Liftings of automorphic representations and L-functions

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I gave a necessary and sufficient condition for the nonvanishing of theta liftings. I constructed regularization of various period integrals of automorphic forms. Applications include a generalization of the theory of tensor product L-functions for general linear groups, a characterization of GL(n,F)-distinguished residual automorphic representations of GL(n,E), necessary conditions for non-vanishing of periods of Gross-Prasad type of generic automorphic representations on products of unitary groups or metaplectic groups.Moreover, we characterize cuspidal automorphic representations of GL(6) whose exterior cube L-functions have a pole and develop a theory of a Rankin-Selberg integral representation representing the symmetric square L-function for general linear groups and characterize its pole in terms of period integrals with respect to exceptional representations of the double cover of general linear groups.