The Siegel-Jacobi operator
The author considers Jacobi forms of several variables with respect to the representation \(\varrho\) and the index \(\mathcal M\), which is a half- integral semipositive matrix, in the sense of \textit{C. Ziegler} [Abh. Math. Semin. Univ. Hamb. 59, 191-224 (1989; Zbl 0707.11035)]. A Siegel \(\Phi\)-operator for these spaces of Jacobi forms is introduced. In the particular case, when \(2{\mathcal M}\) is unimodular, sufficient conditions on the weight of \(\varrho\) are derived such that the Siegel \(\Phi\)-operator is injective resp. surjective. Moreover the commutation relations with Hecke operators are described.
scientific article; zbMATH DE number 492589
scientific article