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

###### David Ginzburg, Stephen Rallis, and David Soudry
Publisher:
World Scientific
Publication Date:
2011
Number of Pages:
339
Format:
Hardcover
Price:
99.00
ISBN:
978-981-4304-98-6
Category:
Monograph
We do not plan to review this book.
• Introduction:
• Overview
• Formulas for the Weil representation
• The case, where H is unitary and the place ν splits in E
• On Certain Residual Representations:
• The groups
• The Eisenstein series to be considered
• L-groups and representations related to Pφ
• The residue representation
• The case of a maximal parabolic subgroup (r = 1)
• A preliminary lemma on Eisenstein series on GLn
• Constant terms of E(h, ƒτ,$ar s$)
• Description of W(Mφ,Dk)
• Continuation of the proof of Theorem 2.1
• Coefficients of Gelfand-Graev Type, of Fourier-Jacobi Type, and Descent:
• Gelfand-Graev coefficients
• Fourier-Jacobi coefficients
• Nilpotent orbits
• Global integrals representing L-functions I
• Global integrals representing L-functions II
• Definition of the descent
• Definition of Jacquet modules corresponding to Gelfand-Graev characters
• Definition of Jacquet modules corresponding to Fourier-Jacobi characters
• Some double coset decompositions:
• The space Qjh(V)k/Q
• A set of representatives for Qjh(V)k/Q
• Stabilizers
• The set Qh(Wm,ℓ)k/L0
• Jacquet modules of parabolic inductions: Gelfand-Graev characters:
• The case where K is a field
• The case K = kk
• Jacquet modules of parabolic inductions: Fourier-Jacobi characters:
• The case where K is a field
• The case K = kk
• The tower property:
• A general lemma on “exchanging roots”
• A formula for constant terms of Gelfand-Graev coefficients
• Global Gelfand-Graev models for cuspidal representations
• The general case: H is neither split nor quasi-split
• Global Gelfand-Graev models for the residual representations E$ar τ$
• A formula for constant terms of Fourier-Jacobi coefficients
• Global Fourier-Jacobi models for cuspidal representations
• Global Fourier-Jacobi models for the residual representations E$ar τ$
• Non-vanishing of the descent I:
• The Fourier coefficient corresponding to the partition (m,m,m′ — 2m)
• Conjugation of Sm by the element αm
• Exchanging the roots y1,2 and x1,1 (dimEV = 2m , m > 2)
• First induction step: exchanging the roots yi,j and xj–1,i, for 1 ≤ i < j ≤ $[frac{m+1}{2}]$; dimEV = 2m
• First induction step: odd orthogonal groups
• Second induction step: exchanging the roots y1,2 and xj–1,i, for i + j ≤ m+ 1, j > $[frac{m+1}{2}]$ (dimEV = 2m)
• Completion of the proof of Theorems 8.1, 8.2; dimEV = 2m
• Completion of the proof of Theorem 8.3
• Second induction step: odd orthogonal groups
• Completion of the proof of Theorems 8.1, 8.2; h(V) odd orthogonal
• Non-vanishing of the descent II:
• The case H�� = $widetilde{ m Sp}$4n+2(��)
• The case H = SO4n+2
• Whittaker coefficients of the descent corresponding to Gelfand-Graev coefficients: the unipotent group and its character; h(V) ≠ SO4n+2
• Conjugation by the element $hat{eta}m$
• Exchanging roots: h(V) = SO4n, U4n
• Nonvanishing of the Whittaker coefficient of the descent corresponding to Gelfand-Graev coefficients: SO4n, U4n
• Nonvanishing of the Whittaker coefficient of the descent corresponding to Gelfand-Graev coefficients: h(V) = U4n+2, SO4n+3
• The Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients: H�� ≠ $widetilde{ m Sp}$4n+2(��)
• The nonvanishing of the Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients: H�� = Sp4n(��),$widetilde{ m Sp}$4n(��), U4n(��)
• Nonvanishing of the Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients: h(V) = U4n+2
• Global genericity of the descent and global integrals:
• Statement of the theorems
• Proof of Theorem 10.3
• Proof of Theorem 10.4
• A family of dual global integrals I
• A family of dual global integrals II
• L-functions
• Non-vanishing of the descent II:
• The cuspidal part of the weak lift
• The image of the weak lift
• On generalized endoscopy
• Base change
• Automorphic induction