## The Descent Map from Automorphic Representations of GL(n) to Classical GroupsThis book introduces the method of automorphic descent, providing an explicit inverse map to the (weak) Langlands functorial lift from generic, cuspidal representations on classical groups to general linear groups. The essence of this method is the study of certain Fourier coefficients of GelfandOCoGraev type, or of FourierOCoJacobi type when applied to certain residual Eisenstein series. This book contains a complete account of this automorphic descent, with complete, detailed proofs. The book will be of interest to graduate students and mathematicians, who specialize in automorphic forms and in representation theory of reductive groups over local fields. Relatively self-contained, the content of some of the chapters can serve as topics for graduate students seminars. |

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### Contents

1 Introduction | 1 |

2 On Certain Residual Representations | 17 |

3 Coefficients of GelfandGraev Type of FourierJacobi Type and Descent | 41 |

4 Some double coset decompositions | 65 |

GelfandGraev characters | 81 |

FourierJacobi characters | 121 |

7 The tower property | 151 |

8 Nonvanishing of the descent I | 187 |

9 Nonvanishing of the descent II | 235 |

10 Global genericity of the descent and global integrals | 281 |

11 Langlands weak functorial lift and descent | 313 |

335 | |

339 | |

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### Common terms and phrases

Asai Assume that h(V assumptions automorphic representation Borel subgroup Chapter coefficients completes the proof conclude conjugation consider constant term corresponding cuspidal representation Deﬁne deﬁnition Denote descent diagonal dimEV double cosets Eisenstein series ﬁnite ﬁrst ﬁxed Fourier coefﬁcient Fourier-Jacobi g i g r Ginzburg GLm(AE global integrals H is metaplectic Heisenberg group hence holomorphic identically zero inducing representation irreducible summand isomorphic isotropic subspace Jacquet modules L-functions Lemma Let f Levi subgroup LS(o LS(T matrix maximal isotropic subspace metaplectic group nilpotent nontrivial notation Note odd orthogonal orthogonal group pairwise parabolically induced pole proof of Theorem Proposition proves quasi-split Rallis and Soudry Recall replace residual representation resp respect satisﬁes smooth automorphic function SO4n space split standard parabolic subgroup symplectic Theorem 2.1 trivial uniform moderate growth unipotent radical unitary group unramiﬁed constituent vector Weyl element Whittaker character Whittaker coefﬁcient Whittaker function Witt index write