Quantizations and Double Affine Representation Theory

表示论被广泛理解为对对称性的研究。更确切地说，它是一个研究的方式，为一个给定的代数对象具体实现通过线性对称。量子化是从经典物理学到量子物理学的一个过渡。表示论和量子化之间的相互作用对数学和物理都非常重要。这是一个既要解决表示论和量子化接口上的一些长期存在的经典问题，又要揭示和研究一种新的“双仿射”对称性的项目。主要目标是充分描述量子化产生的特殊类型的对称性，并确定其关键的数值特征。该项目为研究生提供了研究培训机会。本项目由两部分组成。第一部分围绕轨道方法，一个想法可以追溯到20世纪60年代的基里洛夫，这表明有趣的表示应该从与合适的群作用相关的几何数据中构造出来。PI计划对经典李代数中幂零轨道的等变覆盖的量子化进行分类，并使用这种分类来解决李表示理论中的几个重要问题。PI还计划对某些有趣的Harish-Chandra模块进行分类。第二部分主要研究双仿射型表示理论，包括大正特征域上的有理Cherednik代数和A型量子仿射代数的表示理论。这部分项目的主要目标是获得相应类别的字符公式。该奖项反映了NSF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。

Representation theory is broadly understood as a study of symmetry. More precisely, it is a study of ways for a given algebraic object to be realized concretely via linear symmetries. Quantization means a passage from Classical Physics to Quantum Physics. An interplay between Representation theory and Quantization has been, and still is, of enormous importance for both mathematics and physics. This is a project to both solve some long-tanding classical problems at the interface of Representation theory and Quantization and to uncover and study a new kind of "double affine" symmetry. The principal goals are to fully describe special kinds of symmetries arising from Quantization and determine their crucial numerical characteristics. This project provides research training opportunities for graduate students. This project consists of two parts. The first part centers around the Orbit method, an idea going back to Kirillov in the 1960's, that suggests that interesting representations should be constructed from geometric data related to suitable group actions. The PI plans to classify quantizations of equivariant covers of nilpotent orbits in classical Lie algebras and use this classification to solve several important problems in Lie representation theory. The PI also plans to classify certain interesting Harish-Chandra modules. The second part concentrates on the study of the representation theory of double affine type including that of the rational Cherednik algebras over fields of large positive characteristic and of quantum affine algebras of type A. The primary goal of this part of the project is to obtain character formulas for the corresponding categories.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.