Publisher:

World Scientific

Number of Pages:

339

Price:

99.00

ISBN:

978-981-4304-98-6

Date Received:

Thursday, February 2, 2012

Reviewable:

No

Reviewer Email Address:

Publication Date:

2011

Format:

Hardcover

Audience:

Category:

Monograph

*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*P*_{φ}

- The residue representation

- The case of a maximal parabolic subgroup (r = 1)

- A preliminary lemma on Eisenstein series on GL
_{n}

- Constant terms of
*E*(*h*, ƒ_{τ,$ar s$})

- Description of
*W*(*M*_{φ},D_{k})

- 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
*Q*_{j}*h*(*V*)_{k}/*Q*_{ℓ}

- A set of representatives for
*Q*_{j}*h*(*V*)_{k}/*Q*_{ℓ}

- Stabilizers

- The set
*Q**h*(*W*_{m,ℓ})_{k}/*L*_{ℓ},ω_{0}

- The space
*Jacquet modules of parabolic inductions: Gelfand-Graev characters:*- The case where
*K*is a field

- The case
*K*=*k*⊕*k*

- The case where
*Jacquet modules of parabolic inductions: Fourier-Jacobi characters:*- The case where
*K*is a field

- The case
*K*=*k*⊕*k*

- The case where
*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
*S*_{m}by the element α_{m}

- Exchanging the roots
*y*_{1,2}and*x*_{1,1}(dim= 2m , m > 2)_{E}V

- First induction step: exchanging the roots
*y*_{i,j}and*x*_{j–1,i}, for 1 ≤*i*<*j*≤ $[frac{m+1}{2}]$; dim= 2m_{E}V

- First induction step: odd orthogonal groups

- Second induction step: exchanging the roots
*y*_{1,2}and*x*_{j–1,i}, for*i*+*j*≤ m+ 1,*j*> $[frac{m+1}{2}]$ (dim= 2m)_{E}V

- Completion of the proof of Theorems 8.1, 8.2; dim
= 2m_{E}V

- 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*= SO_{4n+2}

- Whittaker coefficients of the descent corresponding to Gelfand-Graev coefficients: the unipotent group and its character;
*h*(*V*) ≠ SO_{4n+2}

- Conjugation by the element $hat{eta}
_{m}$

- Exchanging roots:
*h*(*V*) = SO_{4n}, U_{4n}

- Nonvanishing of the Whittaker coefficient of the descent corresponding to Gelfand-Graev coefficients: SO
_{4n}, U_{4n}

- Nonvanishing of the Whittaker coefficient of the descent corresponding to Gelfand-Graev coefficients:
*h*(*V*) = U_{4n+2}, SO_{4n+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*_{��}= Sp_{4n}(��),$widetilde{ m Sp}$_{4n}(��), U_{4n}(��)

- Nonvanishing of the Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients:
*h*(*V*) = U_{4n+2}

- The case
*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

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