Around Langlands duality for representations of affine Kac-Moody groups

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In the current proposal the PI proposes to study several problems related to algebraic groups over 2-dimensional local and global fields, their representations and related geometric problems. More specifically the PI intends to continue his study of Hecke algebras of affine Kac-Moody groups over a local non-archomedian field. He plans to apply those ideas to the theory of affine Eisenstein series; this should produce applications to the theory of (usual) automorphic L-functions. In the second part of his project the PI suggests to attack several (mathematically well-posed) problems related to 4-dimensional gauge theory. Both subjects can be reformulated in terms of various questions about G-bundles on algebraic surfaces. The PI believes that above questions may also be connected to the (not yet formulated) 2-dimensional geometric Langlands duality.The proposed research lies on the border of such fields as number theory, algebraic geometry, representation theory and mathematical physics; succesful implementation of the project might shed some new light on the connection between these fields. For example, number theory is perhaps one of the oldest mathematical subjects and one of its most important parts is called the Langlands program. Recently it has been realized that geometric aspects of the Langlands program have many connections with modern mathematical physics (such as 4-dimensional quantum field theory). The proposed research project should confirm the existence of such links as well broaden and generalize them.