The descent map from automorphic representations of GL(n) to classical groups / David Ginzburg, Stephen Rallis, David Soudry.

Format:

Book

Author/Creator:

Ginzburg, D. (David)

Contributor:

Rallis, Stephen, 1942-

Soudry, David, 1956-

Language:

English

Subjects (All):

L-functions.

Automorphic forms.

Representations of groups.

Physical Description:

1 online resource (350 p.)

Edition:

1st ed.

Place of Publication:

Singapore : World Scientific Pub., c2011.

Language Note:

English

Summary:

This 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 Gelfand-Graev type, or of Fourier-Jacobi 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

Contents:

Preface; Contents; 1. Introduction; 1.1 Overview; 1.2 Formulas for the Weil representation; 1.3 The case, where H is unitary and the place v splits in E; 2. On Certain Residual Representations; 2.1 The groups; 2.2 The Eisenstein series to be considered; 2.3 L-groups and representations related to P; 2.4 The residue representation; 2.5 The case of a maximal parabolic subgroup (r = 1); 2.6 A preliminary lemma on Eisenstein series on GLn; 2.7 Constant terms of E(h, f , ); 2.8 Description of W(M ,D ); 2.9 Continuation of the proof of Theorem 2.1

3. Coefficients of Gelfand-Graev Type, of Fourier-Jacobi Type, and Descent3.1 Gelfand-Graev coefficients; 3.2 Fourier-Jacobi coefficients; 3.3 Nilpotent orbits; 3.4 Global integrals representing L-functions I; 3.5 Global integrals representing L-functions II; 3.6 Definition of the descent; 3.7 Definition of Jacquet modules corresponding to Gelfand-Graev characters; 3.8 Definition of Jacquet modules corresponding to Fourier-Jacobi characters; 4. Some double coset decompositions; 4.1 The space Q \h (V ) /Q; 1. The case where K is a field; 2. The case where K = k k

4.2 A set of representatives for Q \h(V ) /Q1. The case where K is a field and h(Vk) is not even orthogonal and split; 2. The case where h(Vk) is even orthogonal and split; 3. The case K = k k; 4.3 Stabilizers; 1. The case where K is a field and h(V ) is not even orthogonal and split; 2. The case where h(V ) is even orthogonal and split; 3. The case K = k k; 4.4 The set Q\h(W , ) /L ,; 1. The case where K is a field and w is anisotropic; 2. The case where K = k k (and w - anisotropic); 5. Jacquet modules of parabolic inductions: Gelfand-Graev characters

5.1 The case where K is a field5.2 The case K = k k; 6. Jacquet modules of parabolic inductions: Fourier-Jacobi characters; 6.1 The case where K is a field; 6.2 The case K = k k; 7. The tower property; 7.1 A general lemma on "exchanging roots"; 7.2 A formula for constant terms of Gelfand-Graev coefficients; 7.3 Global Gelfand-Graev models for cuspidal representations; 7.4 The general case: H is neither split nor quasi-split; 7.5 Global Gelfand-Graev models for the residual representations E; 7.6 A formula for constant terms of Fourier-Jacobi coefficients

7.7 Global Fourier-Jacobi models for cuspidal representations7.8 Global Fourier-Jacobi models for the residual representations E; 8. Non-vanishing of the descent I; 8.1 The Fourier coefficient corresponding to the partition (m,m,m' - 2m); 8.2 Conjugation of Sm by the element α; 8.3 Exchanging the roots y , and x , (dim V = 2m , m > 2); 8.4 First induction step: exchanging the roots y and x , , for 1 i j [m+1 ]; dim V = 2m; 8.5 First induction step: odd orthogonal groups; 8.6 Second induction step: exchanging the roots y and x , , for i + j m+ 1, j > [m+1 ] (dim V = 2m)

8.7 Completion of the proof of Theorems 8.1, 8.2

Notes:

Description based upon print version of record.

Includes bibliographical references (p. 335-338) and index.

ISBN:

9786613433398

9781283433396

1283433397

9789814304993

9814304999

OCLC:

858228212