Unitary Representations and Generalized Harish-Chandra Modules

酉表示和广义 Harish-Chandra 模

• 批准号: 341504-2013
• 负责人: Yee, WaiLing
• 金额: $ 0.95万
• 依托单位: University of Windsor
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572277
• 关键词: Unitary; Representations; Generalized; Harish; Chandra

Representation theory is a subject with numerous applications to other areas of mathematics, physics, and more generally the sciences. The two problems covered by this proposal are unitary representations and generalized Harish-Chandra modules. The Unitary Dual Problem (UDP) is one of the most important open problems in mathematics. For example, solvability of certain differential equations by Fourier analysis techniques corresponds to groups for which the UDP has been solved. Philosophically, the idea behind Fourier analysis is to decompose function spaces into subspaces where the problem is tractable. More generally, I.M. Gelfand's abstract harmonic analysis programme (for which a solution to the UDP is necessary) transfers difficult problems in diverse areas of mathematics to more tractable (though possibly infinitely many) problems in algebra. In addition to numerous areas of mathematics, the UDP has applications in physics and engineering. I propose to extend work that I have done on unitarity of representations by: 1) Simplifying formulas for signature characters. 2) Studying the singular infinitesimal character case. 3) Applying cohomological induction to obtain results for Harish-Chandra modules. Since representations of abelian Lie algebras and compact groups are well-understood, one of the basic philosophies for studying more complicated groups is to restrict one's attention to such a subgroup or subalgebra. This leads to Category O and Harish-Chandra modules. By restricting instead to mixed subgroups, one generalizes these categories and permits the philosophies of these categories to be combined. I propose to continue work on generalized Harish-Chandra modules for mixed subgroups (joint with Paul-Sahi) by studying Kazhdan-Lusztig-Vogan polynomials for mixed subgroups, relating them to polynomials for the Harish-Chandra and parabolic Cateogory O cases, and by implementing a generalized induction functor, thereby establishing a module-theoretic bijection of what is known as S-equivalence.

表示论是一门在数学、物理和更广泛的科学领域有着众多应用的学科。这个建议所涵盖的两个问题是酉表示和广义Harish-Chandra模。 酉对偶问题是数学中最重要的开放问题之一。例如，通过傅立叶分析技术的某些微分方程的可解性对应于UDP已经被求解的组。从哲学上讲，傅立叶分析背后的思想是将函数空间分解成问题易于处理的子空间。一般来说，I.M. Gelfand的抽象谐波分析程序（其中一个解决方案的UDP是必要的）转移困难的问题，在不同领域的数学更易于处理的（虽然可能无限多）问题的代数。除了数学的许多领域，UDP在物理和工程中也有应用。 我建议扩展我在表示的酉性方面所做的工作：1）简化签名字符的公式。2)研究奇异无穷小特征标的情形。3)应用上同调归纳法得到Harish-Chandra模的一些结果。 由于交换李代数和紧群的表示是很好理解的，研究更复杂群的基本哲学之一是将注意力限制在这样的子群或子代数上。这导致了O类和Harish-Chandra模。通过限制，而不是混合子群，一个推广这些范畴，并允许这些范畴的哲学相结合。我建议继续研究广义哈里什-钱德拉模的混合子群（与保罗-萨希联合）通过研究Kazhdan-Lusztig-Vogan多项式的混合子群，将它们与多项式的哈里什-钱德拉和抛物Cateogory O的情况下，并通过实现一个广义归纳函子，从而建立一个模理论的双射什么是所谓的S-等价。