Let $G$ be a connected reductive group and $X$ a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Let $Bun_G$ 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 $coeff_{G}$, from the DG category of D-modules on $Bun_G$ to a certain DG category $Wh(G, ext)$, called the extended Whittaker category. Combined with work in progress by other mathematicians and the speaker, this construction allows to formulate the compatibility of the Langlands duality functor $$\mathbb{L}_G : \operatorname{IndCoh}_{N}(LocSys_{\check{G}} ) \to D(Bun_G)$$ with the Whittaker model. For $G = GL_n$ and $G = PGL_n$, we prove that $coeff_G$ is fully faithful. This result guarantees that, for those groups, $\mathbb{L}_G$ is unique (if it exists) and necessarily fully faithful.
