Pullbacks of Klingen-Eisenstein Series Attached to Jacobi Cusp Forms

Abstract

Let F be a Siegel cusp form of degree n :::_ 2 and ¢ be a Jacobi cusp form of degree r ( < 11) and index T, where T is a kernel form of size n - r. Suppose F and¢ are eigenfunctions of the Hecke operators. Let [¢ ]~ ((Z, w ), s) be the Klingen-Eisenstein series of degree II attached to¢. We show that the Petersson inner product ([¢]~((Z, 0), s), F(Z)) is essentially equal to the quotient of the standard L-function of F and that of¢. Our result is a generalization of the result of Heim [9] which treated the case 11 = 2, r = l .