A generalization of vector valued Jacobi forms (Q645099)

The authors consider Jacobi forms on \(\mathbb{H}\times \mathbb{C}\) of index \(m\in \mathbb{N}_0\) with respect to an irreducible representation \(\varrho_{ k,s}=\det^k \otimes \varrho_s\) of \(GL_2 (\mathbb{C})\) of dimension \(s+1\). This does not only generalize the classical work by Eichler/Zagier, but is moreover more general than \textit{C. Ziegler}'s approach [Abh. Math. Semin. Univ. Hamb. 59, 191--224 (1989; Zbl 0707.11035)]. They show that the space of Jacobi forms is isomorphic to a product of spaces of ordinary scalar valued Jacobi forms of various weights. The isomorphism is constructed by particular linear holomorphic differential operators. In the final section the results are modified in order to deal with half-integral weight.