Representation theory, geometric Langlands duality and categorification

The representation theory of reductive groups, such as the group GLn of invertible complex matrices, is an important topic, with applications to number theory, algebraic geometry, mathematical physics, and quantum topology. One way to study this representation theory is through the geometric Satake correspondence (also known as geometric Langlands duality). This correspondence relates the geometry of spaces called affine Grassmannians with the representation theory of reductive groups. This correspondence was originally developed from the viewpoint of the geometric Langlands program, but it has many other interesting applications. For example, this correspondence can be used to construct knot homology theories in the framework of categorification. In these lectures, we will begin by explaining the representation theory of GLn, beginning with classification of irreducible representations. We will also give a presentation of the category of representations; such a presentation is known as “skein” or “spider” theory. We will also discuss quantum GLn and see how it can be used to define knot invariants. We will then define the affine Grassmannian for GLn and explain the geometric Satake correspondence from the perspective of skein theory. We will conclude by explaining how these ideas, along with the theory of derived categories of coherent sheaves, can be used to construct knot homology theory. 1 Representation theory of GLn We will begin by reviewing the representation theory of the group GLn of invertible complex matrices. Good references are [FH] and [GW]. 1.1 Preliminaries and examples Definition 1.1. An representation of GLn is a pair (V, ρ) where V is a finitedimensional complex vector space V and ρ : GLn → GL(V ) is a group homomorphism which is also a morphism of algebraic varieties. Saying that ρ is a morphism of algebraic varieties means the following. Given any v ∈ V and α ∈ V ∗, we can define a map ρv,α : GLn → C, ρv,α(g) = α(ρ(g)(v)).