特克模型，也将被研究。对称是研究数学和物理对象的有效方法。群体是对称的集合，表征理论研究的是这些对称如何以不同的方式表现出来。举个简单的例子，有一个由五个物体的排列组成的60个对称组，这个组也表现为一个二十面体和富勒烯分子的60个对称。研究者研究无限维对称的某些无限群。在过去的四十年里，数学家和物理学家猜测，在这种对称性和尚未被发现的数字的基本性质之间可能存在着深刻的关系。研究者试图证实这些猜想的一部分，并使它们更加明确。

The investigator attempts to classify square-integrable tamely ramified representations of split reductive p-adic groups G and their inner forms, by means of homomorphisms from the tame Weil-Deligne group into the dual group of G. This project is part of the local Langlands conjecture for such groups. The representations are partitioned into finite sets, called ``L-packets". The method is a p-adic analogue of Lusztig's Jordan decomposition for finite reductive groups. From a functorial point of view, the tame L-packets are bijective lifts of unipotent L-packets for quasi-split groups and their inner forms. Properties of the L-packets, such as formal degrees and Whittaker models, will also be investigated.Symmetry is an effective way to study mathematical and physical objects. A Group is a collection of symmetries, and Representation Theory is the study of how these symmetries manifest in different ways. As a simple example, there is a group of sixty symmetries consisting of permutations of five objects, and this group also manifests as the sixty symmetries of an icosohedron, and of the fullerene molecule. The investigator studies certain infinite groups of infinite dimensional symmetries. In the past forty years, mathematicians and physicists have guessed that there may be deep relations between such symmetries and fundamental, as yet undiscovered, properties of numbers. The investigator attempts to confirm part of these conjectures, and make them more explicit.