One of the central themes in representation theory is the description of the collection of representations of a group in terms of the geometry of a "dual" group. For abelian groups, duality is given by the Fourier transform (Pontrjagin duality), for solvable groups duality is given by the orbit method, and for semisimple groups over local fields (such as the p-adics, reals and complexes) duality is provided by the Langlands program. In this talk I will describe work with D. Nadler in which we develop a new geometric approach to the representation theory of real and complex semisimple Lie groups. The main idea is to consider Lie groups from the point of view of their loop spaces, where miraculously duality becomes far more transparent. Among the applications are a duality for Lusztig's theory of character sheaves and a generalization of Vogan duality.