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Jacobi forms of several variables and the Maaß space. (English) Zbl 0879.11020
Let \(V\) be a real vector space of dimension \(n\) equipped with a positive definite symmetric bilinear from \(\sigma\), and let \(\Lambda\) be an even lattice in \(V\). The author extends V. A. Gritsenko’s approach [J. Sov. Math. 53, 243-252 (1991); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 168, 32-44 (1988; Zbl 0723.11021)] and defines Jacobi forms of weight \(k\) with respect to \((\Lambda,\sigma)\). They are functions on \(H\times\mathbb{C}^n\), where \(H\) denotes the upper half plane in \(\mathbb{C}\). Jacobi forms for various \(\Lambda\) and \(\sigma\) occur as coefficients in the Fourier-Jacobi expansion of all kinds of modular forms. For the case of Siegel modular forms, see H. Klingen [Math. Ann. 285, 405-416 (1989; Zbl 0695.10023)], for example. The space \(J_k(\Lambda,\sigma)\) of these Jacobi forms is isomorphic with a certain space of vector valued elliptic modular forms of weight \(k-{n\over 2}\).
The author introduces a subgroup \(\Gamma_S\) of an orthogonal group of signature \((1,n+1)\). Here, \(S\) is a real positive definite even \(n\times n\) matrix. Modular forms with respect to \(\Gamma_S\) are functions on a certain half space \(H_S\) in \(\mathbb{C}^{n+2}\). For their Fourier-Jacobi expansion, the Koecher effect is established, and it is shown that the coefficients are Jacobi forms with respect to \(\mathbb{Z}^n\) and suitable \(\sigma\). The Maaß space \(M^*_k (\Gamma_S)\) is contained in \(M_k(\Gamma_S)\), the space of all modular forms of weight \(k\) for \(\Gamma_S\). It is defined by conditions similar to those in the original setting of Siegel modular forms, and it is shown that \(M^*_k(\Gamma_S)\) is isomorphic to \(J_k(\mathbb{Z}^n, \sigma_S)\), where \(\sigma_S(a,b) =a^tSb\) for real (column) vectors \(a,b\). An application yields singular modular forms in \(M_{n/2} (\Gamma_S)\).

MSC:
11F50 Jacobi forms
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