# Jacobi forms over complex quadratic fields via the cubic Casimir operators

### Kathrin Bringmann

Universität Köln, Germany### Charles H. Conley

University of North Texas, Denton, USA### Olav K. Richter

University of North Texas, Denton, USA

## 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 $K=\mathbb{Q}(i)$.

## Cite this article

Kathrin Bringmann, Charles H. Conley, Olav K. Richter, Jacobi forms over complex quadratic fields via the cubic Casimir operators. Comment. Math. Helv. 87 (2012), no. 4, pp. 825–859

DOI 10.4171/CMH/270