On the extended Whittaker category

Let $G$ be a connected reductive group, with connected center, and $X$ a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Let ${\mathrm{Bun}}_G$ denote the stack of $G$-bundles on $X$. In analogy with the classical theory of Whittaker coefficients for automorphic functions, we construct a “Fourier transform” functor, called ${\mathsf{c}\mathsf{o}\mathsf{e}\mathsf{f}\mathsf{f}}_{G,\mathsf{e}\mathsf{x}\mathsf{t}}$, from the DG category of $\mathfrak{D}$-modules on ${\mathrm{Bun}}_G$ to a certain DG category $\mathcal{W}h(G,\mathsf{e}\mathsf{x}\mathsf{t})$, called the \emph{extended Whittaker category}. Combined with work in progress by other mathematicians and the author, this construction allows to formulate the compatibility of the Langlands duality functor $$\mathbb{L}$G: \operatorname{IndCoh}{\mathcal N}(\operatorname{LocSys}_{\check{G}}) \to \mathfrak{D}(\operatorname{Bun}_G)$ with the Whittaker model. For $G=G{L}_n$ and $G=PG{L}_n$, we prove that ${\mathsf{c}\mathsf{o}\mathsf{e}\mathsf{f}\mathsf{f}}_{G,\mathsf{e}\mathsf{x}\mathsf{t}}$ is fully faithful. This result guarantees that, for those groups, ${\mathbb{L}}_G$ is unique (if it exists) and necessarily fully faithful.