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The CAP representations indexed by Hilbert cusp forms

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.