Fundamental Lemma

We give a geometric analog 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, which is a gerbe over the affine grassmanian of G=Sp_{2n}. We give a tannakian description of the Langlands dual to the metaplectic group. Namely, we introduce a categorical version Sph of the (nonramified) Hecke algebra of the metaplectic group and describe it as a tensor category. The tensor category Sph acts on the derived category D(\tilde\Bun_G) by Hecke operators. Further, we construct a perverse sheaf on \tilde\Bun_G corresponding to the Weil representation and show that it is a Hecke eigensheaf with respect to Sph.