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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.