Loop group actions on categories and Whittaker invariants

In the third part, we construct a functor Θ:CN((t)),χ→CN((t)),χ, which depends on a choice of dimension theory for G((t)). We conjecture this functor to be an equivalence. After developing the Fourier-Deligne transform for Tate vector spaces, we prove this conjecture for G=GLn. We show that both Whittaker categories can be obtained by taking invariants of C with respect to a very explicit pro-unipotent group subscheme (not indscheme!) of G((t)).