Nonabelian Jacobian of Projective Surfaces : Geometry and Representation Theory 1 Introduction 2 Nonabelian Jacobian J(X; L; d): main properties 3 Some properties of the filtration H 4 The sheaf of Lie algebras G 5 Period maps and Torelli problems 6 sl2 structures on F 7 sl2 structures on G 8 Involution on G 9 Stratification of T 10 Configurations and theirs equations 11 Representation theoretic constructions 12 J(X; L; d) and the Langlands Duality. The Jacobian of a smooth projective curve is undoubtedly one of the most remarkable and beautiful objects in algebraic geometry. This work is an attempt to develop an analogous theory for smooth projective surfaces a theory of the nonabelian Jacobian of smooth projective surfaces. Just like its classical counterpart our nonabelian Jacobian relates to vector bundles (of rank 2) on a surface as well as its Hilbert scheme of points. But it also comes equipped with the variation of Hodge like structures which produces a sheaf of reductive Lie algebras naturally attached to our Jacobian. This constitutes a nonabelian analogue of the (abelian) Lie algebra structure of the classical Jacobian. This feature naturally relates geometry of surfaces with the representation theory of reductive Lie algebras groups. This work’s main focus is on providing an in depth study of various aspects of this relation. It presents a substantial body of evidence that the sheaf of Lie algebras on the nonabelian Jacobian is an efficient tool for using the representation theory to systematically address various algebro geometric problems. It also shows how to construct new invariants of representation theoretic origin on smooth projective surfaces.