  We combine the Duke-Imamoglu-Ikeda lifting with the theta lifting to produce
new CAP representations of metaplectic, symplectic and orthogonal groups. These
constructions partially generalize the theories of Waldspurger on the Shimura
correspondence and of Piatetski-Shapiro on the Saito-Kurokawa lifting to higher
dimensions. Applications include a relation between Fourier coefficients of
Hilbert cusp forms of weight k+1/2 and a weighted sum of the representation
numbers of a quadratic form of rank 2k by a quadratic form of rank 4k.
