L-functions for symplectic groups

We construct global integrals of Shimura type, which represent the standard (partial) L-function L-S(pi x sigma,s), for pi x sigma, an irreducible, automorphic, cuspidal and generic representation of Sp(2n)(A) x GL(k)(A). We present two different constructions : one for the case n > k and one for the case n less than or equal to k. These constructions are, in a certain sense, dual to each other. We also study the (completely analogous) case where a is a representation of the metaplectic group (Sp) over tilde(2n)(A). Here we have to first fix a choice of a non-trivial additive character psi, in order to define the L-function L-psi(S)( pi x sigma,s). The integrals depend on a cusp form of pi, a theta series on (Sp) over tilde p(2n)(A) (l = min(n, k)) and an Eisenstein series on Sp(2k)(A) (Or (Sp) over tilde(2k)(A)) induced from sigma.