Jacobi forms over complex quadratic fields via the cubic Casimir operators

Abstract

We dedicate this article to Harold Stark on the occasion of his 70 th birthday.Abstract. We prove that the center of the algebra of differential operators invariant under the action of the Jacobi group over a complex quadratic field is generated by two cubic Casimir operators, which we compute explicitly. In the spirit of Borel, we consider Jacobi forms over complex quadratic fields that are also eigenfunctions of these Casimir operators, a new approach in the complex case. Theta functions and Eisenstein series provide standard examples. In addition, we introduce an analog of Kohnen's plus space for modular forms of half-integral weight over K = Q(i), and provide a lift from it to the space of Jacobi forms over K = Q(i).

Centers and characters of Jacobi group-invariant differential operator algebras

“…Our first result is that for all N , the center Z(D(G J N × K J N V )) is precisely the image of Z(g J N ). This was previously known for N = 1 [BCR12] and N = 2 [Da13]. The structure of Z(g J N ) was deduced for N = 1 in [BCR12] and for N > 1 in [CR]: it is generated by the center z(g J N ) of g J N itself and in addition a single "Casimir element" of degree N + 2, which acts by an IDO of degree 3 for N = 1 and of degree 4 for N > 1.…”

“…In Section 3 we define the relevant homogeneous space, its scalar slash actions, and their IDO algebras. Most of these first two sections is a recapitulation of material in [BCR12] and [CR]. Our results are given in Sections 4, 5, and 6.…”

“…This definition was used in the study of nonholomorphic Maaß-Jacobi forms in [Pi09] and [BR10]. At this time it was realized that the center of the universal enveloping algebra of G J 1 in fact acts by C[C] [BCR12]. Thus the definition from [BS98] coincides with the classical one.…”

“…We will use the algebraic description of the IDO algebras developed by Helgason; see for example [He77,He79,BCR12]. This description depends only on the representation of K J N defining the vector bundle, so we will not need the explicit descriptions of the slash actions given in [CR].…”

A Note on Maass-Jacobi Forms II

“…Remark 9.2. In [3], Bringmann, Conley and Richter proved that the center of the algebra of differential operators invariant under the action of the Jacobi group over a complex quadratic field is generated by two Casimir operators of degree three. They also introduce an analogue of Kohnen's plus space for modular forms of halfintegral weight over K = Q(i), and provide a lift from it to the space of Jacobi forms over K. Definition 9.3.…”