On the extended Whittaker category

Abstract

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 ${\text {Bun}}_G$ 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 $\mathsf {coeff}_{G,\mathsf {ext}}$ from the DG category of ${\mathfrak {D}}$-modules on ${\text {Bun}}_G$ to a certain DG category ${{\mathcal {W}}h}(G,\mathsf {ext})$, called the extended Whittaker category. This construction allows to formulate the compatibility of the Langlands duality functor $\mathbb {L}_G: {\mathsf {IndCoh}}_{\mathcal {N}}({\text {LocSys}}_{{\check{G}}}) \rightarrow \mathfrak {D}({\text {Bun}}_G)$ with the Whittaker model. For $G=GL_n$ and $G=PGL_n$, we prove that $\mathsf {coeff}_{G,\mathsf {ext}}$ is fully faithful. This result guarantees that, for those groups, $\mathbb {L}_G$ is unique (if it exists) and necessarily fully faithful.