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