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In this paper, we give a new definition for the space of non-holomorphic Jacobi Maaß forms (denoted by J nhk,m ) of weight k∈ℤ and index m∈ℕ as eigenfunctions of a degree three differential operator \(\mathcal{C}^{k,m}\) . We show that the three main examples of Jacobi forms known in the literature: holomorphic, skew-holomorphic and real-analytic Eisenstein series, are contained in J nhk,m . We construct new examples of cuspidal Jacobi Maaß forms F f of weight k∈2ℤ and index 1 from weight k−1/2 Maaß forms f with respect to Γ0(4) and show that the map f F f is Hecke equivariant. We also show that the above map is compatible with the well-known representation theory of the Jacobi group. In addition, we show that all of J nhk,m can be “essentially” obtained from scalar or vector valued half integer weight Maaß forms.

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Correspondence to Ameya Pitale.

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Communicated by U. Kühn.

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Pitale, A. Jacobi Maaß forms. Abh. Math. Semin. Univ. Hambg. 79, 87–111 (2009).

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