# On the extended Whittaker category

Research paper by Dario Beraldo

Indexed on: 02 Apr '19Published on: 02 Apr '19Published in: Selecta Mathematica

#### 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.