Loop group actions on categories and Whittaker invariants

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:N t→G a, of , and a category C acted upon by N(( t)), there are two possible notions of the category of ( N((t)),chi)-objects: the invariant category CNt ,c and the coinvariant category. These are the Whittaker categories of C, which are in general not equivalent. 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)).