Elements of the Representation Theory of the Jacobi Group

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 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