Representations of Infinite-Dimensional Algebras and Related Topics

主要研究者打算探索在局部非阿基米德域上的约化代数群的表示理论和仿射卡茨-穆迪代数的表示理论中的共同趋势。特别是，他有兴趣找到一个统一的描述惠特克泛函在这两个代表性的理论使用几何。 理解惠特克函数的基本几何应该提供新的见解朗兰兹对应领域的有理函数的代数曲线，无论是在有限域或复数域。 一个密切相关的项目是证明几何朗兰兹猜想使用理论的“惠特克sheets '”上的德林费尔德模空间。 另一方面，研究者打算继续研究变形W-代数及其与量子仿射代数表示理论的联系。 总之，本研究的主要思想是，数学中看似完全不同的领域，如表示论和几何学，都包含着各种隐藏的共同趋势。 最近，一些深层次的现象已经被观察到，如朗兰兹对偶。 研究者认为，这些现象是如此基本，以至于它们应该以这样或那样的方式在数学的许多不同领域中表现出来。 我们的目标是在他们仍然未知的地方发现他们，并通过这种方式了解这种现象的确切性质。 这也可能揭示不同领域之间新的和意想不到的联系，通过这些联系，可以获得对数学基本问题的新见解。

The principal investigator intends to explore common trends in representation theory of reductive algebraic groups over local non-Archimedian fields and representation theory of affine Kac-Moody algebras. In particular, he is interested in finding a unified description of the Whittaker functionals in these two representation theories using geometry. Understanding the underlying geometry of the Whittaker functions should provide new insights into the Langlands correspondence over the field of rational functions on an algebraic curve, either over a finite field or over the field of complex numbers. A closely related project is to prove the geometric Langlands conjecture using the theory of ``Whittaker sheaves'' on the Drinfeld moduli spaces. On the other hand, the investigator intends to continue his study of deformed W-algebras and their connections with representation theory of quantum affine algebras. It is expected that this study will lead to better understanding of the nature of the Langlands duality.In summary, the main idea of the proposed research is that seemingly disparate areas of mathematics, such as representations theory and geometry contain various hidden common trends. Recently, several deep phenomena have been observed, such as the Langlands duality. The investigator believes that these phenomena are so fundamental that they should manifest themselves in one way or another in many different areas of mathematics. The goal is to uncover them where they are still unknown, and this way understand the precise nature of the phenomenon. This may also reveal new and unexpected connections between different fields, through which new insights may be gained into fundamental problems of mathematics.