94 research outputs found
Geometric Langlands in prime characteristic
Let be a semisimple algebraic group over an algebraically closed field
, whose characteristic is positive and does not divide the order of the Weyl
group of , and let be its Langlands dual group over . Let
be a smooth projective curve over . Denote by \Bun_G the moduli stack of
-bundles on and \Loc_{\breve G} the moduli stack of -local
systems on . Let D_{\Bun_G} be the sheaf of crystalline differential
operators on \Bun_G. In this paper we construct an equivalence between the
bounded derived category D^b(\on{QCoh}(\Loc_{\breve G}^0)) of quasi-coherent
sheaves on some open subset \Loc_{\breve G}^0\subset\Loc_{\breve G} and
bounded derived category D^b(D_{\Bun_G}^0\on{-mod}) of modules over some
localization D_{\Bun_G}^0 of D_{\Bun_G}. This generalizes the work of
Bezrukavnikov-Braverman in the \GL_n case.Comment: 57 pages, corrected some arguments in section 3.6 and 3.7, to appear
in Compositio Mat
Affine Matsuki correspondence for sheaves
We lift the affine Matsuki correspondence between real and symmetric loop
group orbits in affine Grassmannians to an equivalence of derived categories of
sheaves. In analogy with the finite-dimensional setting, our arguments depend
upon the Morse theory of energy functions obtained from symmetrizations of
coadjoint orbits. The additional fusion structures of the affine setting lead
to further equivalences with Schubert constructible derived categories of
sheaves on real affine Grassmannians
Non-abelian Hodge theory for algebraic curves in characteristic p
Let G be a reductive group over an algebraically closed field of positive
characteristic. Let C be a smooth projective curve over k. We give a
description of the moduli space of flat G-bundles in terms of the moduli space
of G-Higgs bundles over the Frobenius twist C' of C. This description can be
regarded as the non-abelian Hodge theory for curves in positive characteristic.Comment: The introduction and the example for GL_n are partially rewritten.
Section 3 is re-organized. Various typos are corrected. To appear in GAF
A Formula for the Geometric Jacquet Functor and its Character Sheaf Analogue
Let (G,K) be a symmetric pair over the complex numbers, and let X=K\G be the
corresponding symmetric space. In this paper we study a nearby cycles functor
associated to a degeneration of X to MN\G, which we call the "wonderful
degeneration". We show that on the category of character sheaves on X, this
functor is isomorphic to a composition of two averaging functors (a parallel
result, on the level of functions in the p-adic setting, was obtained in [BK,
SV]). As an application, we obtain a formula for the geometric Jacquet functor
of [ENV] and use this formula to give a geometric proof of the celebrated
Casselman's submodule theorem and establish a second adjointness theorem for
Harish-Chandra modules.Comment: Revised version. Equivariancy replaces stratification arguments, so
that the results are applicable to all sheaf setting
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