Elements of the Representation Theory of the Jacobi Group / Edition 1

Elements of the Representation Theory of the Jacobi Group / Edition 1

by Rolf Berndt, Ralf Schmidt
     
 

ISBN-10: 3764359226

ISBN-13: 9783764359225

Pub. Date: 07/01/1998

Publisher: Birkhauser Basel

The Jacobi group is a semidirect product of a symplectic group with a Heisenberg group. It is an important example for a non-reductive group and sets the frame within which to treat theta functions as well as elliptic functions - in particular, the universal elliptic curve.

This text gathers for the first time material from the representation theory of this

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Overview

The Jacobi group is a semidirect product of a symplectic group with a Heisenberg group. It is an important example for a non-reductive group and sets the frame within which to treat theta functions as well as elliptic functions - in particular, the universal elliptic curve.

This text gathers for the first time material from the representation theory of this group in both local (archimedean and non-archimedean) cases and in the global number field case. Via a bridge to Waldspurger's theory for the metaplectic group, complete classification theorems for irreducible representations are obtained. Further topics include differential operators, Whittaker models, Hecke operators, spherical representations and theta functions. The global theory is aimed at the correspondence between automorphic representations and Jacobi forms. This volume is thus a complement to the seminal book on Jacobi forms by M. Eichler and D. Zagier.

Incorporating results of the authors' original research, this exposition is meant for researchers and graduate students interested in algebraic groups and number theory, in particular, modular and automorphic forms.

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Product Details

ISBN-13:
9783764359225
Publisher:
Birkhauser Basel
Publication date:
07/01/1998
Series:
Progress in Mathematics Series, #163
Pages:
236
Product dimensions:
9.21(w) x 6.14(h) x 0.56(d)

Table of Contents

Introduction
1The Jacobi Group
2Basic Representation Theory of the Jacobi Group
3Local Representations: The Real Case
4The Space L[superscript 2](T[superscript J]\G[superscript J](R)) and its Decomposition
5Local Representations: The p-adic Case
6Spherical Representations
7Global Considerations
Bibliography
Index of Notations
Index

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