Representations of affine Kac-Moody algebras and representations of Groups over a 2-dimensional local field

Harvard University, Cambridge MA

Investigators

Abstract

Let G be a complex semi-simple group, and let g be its Lie algebra. Our main object of study is the Kac-Moody extension g' of the loop algebra g((t)); more specifically, the category of representations of g' at the critical level, denoted g'_(crit)-mod. Our goal is to develop a localization theory for this category, i.e., to relate it to other categories that are constructed locally out of spaces, endowed with an action of the loop group G((t)), such as the affine Grassmannian. Ultimately, we believe that the structure of g'_(crit)-mod is governed by a pattern of local Langlands correspondence, i.e., its extrinsic structure can be described solely in terms of the space of local systems on the formal punctured disc with respect to the Langlands dual group. The second (and, so far, unrelated) part of the project is the development of representation theory for groups associated to G and 2-dimensional local fields. The object of study of the current project is representation theory. Starting from the 1930's it was realized that there is fairly small list of types of possible fundamental symmetries that occur in Nature. These types are called "root data", and the corresponding concrete mathematical objects are called semi-simple groups. In order to have a representation theory one has to pick one of those semi-simple groups and couple it with a data of algebraic or analytic nature, such as what is known in mathematics as a "field" or "ring", e.g., real or complex numbers. Having a representation theory, the objects that one deals with look fairly complicated, and most of the questions that one can ask are not amenable to complete answers. However, we hope to find a point of view on representations, which, on the one hand, would answer some deep questions about their structure, and, on the other hand, will ultimately reduce to questions about the initial data, i.e., the root system.