Mathematical Sciences: Representation Theory of p-adic Groups

通过对紧开子群的限制，研究了p进群的光滑表示范畴的分类。这种分类本身就很有趣，而且因为p进群的表示理论在朗兰兹（R.P. Langlands）为解决数论中的某些基本问题所概述的程序中发挥了基本作用。在与C.J. Bushnell合作的一系列论文中，首席研究员挑选出了一些被称为类型的对，它们由紧致开子群和这个子群的不可约表示组成。这种类型的存在导致了用赫克代数方法分析表征的可能性；在GL(N)群的情况下，这种类型的完整集合的存在已经支付了可观的红利。这个项目将类型理论扩展到其他p进群。本文还从类型理论的角度考虑了局部朗兰兹对应，以及利用类型计算局部因子的可能性。本研究属于p进代数群的广义范畴。从历史上看，代数群是为了描述n维空间的所有变换或对称而产生的。然而，这里所考虑的空间并不是我们所熟悉的实空间，如直线空间和平面空间，而是在数论和纯代数中很重要的相关对象。

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.