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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

Dummy View - NOT TO BE DELETED