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Geometric Langlands in prime characteristic. (English) Zbl 1390.14044
The geometric Langlands conjecture proposed by Beilinson and Drinfeld, asserts that the derived category \(D(Bund_G^0)\) of D-modules on moduli stack \(Bun_G\) on a smooth projective curve, is equivalent to the derived cateogry \(D(QCoh(LocSys^0_{\check{G}}))\) of quasi-coherent sheaves on the moduli stack \(LocSys_{\check{G}}\) of \(\check{G}\)-local systems on \(C\), where \(\check{G}\) is the Langlands dual group of \(G\). The original conjecture is in the setting of geometry over \(\mathbb{C}\). This paper estabished a similar duality “generically” when the field is of Characteristic \(p\), where \(G\) is assumed to be semisimple and \(C\) is at least of genus \(2\). This work is a generalization of the duality in characteristic \(p\) established by Bezrukavnikov and Braverman [4] when \(G=\mathrm{GL}_n\).
The classical limit of geometric Langlands is a Fourier-Mukai type duality between Hitchin fibrations. It was established “generically” by Donagi and Pantev [14] when the field is \(\mathbb{C}\) by using transcendental methods. A feature of characteristic \(p\) D-module, is that there is a bridge between D-module and certain twisted sheaf on Hitchin fibration. This is exactly the idea of this paper (and also the work by Bezrukavnikov and Braverman [loc. cit.]), to deduce the geometric Langlands duality in characteristic p from Hitchin fibration duality. In the case of \(\mathrm{GL}_n\), the Hitchin fibration can be reduced to a Fourier-Mukai duality on Jacobian of curves, since generailly the Hitchin fiber is the Jacobian of the asccoiated spectral curve. For general \(G\), the story is more complicated. The authors established a Fourier-Mukai type duality on Beilinson 1-motive, which is a slightly generalization of abelian variety.
The equivalence between \(D(Bun_G^0)\) and \(D(QCoh(LocSys^0_{\check{G}}))\) established by the authors, is still not completely confirmed to be the geometric Langlands transform (i.e. satisfy appropriate Hecke operators conditions) conjectured by Beilinson and Drinfeld [3].

14D24 Geometric Langlands program (algebro-geometric aspects)
22E57 Geometric Langlands program: representation-theoretic aspects
Full Text: DOI arXiv
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