Jacobi forms over complex quadratic fields via the cubic Casimir operators

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 $K=\mathbb{Q}(i)$, and provide a lift from it to the space of Jacobi forms over $K=\mathbb{Q}(i)$.