Loop group actions on categories and Whittaker invariants

© 2017 Elsevier Inc. The present paper is divided in three parts. In the first one, we develop the theory of D-modules on ind-schemes of pro-finite type. This allows to define D-modules on (algebraic) loop groups and, consequently, the notion of strong loop group action on a DG category. In the second part, we construct the functors of Whittaker invariants and Whittaker coinvariants, which take as input a DG category acted on by G((t)), the loop group of a reductive group G. Roughly speaking, the Whittaker invariant category of C is the full subcategory CN((t)),χ⊆C consisting of objects that are N((t))-invariant against a fixed non-degenerate character χ:N((t))→Gaof conductor zero. (Here N is the maximal unipotent subgroup of G.) The Whittaker coinvariant category CN((t)),χis defined by a dual construction. 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)).