Mathematical Sciences: Representation Theory of p-adic Groups

本研究通过对紧开子群的限制，对p-add群的光滑表示范畴进行分类。这种分类在本质上是有意义的，也是因为p-进群的表示理论在R.P.朗兰兹为解决数论中的某些基本问题而概述的纲领中所扮演的基本角色。在与C.J.布什内尔的一系列论文中，首席研究员挑出了某些称为类型的对，由一个紧致的开子群和这个子群的一个不可约表示组成。这种类型的存在导致了通过Hecke代数方法分析表示的可能性；在群GL(N)的情况下，这种类型的完全集的存在已经带来了很大的红利。本项目将类型理论推广到其他p-进群上。根据类型理论，也考虑了局部朗兰兹对应，以及使用类型计算局部因子的可能性。这项研究属于广义的p-进代数群范畴。在历史上，代数群的出现是为了描述n维空间的所有变换或对称。然而，这里考虑的空间不是我们熟悉的像直线和平面这样的实空间，而是在数论和纯代数中重要的相关对象。

This research concerns the classification of the category of smooth representations of a p-adic group via restriction to compact, open subgroups. This classification is of interest both intrinsically and because of the fundamental role that the representation theory of p-adic groups plays in the program outlined by R.P. Langlands for tackling certain basic questions in the theory of numbers. In a series of papers with C.J. Bushnell, the principal investigator has singled out certain pairs, called types, consisting of a compact, open subgroup and an irreducible representation of this subgroup. The existence of such a type leads to the possibility of analyzing representations via Hecke algebra methods; the existence of a complete set of such types in the case of the groups GL(N) has already paid substantial dividends. This project extends the theory of types to others p-adic groups. The local Langlands correspondence is also considered in light of the theory of types, as is the possibility of using types to compute local factors. This research falls into the broad category of p-adic algebraic groups. Historically, algebraic groups arose in an effort to describe all the transformations or symmetries of an n-dimensional space. The spaces under consideration here, however, are not the familiar real spaces like the line and the plane, but related objects which are important in number theory and pure algebra.