Pullbacks of Klingen-Eisenstein series attached to Jacobi cusp forms

Let F be a Siegel cusp form of degree n >= 2 and phi be a Jacobi cusp form of degree r ( < n) and index T, where T is a kernel form of size n - r. Suppose F and phi are eigenfunctions of the Hecke operators. Let [phi](r)(n) ((Z, w), s) be the Klingen-Eisenstein series of degree n attached to phi. We show that the Petersson inner product ([phi](r)(n)((Z, 0), s), F(Z)) is essentially equal to the quotient of the standard L-function of F and that of phi. Our result is a generalization of the result of Heim [9] which treated the case n = 2, r = 1.