For a group G, its dual appears via the Tannakian duality. In other words, the group ^LG will not be given in terms of generators and relations, but rather by means of the category of its finite dimensional representations.
It is this category that admits a geometric description. Namely, it can be realized by *perverse sheaves* on a certain geometric object attached to G, called the affine Grassmannian. Such a description of ^LG allows us to state the geometric Langlands conjecture for an arbitrary group G (and not just GL(n)), which we will do at the end of the lecture.