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

仿射 Hecke 代数和还原 p-adic 群的酉表示

• 批准号: 1620329
• 负责人: Baiying Liu
• 金额: $ 2.29万
• 依托单位: Institute For Advanced Study
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-10-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620329&HistoricalAwards=false
• 关键词: Unitary; representations; affine; Hecke; algebras

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.

该项目侧重于通过仿射Hecke代数研究可约p进群的幺正表示。一般的方法是由局部朗兰兹规划激发的，该规划预测了p进群的可容许表示的较大类与由对偶复群定义的某些几何范畴之间的对应关系。第一个研究方向是发展一种算法来计算仿射Hecke代数的不变厄米特形式的特征，这是受到最近在实约化群设置中获得的结果的启发。第二个方向涉及（梯度）仿射Hecke代数的基本抽象调和分析问题，以及狄拉克算子理论在这种情况下的某些应用。该项目属于李群的表示理论领域。李群，以挪威数学家Sophus Lie的名字命名，是系统中固有对称性的数学对象，它们的表示，即李群表现自己的方式，在理论物理和数论中有着重要的影响。本研究将产生可构成博士或硕士论文基础的课题；一些与项目相关的组合性问题适合通过REU项目进行本科生研究。