Whittaker functors for G S p 4

1.1 One of the important technical tools in D. Gaitsgory’s proof of the Vanishing Conjecture appearing in the geometric Langlands correspondence ([3]) is the theory of Whittaker functors for GLn. In this paper we define analogous functors for GSp4 and study their properties. Let us first review the situation at the level of automorphic forms on G = Sp4. Let X be a smooth projective absolutely irreducible curve over Fq, F = Fq(X) and A be the adeles ring of F . Let B be a Borel subgroup of G and U ⊂ B its unipotent radical. For a character ψ : U(F )\U(A) → C∗ one has a global Whittaker module over G(A) WMψ = {f : U(F )\G(A) → C | f(ug) = ψ(u)f(g) for u ∈ U(A), f is smooth} Let Acusp(G(F )\G(A)) be the space of cusp forms on G(F )\G(A). The usual Whittaker operator Wψ : Acusp(G(F )\G(A)) → WMψ is given by