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

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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 loopalgebra g((t)); more specifically, the category of representations ofg' at the critical level, denoted g'_(crit)-mod. Our goal is to develop alocalization theory for this category, i.e., to relate it to other categoriesthat are constructed locally out of spaces, endowed with an actionof the loop group G((t)), such as the affine Grassmannian. Ultimately,we believe that the structure of g'_(crit)-mod is governed by a patternof local Langlands correspondence, i.e., its extrinsic structure can bedescribed solely in terms of the space of local systems on the formalpunctured disc with respect to the Langlands dual group.The second (and, so far, unrelated) part of the project is the developmentof representation theory for groups associated to G and 2-dimensional localfields.The object of study of the current project is representation theory.Starting from the 1930's it was realized that there is fairly small listof types of possible fundamental symmetries that occur in Nature.These types are called "root data", and the corresponding concretemathematical objects are called semi-simple groups. In order to havea representation theory one has to pick one of those semi-simplegroups and couple it with a data of algebraic or analytic nature, suchas what is known in mathematics as a "field" or "ring", e.g., real orcomplex numbers. Having a representation theory, the objects thatone deals with look fairly complicated, and most of the questions thatone can ask are not amenable to complete answers. However, wehope to find a point of view on representations, which, on the onehand, would answer some deep questions about their structure, and,on the other hand, will ultimately reduce to questions about theinitial data, i.e., the root system.