Jacobi forms of several variables and the Maaß space

Abstract

In this paper we describe the space of Jacobi forms on ℋ × ℂn. This type of Jacobi forms appears in the Fourier-Jacobi expansions of all kinds of modular forms of several variables. We can characterize the space of Jacobi forms of weight k by vector valued elliptic modular forms of weight k - n/2. Then we use the Jordan theoretic language in order to describe modular forms on the orthogonal group O(2, n + 2), whose Fourier-Jacobi expansions also yield Jacobi forms of our type. Finally we determine a Maaß space as the isomorphic image of a particular space of Jacobi forms. Especially our procedure guarantees the existence of certain non-trivial singular modular forms. © 1996 Academic Press, Inc.