We develop some aspects of the theory of D-modules on schemes and indschemes of pro-finite type. These notions are used to define D-modules on (algebraic) loop groups and, consequently, actions of loop groups on DG categories. We also extend the Fourier-Deligne transform to Tate vector spaces.

Let N be the maximal unipotent subgroup of a reductive group G. For a non-degenerate character c of N((t)), and a category C acted upon by N((t)), there are two possible notions of the category of (N((t)),c)-objects: the invariant category and the coinvariant category. These are the Whittaker categories of C, which are in general not equiva- lent.

However, there is always a natural functor T from the coinvariant category to the invariant category. We conjecture that T is an equivalence, provided that the N((t))-action on C is the restriction of a G((t))-action.

We prove this conjecture for G=GLn and show that the 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)).