Differential Operators on Jacobi Forms of Several Variables

The theory of the classical Jacobi forms on H × C has been studied extensively by Eichler and Zagier[?]. Ziegler[?] developed a more general approach of Jacobi forms of higher degree. In [?] and [?], Gritsenko and Krieg studied Jacobi forms on H × Cn and showed that these kinds of Jacobi forms naturally arise in the Jacobi Fourier expansions of all kinds of automorphic forms in several variables. Krieg[?] also considered modular forms on the orthogonal group O(2, n+ 2). On the other hand, there are many interesting connections between differential operators and the theory of elliptic modular forms and many interesting results have been explored. In particular, it has been known for sometime how to obtain an elliptic modular form from the derivatives of N elliptic modular forms, which has already been studied in detail by Rankin [?]. When N = 2, as a special case of Rankin’s result in [?], Cohen has constructed certain covariant bilinear operators which he used to obtain modular forms with interesting Fourier coefficients[?]. Later, these covariant bilinear operators were called Rankin-Cohen operators by Zagier who studied their algebraic relations[?]. Recently, the Rankin-Cohen type bracket operators on the classical Jacobi forms on H × C and Siegel forms of genus 2 have been studied using the heat operator and differential operator, respectively[?, ?, ?, ?]. In this paper we study differential operators on the space of Jacobi forms on H×Cn. It generalizes the results for Jacobi forms on H × C to those on H × Cn. This paper organized as follows. In section 2, we recall the definition of Jacobi forms and a heat operator. In section 3, we show how to construct Jacobi forms on H × Cn using a formal power series satisfying a certain functional equation. This method has been first studied by Zagier[?] to construct elliptic modular forms from a formal power series satisfying a certain functional equation and modified to construct the classical Jacobi forms as well as Siegel forms of genus 2 in [?, ?]. In section 4, the Rankin-Cohen type of bilinear differential operators, more generally, multilinear differential operators on the space of Jacobi forms on