Moduli of metaplectic bundles on curves and Theta-sheaves

Historically θ-series have been one of the major methods of constructing automorphic forms. A representation-theoretic appoach to the theory of θ-series, as discoved by A. Weil [18] and extended by R. Howe [12], is based on the oscillator representation of the metaplectic group (cf. [17] for a recent survey). In this paper we propose a geometric interpretation this representation (in the nonramified case) placing it in the framework of the geometric Langlands program. Let X be a smooth projective curve over an algebraically closed field of characteristic > 2. Let G denote the sheaf of automorphisms of On X ⊕Ω n (here Ω is the canonical line bundle on X) preserving the symplectic form ∧2(On X ⊕Ω n) → Ω, so G is a twisted form of Sp2n. We introduce an algebraic stack B̃unG, which we think of as the moduli stack of metaplectic bundles on X. It also has a local version G̃rG, which is a gerb over the affine grassmanian GrG. We introduce certain category Sph(G̃rG) of l-adic perverse sheaves on G̃rG, which naturally act on B̃unG by Hecke operators. This is a geometric analog of (a part of) the Hecke algebra of the metaplectic group (over a local non-archimedian field). We equip Sph(G̃rG) with the structure of the tensor category and prove a version of the Satake equivalence. Namely, there exists a reductive group Ǧ over Q̄l and a canonical equivalence between Sph(G̃rG) and the category Rep(Ǧ) of Q̄l-representations of Ǧ. After an additional choice, Ǧ identifies with Sp2n. We construct a perverse sheaf Aut on B̃unG, which we think of as a geometric analog of the Weil representation. We calculate the fibres of Aut and its constant terms for maximal parabolic subgroups of G. Finally, we argue that Aut is a Hecke eigensheaf on B̃unG with eigenvalue