  We develop some aspects of the theory of $D$-modules on ind-schemes 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.
  Let $N$ be the maximal unipotent subgroup of a reductive group $G$. For a
non-degenerate character $\chi: N(\!(t)\!) \to \mathbb{G}_a$ and a category
$\mathcal{C}$ acted upon by $N(\!(t)\!) $, we define the category
$\mathcal{C}^{N(\!(t)\!), \chi}$ of $(N(\!(t)\!), \chi)$-invariant objects,
along with the coinvariant category $\mathcal{C}_{N(\!(t)\!), \chi}$. These are
the Whittaker categories of $\mathcal{C}$, which are in general not equivalent.
However, there is always a family of functors $\Theta_k:
\mathcal{C}_{N(\!(t)\!), \chi} \to \mathcal{C}^{N(\!(t)\!), \chi}$,
parametrized by $k \in \mathbb{Z}$.
  We conjecture that each $\Theta_k$ is an equivalence, provided that the
$N(\!(t)\!)$-action on $\mathcal{C}$ extends to a $G(\!(t)\!)$-action. Using
the Fourier-Deligne transform (adapted to Tate vector spaces), we prove this
conjecture for $G= GL_n$ and show that the Whittaker categories can be obtained
by taking invariants of $\mathcal{C}$ with respect to a very explicit
pro-unipotent group subscheme (not ind-scheme) of $G(\!(t)\!)$.
