Elements of the Representation Theory of the Jacobi Group

After Pyatetski-Shapiro [PSI] and Satake [Sal] introduced, independent of one another, an early form of the Jacobi Theory in 1969 (while not naming it as such), this theory was given a definite push by the book The Theory of Jacobi Forms by Eichler and Zagier in 1985. Now, there are some overview articles describing the developments in the theory of the Jacobi group and its automor- phic forms, for instance by Skoruppa [Sk2], Berndt [Be5] and Kohnen [Ko]. We refer to these for more historical details and many more names of authors active in this theory, which stretches now from number theory and algebraic geometry to theoretical physics. But let us only briefly indicate several - sometimes very closely related - topics touched by Jacobi theory as we see it: - fields of meromorphic and rational functions on the universal elliptic curve resp. universal abelian variety - structure and projective embeddings of certain algebraic varieties and homogeneous spaces - correspondences between different kinds of modular forms - L-functions associated to different kinds of modular forms and autom- phic representations - induced representations - invariant differential operators - structure of Hecke algebras - determination of generalized Kac-Moody algebras and as a final goal related to the here first mentioned - mixed Shimura varieties and mixed motives.

"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