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 timematerial 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 numbertheory, in particular, modular and automorphic forms. Very well written monograph combining algebraic groups and number theory. Recommended reading for researchers of modular and automorphic forms. Up to date and structured collection of known results. INDICE: Introduction. 1 The Jacobi Group. 2 Basic Representation Theory ofthe Jacobi Group. 3 Local Representations: The Real Case. 4 The Space and itsDecomposition. 5 Local Representations: The p-adic Case. 6 Spherical Representations. 7 Global Considerations. Bibliography. Index of Notations. Index.

• ISBN: 978-3-0348-0282-6
• Editorial: Springer Basel
• Encuadernacion: Rústica
• Páginas: 214
• Fecha Publicación: 30/11/2011
• Nº Volúmenes: 1
• Idioma: Inglés