Geometric Waldspurger periods

1.0 This paper, which is a sequel to [6], is a step towards a geometric version of the Howe correspondence (an analogue of the theta-lifting in the framework of the geometric Langlands program). We consider only the (unramified) dual reductive pair (H = GO2m, G = GSp2n) over a smooth projective connected curve X. Let BunG (resp., BunH) denote the stack of Gtorsors (resp., H-torsors) on X. Using the theta-sheaf introduced in [6], we define functors FG : D(BunH) → D(BunG) and FH : D(BunG) → D(BunH) between the corresponding derived categories, which are geometric analogs of the theta-lifting operators. One of our main results is the geometric Langlands functoriality for the dual pair (GO2,GL2), where GO2 = π∗Gm is a group scheme over X, here π : X̃ → X is a nontrivial étale two-sheeted covering. For H = GO4 and G = GSp4 we also get some partial results described below. Let us now formulate just one consequence of them, which we find striking. Assume that the ground field k = Fq is finite of q elements (with q odd). Set G = GL2. Let E be a rank 2 irreducible l-adic local system on X. Write AutE for the corresponding automorphic sheaf on BunG (cf. [4]). Let fE : BunG(k) → Q̄l denote the function ‘trace of Frobenius’ of AutE. Let φ : Y → X be a nontrivial étale two-sheeted covering. Write PicY for the Picard stack of Y . Let J be a rank one local system on Y equipped with an isomorphism N(J ) →̃ detE, where N(J ) is the norm of J (cf. A.1). Write fJ : (PicY )(k) → Q̄l for the corresponding character (the trace of Frobenius of the automorphic local system AJ corresponding to J ). The Waldspurger period of fE is ∫