Moduli of metaplectic bundles on curves and theta-sheaves

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)). (c) 2006 Elsevier Masson SAS.