Representations of Infinite-Dimensional Algebras and Related Topics

研究了局部非阿基米德域上约化代数群的表示理论和仿射Kac-Moody代数的表示理论的共同趋势。他特别感兴趣的是利用几何学在这两种表示理论中找到惠特克泛函的统一描述。理解惠特克函数的基本几何应该提供新的见解在代数曲线上的有理函数域上的朗兰兹对应，无论是在有限域上还是在复数域上。一个密切相关的项目是在德林菲尔德模空间上用“惠特克束”理论证明几何朗兰兹猜想。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。期望这项研究将有助于更好地理解朗兰兹二象性的本质。综上所述，这项研究的主要思想是，看似不同的数学领域，如表示理论和几何，包含着各种隐藏的共同趋势。最近，一些深层现象被观察到，如朗兰兹对偶。研究者认为，这些现象是如此基本，它们应该以这样或那样的方式在许多不同的数学领域表现出来。我们的目标是在它们尚不为人知的地方发现它们，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，通过它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.