Geometric representation theory and moduli spaces of bundles

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The proposal lies in the field of geometric representation theory. The proposed research essentially consists of 4 subjects; the unifying theme for all of them is the relation between representation theory and geometry of moduli space of bundles on algebraic curves and algebraic surfaces. ***The first part has to do with Hecke algebras for affine Kac-Moody groups over local fields (this subject has been initiated by A.Braverman and D.Kazhdan; we propose to study some further questions. In particular, we plan to develop an analog of the theory of Kazhdan-Lusztig polynomials for double affine Hecke algebras).***In the second part we propose a new uniform geometric definition of local L-functions for representations of a reductive group G over local fields of positive characteristic (attached to an arbitrary finite-dimensional representation of the Langlands dual group) . The definition follows the classical pattern of Godement and Jacquet, but the definition of the relevant Schwartz space is much more involved and it is based on the geometry of perverse sheaves of certain infinite-dimensional spaces attached to the loop group of G. The plan is to show that our construction is well-defined and that it is compatible with some known results on the local Langlands correspondence (such as Lusztig's classificiation of unipotent representations).***In the 3rd part we propose a mathematical definition of the so called Coulomb branch of 3-dimensional N=4 super-symmetric gauge theories and its quantization. The definition is based on the geometry of the affine Grassmannian of the corresponding gauge group. We plan to apply our construction to the proof of the conjectural categorical version of the so called "symplectic duality" due to Braden, Licata, Proudfoot and Webster.***In the 4th part we propose to give a new proof of the Kazhdan-Lusztig conjecture for simple finite-dimensional and affine Lie algebras (in partucular, covering some new cases - such as the case of representations of affine Lie algebras on critical level). The proposed proof is based on the geometry of the so called Zastava spaces (in the finite-dimensional case) and Uhlenbeck spaces of bundles (in the affine case).********