Elements of the Representation Theory of the Jacobi Group

ISBN-13: 9783034802826 / Angielski / Miękka / 2012 / 213 str.

Rolf Berndt; Ralf Schmidt

Opis

After Pyatetski-Shapiro PS1] and Satake Sa1] 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 de?nite 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 Jacobigroupandits autom- phic forms, for instance by Skoruppa Sk2], Berndt Be5] and Kohnen Ko]. We refertotheseformorehistoricaldetailsandmanymorenamesofauthorsactive inthistheory, whichstretchesnowfromnumbertheoryandalgebraicgeometry to theoretical physics. But let us only brie?y indicate several- sometimes very closely related - topics touched by Jacobi theory as we see it: eldsofmeromorphicandrationalfunctionsontheuniversalellipticcurve resp. universal abelian variety structure and projective embeddings of certain algebraic varieties and homogeneous spaces correspondences between di?erent kinds of modular forms L-functions associated to di?erent kinds of modular forms and autom- phic representations induced representations invariant di?erential operators structure of Hecke algebras determination of generalized Kac-Moody algebras and as a ?nal goal related to the here ?rst mentioned mixed Shimura varieties and mixed motives. Now, letting completely aside the arithmetical and algebraic geometrical - proach to Jacobi forms developed and instrumentalized by Kramer Kr], we ix x Introduction will treat here a certain representation theoretic point of view for the Jacobi theory parallel to the theory of Jacquet-Langlands JL] for GL(2) as reported by Godement Go2], Gelbart Ge1] and, recently, Bump Bu].

Spis treści

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.

O autorze

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.

Blurb

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.

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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)