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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 ).

Following G. Laumon [12], to a nonramified -adic local system E of rank n on a curve X one associates a complex of -adic sheaves n K E on the moduli stack of rank n vector bundles on X with a section, which is cuspidal and satisfies the Hecke property for E. This is a geometric counterpart of the well-known construction due to J. Shalika [19] and I. Piatetski-Shapiro [18]. We express the cohomology of the tensor product n K E 1 ⊗ n K E 2 in terms of cohomology of the symmetric powers of X . This may be considered as a geometric interpretation of the local part of the classical Rankin-Selberg method for GL(n) in the framework of the geometric Langlands program.

Let k be an algebraically closed field andFor an almost simple algebraic group G we classify central extensions 1 → Gm → E → G(F ) → 1; any such extension splits canonically over G (O). Fix a positive integer N and a primitive character ζ : µ N (k) →Q * (under some assumption on the characteristic of k). Consider the category of G(O)-bi-invariant perverse sheaves on E with Gm-monodromy ζ. We show that this is a tensor category, which is tensor equivalent to the category of representations of a reductive groupǦ E,N . We compute the root datum ofǦ E,N .