Unipotent Representations and Associated Cycles

自然规律，特别是物理规律，满足基本的对称性。在数学语言中，这些对称性通常用李群及其表示（以19世纪数学家Sophus Lie的名字命名，他在微分方程解的研究中开创了这些概念）来表达。这个项目是关于单位表示的性质。对酉表示的现代研究源于量子物理学。此外，它们在许多其他应用中起着至关重要的作用，如断层扫描、晶体学和信号处理。用更专业的术语来说，这个项目关注的是幂偶表示的性质，幂偶表示是实群和p进约群的酉对偶的组成部分。酉对偶的确定是这类群表示理论中的一个主要问题。PI将对仿射Hecke代数模块的厄米特形式的签名进行形式化和锐化的推测。这是确定p进群的酉对偶所必需的。这个项目的另一个重点是明确在描述单一对偶中的单能表征的作用。除了线性群，PI还将研究数学物理和自同构形式研究中出现的非线性和不连通群的这些概念。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Laws of nature, in particularly those of physics, satisfy basic symmetries. In mathematical language, these symmetries are most often expressed in terms of Lie groups and their representations (named after Sophus Lie, a 19th century mathematician who pioneered these concepts in his study of solutions of differential equations). This project is concerned with properties of unitary representations. The modern study of unitary representations stems from quantum physics. In addition, they play a crucial role in many other applications such as tomography, crystallography, and signal processing.In more technical terms, this project is focused on properties of unipotent representations, which are the building blocks of the unitary duals of real and p-adic reductive groups. The determination of the unitary dual is a major problem in the representation theory of such groups. The PI will formulate and sharpen conjectures on the signatures of hermitian forms for modules of the affine Hecke algebra. These are essential for the determination of the unitary duals of p-adic groups. Another focus of this project is to make explicit the role of unipotent representations in the description of the unitary dual. In addition to linear groups, the PI will study these notions for nonlinear and disconnected groups that arise in mathematical physics and in the study of automorphic forms.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.