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Selecta Mathematica

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On the extended Whittaker category

  • Dario BeraldoEmail author
Article
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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.

Mathematics Subject Classification

14D24 14H60 22E57 

Notes

Acknowledgements

As explained in the introduction, this paper realizes a part of the program indicated in [8]. It is a pleasure to thank Dennis Gaitsgory for several crucial discussions: for instance, the usage of the blow-up in Sect. 7 was inspired by his idea of the proof of Theorem 1.4.1 for \(GL_3\).

I am indebted to Sam Raskin for his help regarding unital structures on sheaves of categories over the Ran space. I am also grateful to Dima Arinkin, Jonathan Barlev, Ian Grojnowski and Constantin Teleman.

The constructions and the results of the present paper were announced during the workshop “Towards the proof of the geometric Langlands conjecture” held at IIAS, Jerusalem, in March 2014. I wish to thank the organizers and the participants of the workshop for their inspiring interest.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut de mathématiques de ToulouseUniversité Paul SabatierToulouseFrance

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