Representations of Infinite-Dimensional Algebras and Related Topics

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

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.