Unitary representations of affine Hecke algebras and reductive p-adic groups

该项目的重点是通过仿射Hecke代数研究约化p-adic群的酉表示。一般的方法是由当地朗兰兹计划，预测之间的对应关系较大类的容许表示的p-adic组和某些几何类别定义的对偶复杂的群体。研究的第一个方向是开发一个算法，用于计算签名不变的埃尔米特形式的仿射Hecke代数，最近取得的成果，在设置的真实的约化群的动机。第二个方向是关于（分次）仿射Hecke代数的基本抽象调和分析问题，以及Dirac算子理论在这一背景下的某些应用。该项目福尔斯属于李群的表示论领域。李群，以挪威数学家Sophus Lie的名字命名，是系统中固有对称性及其表示的数学对象，即，李群可以表现自己的方式，在理论物理和数论中产生了重要影响。这项研究将产生的主题，可以构成博士学位的基础。或硕士论文;与项目相关的组合性质的一些问题适合通过REU计划进行本科研究。

The project focuses on the study of unitary representations of reductive p-adic groups via affine Hecke algebras. The general approach is motivated by the local Langlands program which predicts a correspondence between the larger class of admissible representations of p-adic groups and certain geometric categories defined in terms of the dual complex group. The first direction of research is the development of an algorithm for computing signatures of invariant hermitian forms for the affine Hecke algebra, motivated by the recent results obtained in the setting of real reductive groups. A second direction concerns basic abstract harmonic analysis problems for (graded) affine Hecke algebras, and certain applications of Dirac operator theory in this setting.The project falls in the area of representation theory of Lie groups. Lie groups, named after the Norwegian mathematician Sophus Lie, are mathematical objects underlying the symmetries inherent in a system, and their representations, i.e., the ways in which the Lie groups can manifest themselves, have had an important impact in theoretical physics and number theory. This research will generate topics that can constitute bases for Ph.D. or master theses; some problems of combinatorial nature related to the project are suitable for undergraduate research through REU programs.