# The Descent Map from Automorphic Representations of GL(n) to Classical Groups

World Scientific, 2011 - Mathematics - 352 pages
8. Non-vanishing of the descent I. 8.1. The Fourier coefficient corresponding to the partition (m, m, m' - 2m). 8.2. Conjugation of S[symbol] by the element [symbol]. 8.3. Exchanging the roots y[symbol] and x[symbol]. 8.4. First induction step : exchanging the roots y[symbol] and x[symbol], for 1[symbol]; dim[symbol]V = 2m. 8.5. First induction step : odd orthogonal groups. 8.6. Second induction step : exchanging the roots y[symbol] and x[symbol], for i + j[symbol]. 8.7. Completion of the proof of Theorems 8.1., 8.2.; dim[symbol]V = 2m. 8.8. Completion of the proof of theorem 8.3. 8.9. Second induction step : odd orthogonal groups. 8.10. Completion of the proof of Theorems 8.1, 8.2; h(V) odd orthogonal -- 9. Non-vanishing of the descent II. 9.1. The case H[symbol]. 9.2. The case H = SO[symbol]. 9.3. Whittaker coefficients of the descent corresponding to Gelfand-Graev coefficients : the unipotent group and its character; h(V) [symbol]. 9.4. Conjugation by element [symbol]. 9.5. Exchanging roots : h(V) = SO[symbol]. 9.6. Nonvanishing of the Whittaker coefficient of the descent corresponding to Gelfand-Graev coefficients : h(V) = SO[symbol], U[symbol]. 9.7. Nonvanishing of the Whittaker coefficient of the descent corresponding to Gelfand-Graev coefficients : h(V) = U[symbol], SO[symbol]. 9.8. The Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients : H[symbol]. 9.9. The nonvanishing of the Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients : H[symbol] = S[symbol]. 9.10. Nonvanishing of the Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients : h(V) = U[symbol] -- 10. Global genericity of the descent and global integrals. 10.1. Statement of the theorems. 10.2 Proof of Theorem 10.3. 10.3. Proof of Theorem 10.4. 10.4. A family of dual global integrals I. 10.5. A family of dual global integrals II. 10.6. L-functions -- 11. Langlands (weak) functorial lift and descent. 11.1. The cuspidal part of the weak lift. 11.2. The image of the weak lift. 11.3. On generalized endoscopy. 11.4. Base change. 11.5. Automorphic induction

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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 Bibliography 335 Index 339 Copyright