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.

The book is very well written and gives an up to date collection of the results known. It will be quite useful for everyone working in the field.

(Zentralblatt MATH)

This book is certainly recommended for researchers interested in modular and automorphic forms.

(Mathematical Reviews)

This book complements the book on Jacobi forms by M. Eichler and D. Zagier and includes many of the author's original results. It can be read, however, independently of other sources, and it will be very useful for researchers interested in algebraic groups and number theory.

(Mathematica)

Autorentext

Rolf Berndt is a Professor of Mathematics at the University of Hamburg.

Ralf Schmidt is a Professor of Mathematics at the University of Oklahoma, Norman, OK, USA.

Klappentext

Combining algebraic groups and number theory, this volume gathers material from the representation theory of this group for the first time, doing so for both local (Archimedean and non-Archimedean) cases as well as for the global number field case.

Inhalt

The Jacobi Group.- Basic Representation Theory of the Jacobi Group.- Local Representations: The Real Case.- The Space L2(?J\GJ (?)) and its Decomposition.- Local Representations: The p-adic Case.- Spherical Representations.- Global Considerations.