Compatibility of the Theta correspondence with the Whittaker functors

We prove in this note that the global geometric theta lifting for the pair (H,G) is compatible with the Whittaker normalization, where (H,G) = (SO2n, Sp2n), (Sp2n, SO2n+2), or (GLn, GLn+1). More precisely, let k be an algebraically closed field of characteristic p > 2. Let X be a smooth projective connected curve over k. For a stack S write D(S) for the derived category of étale constructible Q̄l-sheaves on S. For a reductive group G over k write BunG for the stack of G-torsors on X. The usual Whittaker distribution admits a natural geometrization WhitG : D(BunG) → D(Spec k). We construct an isomorphism of functors between WhitG ◦ F and WhitH where F : D(BunH) → D(BunG) is the theta lifting functor (cf. Theorems 1, 2 and 3). This result at the level of functions (on BunH(k) and BunG(k) when k is a finite field) is well known since a long time and the geometrization of the argument is straightforward. We wrote this note for the following reason. Our proof hold also for k = C in the setting of D-modules. In this case for a reductive group G, Beilinson and Drinfeld proposed a conjecture, which (in a form that should be further precised) says that there exist an equivalence αG between the derived category of D-modules on BunG and the derived category of O-modules on LocǦ. Here LocǦ is the stack of Ǧ-local systems on X, and Ǧ is the Langlands dual group to G. Moreover, WhitG should be the composition D(D−mod(BunG)) αG → D(LocǦ,O) RΓ → D(SpecC). A morphism γ : Ȟ → Ǧ gives rise to the extension of scalars morphism γ̄ : LocȞ → LocǦ. The functor γ̄∗ : D(LocȞ ,O) → D(LocǦ,O) should give rise to the Langlands functoriality functor