Moduli of metaplectic bundles on curves and Theta-sheaves

Abstract

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 Bun ˜ G of metaplectic bundles on X. It also has a local version 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 ( Gr ˜ G ) of certain perverse sheaves on Gr ˜ G , which act on Bun ˜ G by Hecke operators. A version of the Satake equivalence is proved describing Sph ( Gr ˜ G ) as a tensor category. Further, we construct a perverse sheaf on Bun ˜ G corresponding to the Weil representation and show that it is a Hecke eigen-sheaf with respect to Sph ( Gr ˜ G ) .