## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 951-1008 |

Number of pages | 58 |

Journal | Israel Journal of Mathematics |

Volume | 192 |

Issue number | 2 |

DOIs | |

State | Published - Nov 2012 |

### Bibliographical note

Funding Information:∗All three authors are partially supported by the USA–Israel Binational Science Foundation, and the second-named author is also supported in part by NSF Grant DMS–0653742 and DMS–1001672. Received November 16, 2010 and in revised form July 12, 2011