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].
