Affine Flag Varieties and Quantization in Postive Characteristic

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The project consists of two parts: the first one is devoted to descriptionof constructible sheaves on homogeneous spaces (mostly of the loop groups)in terms of coherent sheaves; the second one concerns with study of categories of coherent sheaves by the method of quantization in positive characteristic. The first one is a continuation of our previous work inspired by the (localversion of) "geometric Langlands duality" conjecture of Beilinson and Drinfeld.The second one is suggested by our work (with Mirkovic and Rumynin) on geometric approach (especially, use of D-modules) in modular representation theory. In some situations the two constructions lead to the same (t-)structures on the category of coherent sheaves.The results of the project are expected to yield a better understanding of the nature of this intriguing coincidence.From a formal point of view representation theory is a branch of algebra.However, many of its famous advances were due to discovery of connections toother disciplines such as differential or algebraic geometry, where geometricintuition can be applied. In a previous work we developed such geometric methods for a branch of representation theory called modular representationtheory (where the role of numbers is played by residues of integers modulo a fixed prime number). One of the goals of the present project is to "repaythe debt of algebra to geometry" by applying ideas stemming from thiswork to questions in algebraic geometry. The geometric structures arising fromsuch applications also appear in the study of some topological objects relatedto loop groups (whose definition is similar in spirit to constructions of physists' String Theory). This miraculous coincidence has strong technicalconsequences; we hope to get a better understanding of its nature as a resultof the work on the project.