Quantizations and Double Affine Representation Theory

表象理论被广泛理解为对对称性的研究。更确切地说，它是研究给定的代数对象通过线性对称具体实现的方法。量子化意味着从经典物理到量子物理的一次跨越。表象理论和量子化之间的相互作用对于数学和物理都是非常重要的，现在仍然是如此。这是一个既要解决表象理论与量子化交界处的一些长期经典问题，又要揭示和研究一种新的“双仿射”对称性的项目。主要目标是全面描述由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科研培训机会。该项目由两部分组成。第一部分围绕着轨道方法，这一想法可以追溯到基里洛夫在20世纪60年代的S，该方法建议有趣的表示应该由与适当的群体动作相关的几何数据来构建。PI计划对经典李代数中幂零轨道的等变覆盖的量化进行分类，并利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模进行分类。第二部分集中研究双仿射型表示理论，包括大正特征域上的有理Cherednik代数的表示理论和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范畴的特征公式。这一奖项反映了NSF的法定使命，并通过使用基金会的智力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.