Moduli of metaplectic bundles on curves and Theta-sheaves

Details Moduli of metaplectic bundles on curves and Theta-sheaves Moduli of metaplectic bundles on curves and Theta-sheaves

2004-05-02

arxiv.org/abs/math/0405021v5

Mathematics

Algebraic Geometry

LaTeX2e, 54 pages

Scientific paper

We give a geometric interpretation of the Weil representation of the metaplectic group, placing it in the framework of the geometric Langlands program. For a smooth projective curve X we introduce an algebraic stack \tilde\Bun_G of metaplectic bundles on X. It also has a local version \tilde\Gr_G, which is a gerbe over the affine grassmanian of G. We define a categorical version of the (nonramified) Hecke algebra of the metaplectic group. This is a category Sph(\tilde\Gr_G) of certain perverse sheaves on \tilde\Gr_G, which act on \tilde\Bun_G by Hecke operators. A version of the Satake equivalence is proved describing Sph(\tilde\Gr_G) as a tensor category. Further, we construct a perverse sheaf on \tilde\Bun_G corresponding to the Weil representation and show that it is a Hecke eigen-sheaf.

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