Loop group actions on categories and Whittaker invariants

Abstract: 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 $G(!(t)!)$.