Synopses & Reviews
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.
Review
"The book under review collects and regroups results on the representation theory of the Jacobi group of lowest degree mostly due to R.Berndt and his coworkers J.Homrighausen and R.Schmidt. 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. The first chapter introduces the Jacobi group ~GJ, a semi-direct product of SL(2) with a Heisenberg group, and describes different possible coordinates on the group, the Haar measure, the Lie algebra, as well as, over the reals, a (generalized) Iwasawa decomposition and the associated homogeneous space..." ---Zentralblatt Math "This book is an exposition which incorporates results of the authors' research works [that] will be very helpful for those who have some knowledge of the Jacquet-Langlands theory for GL2... Recommended for researchers interested in modular and automorphic forms." ---Mathematical Reviews
Table of Contents
Series: Progress in Mathematics, Vol. 163
Introduction
1 The Jacobi Group
1.1 Definition of GJ
1.2 GJ as an algebraic group
1.3 The Lie algebra of GJ
1.4 GJ over the reals
2 Basic representation theory of the Jacobi group
2.1 Induced representations
2.2 The Schrödinger representation
2.3 Mackey's method for semidirect products
2.4 Representations of GJ with trivial central character
2.5 The Schrödinger-Weil representation
2.6 Representations of GJ with non-trivial central character
3 Local representations: The real case
3.1 Representations of gJC
3.2 Models for infinitesimal representations and unitarizability
3.3 Representations induced from BJ
3.4 Representations induced from KJ and the automorphic factor
3.5 Differential operators on X = H x C
3.6 Representations induced from NJ and Whittaker models
4 The space L2(GJ\GJ(R)) and its decomposition
4.1 Jacobi forms and more general automorphic forms
4.2 The cusp condition for GJ(R)
4.3 The discrete part and the duality theorem
4.4 The continuous part
5 Local representations: The p-adic case
5.1 Smooth and admissible representations
5.2 Whittaker models for the Schrödinger-Weil representation
5.3 Representations of the metaplectic group
5.4 Induced representations
5.5 Supercuspidal representations
5.6 Intertwining operators
5.7 Whittaker models
5.8 Summary and Classification
5.9 Unitary representations
6 Spherical representations
6.1 The Hecke algebra of the Jacobi group
6.2 Structure of the Hecke algebra in the good case
6.3 Spherical representations in the good case
6.4 Spherical Whittaker functions
6.5 Local factors and spherical dual
6.6 The Eichler-Zagier operators
7 Global considerations
7.1 Adelization of GJ
7.2 The global Schrödinger-Weil representation
7.3 Automorphic representations
7.4 Lifting of Jacobi forms
7.5 The representation corresponding to a Jacobi form