Archimedean Zeta Integrals for CL(3) x GL(2) / Miki Hirano, Taku Ishii, Tadashi Miyazaki.

Format:

Book

Author/Creator:

Hirano, Miki.

Contributor:

Ishii, Taku.

Miyazaki, Tadashi.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.278

Language:

English

Subjects (All):

Coulomb functions.

Riemann integral.

Functions, Zeta.

Automorphic forms.

Physical Description:

1 online resource (136 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"In this article, we give explicit formulas of archimedean Whittaker functions on GL(3) and GL(2). Moreover, we apply those to the calculation of archimedean zeta integrals for GL(3) x GL(2), and show that the zeta integral for appropriate Whittaker functions is equal to the associated L-factors"-- Provided by publisher.

Contents:

Cover

Title page

Introduction

Acknowledgments

Part 1. Whittaker functions

Chapter 1. Basic objects

1.1. Notation

1.2. Groups and algebras

1.3. Whittaker functions

1.4. Capelli elements

1.5. The gamma function and the Bessel functions

1.6. Special functions of two variables

Chapter 2. Preliminaries for ( ,\bR)

2.1. Generalized principal series representations

2.2. The elements of \g_{\bC} and (\g_{\bC})

2.3. The eigenvalues of generators of (\g_{\bC})

Chapter 3. Whittaker functions on (2,\bR)

3.1. Representations of (2)

3.2. Principal series representations

3.3. Principal series Whittaker functions

3.4. Essentially discrete series Whittaker functions

Chapter 4. Whittaker functions on (3,\bR)

4.1. Representations of (3)

4.2. Principal series representations

4.3. Principal series Whittaker functions at scalar -types

4.4. Principal series Whittaker functions at 3 dimensional -types

4.5. Generalized principal series representations

4.6. Generalized principal series Whittaker functions

Chapter 5. Preliminaries for ( ,\bC)

5.1. Principal series representations

5.2. The elements of \g_{\bC} and (\g_{\bC})

5.3. The eigenvalues of generators of (\g_{\bC})

Chapter 6. Whittaker functions on (2,\bC)

6.1. Representations of (2)

6.2. Principal series representations

6.3. Principal series Whittaker functions

Chapter 7. Whittaker functions on (3,\bC)

7.1. Representations of (3)

7.2. Principal series representations

7.3. Principal series Whittaker functions

Part 2. Archimedean zeta integrals for (3)× (2)

Chapter 8. Preliminaries

8.1. The aim of Part 2

8.2. Some formulas for the calculation

Chapter 9. The local zeta integrals for (3,\bR)× (2,\bR)

9.1. The local Langlands correspondence for ( ,\bR).

9.2. Preparations for (2)-modules

9.3. Whittaker functions on (2,\bR)

9.4. Whittaker functions on (3,\bR)

9.5. The local zeta integrals for (3,\bR)× (2,\bR)

9.6. The calculation for '= _{( ₁', ₂')}⊠ _{( ₂', ₂')}

9.7. The calculation for '= _{( ₁',1)}⊠ _{( ₂',0)}

9.8. The calculation for '= _{( ', ')}

Chapter 10. The local zeta integrals for (3,\bC)× (2,\bC)

10.1. The local Langlands correspondence for ( ,\bC)

10.2. Preparations for (2)-modules

10.3. Whittaker functions on (2,\bC)

10.4. Whittaker functions on (3,\bC)

10.5. The local zeta integrals for (3,\bC)× (2,\bC)

10.6. The calculation in the case ₂&gt

- ₂'

10.7. The calculation in the case - ₁'&gt

₂

10.8. The calculation in the case - ₂'≥ ₂≥- ₁'

Appendix A. Archimedean zeta integrals for (2)× ( ) ( =1,2)

A.1. The local zeta integrals for (2,\bR)× (1,\bR)

A.2. The local zeta integrals for (2,\bR)× (2,\bR)

A.3. The local zeta integrals for (2,\bC)× (1,\bC)

A.4. The local zeta integrals for (2,\bC)× (2,\bC)

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Hirano, Miki Archimedean Zeta Integrals for

ISBN:

9781470471668

1470471663

OCLC:

1336953688