Pullbacks of Klingen-Eisenstein Series Attached to Jacobi Cusp Forms

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