Towards Langlands duality for affine Kac-Moody groups

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In the current proposal the PI proposes to develop an analog of several aspects of the so-called Langlands duality to affine Kac-Moody groups.More specifically, the PI proposes to study the spherical and Iwahori Hecke algebras for affine Kac-Moody groups as well as their geometric versions (some parts of that program have already been developed in the PI's joint works with D.Kazhdan and M.Finkelberg). The PI also proposes to study an analog automorphic forms for affine Kac-Moody groups as well as some structures generalizing the so-called geometric Langlands correspondence for finite-dimensional reductive groups.The proposed research lies on the border of such fields as number theory, representation theory, algebraic geometry and mathematical physics; successful implementation of the project might shed some new light on the connections 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 algebraic geometry as well as 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.