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Abstract:
We define harmonic Siegel modular forms based on a completely new approach using vector-valued covariant operators. The Fourier expansions of such forms are investigated for two distinct slash actions. Two very different reasons are given why these slash actions are natural. We prove that they are related by xi-operators that generalize the xi-operator for elliptic modular forms. We call them dual slash actions or dual weights, a name which is suggested by the many properties that parallel the elliptic case.
Based on Kohnen's limit process for real-analytic Siegel Eisenstein series, we show that, under mild assumptions, Jacobi forms can be obtained from harmonic Siegel modular forms, generalizing the classical Fourier-Jacobi expansion. The resulting Fourier-Jacobi coefficients are harmonic Maass-Jacobi forms, which are defined in full generality in this work. A compatibility between the various xi-operators for Siegel modular forms, Jacobi forms, and elliptic modular forms is deduced, relating all three kinds of modular forms.
