On correspondences between certain automorphic forms on Sp_4n(A) and ̃Sp_2n(A)

In 2005, Ginzburg, Rallis and Soudry constructed, in terms of residues of certain Eisenstein series, and by use of the descent method, families of nontempered automorphic representations of Sp_4nm(A) and ̃Sp_2n(2m-1)(A), which generalized the classical work of Piatetski-Shapiro on Saito-Kurokawa liftings. In this paper, we introduce a new framework (Diagrams of Constructions) in order to establish explicit relations among the representations introduced in [GRS05]. In particular, we prove that these constructions yield bijections between a certain set of cuspidal automorphic forms on ̃Sp_2n(A) and a certain set of square-integrable automorphic forms of Sp_4n(A). The proofs use new interpretations of composition of two consecutive descents with explicit identities, which we expect to be very useful to further investigation of the automorphic discrete spectrum of classical groups.