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On holomorphic theta functions associated with rank r isotropic discrete subgroups

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Abstract

We investigate some basic analytic properties of the \(L^2\)-holomorphic automorphic functions on a g-complex vector space associated with isotropic discrete subgroups \(\Gamma _r\) of rank \(r\le g\). We show that each of these spaces form an infinite dimensional reproducing kernel Hilbert space which looks like a tensor product of a theta Bargmann–Fock space on \(Span_{{\mathbb {C}}}(\Gamma _r)\) and the classical Bargmann–Fock space on a \((g-r)\)-complex space. An explicit orthonormal basis using Fourier series is constructed and the explicit expression of its reproducing kernel function is given in terms of several variables Riemann theta function of particular characteristics.

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Acknowledgements

The authors express deep thanks to the anonymous referee for his careful reading, suggestions, remarks, and comments, helping to improve the content and the presentation of the paper. Some of the obtained results were presented in the international conference 2014 organized in honor of Professors Ahmed Intissar (passed away in 2017) and Takeshi Kawazoe. The authors are thankful to K. Koufany and Keio univ, Yokohaman, where part of the present paper was appeared (in a different form) in the proceeding “Seminar MS 39 Keio univ, Yokohama, 2016.”

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Correspondence to A. Ghanmi.

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To the memory of Professor Ahmed Intissar (who passed away July 2017)

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This paper was a part of the special proceeding “Seminar MS 39 Keio univ, Yokohama, 2016” in honor of Professors A. Intissar and T. Kawazoe.

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Ghanmi, A., Intissar, A. & Souid El Ainin, M. On holomorphic theta functions associated with rank r isotropic discrete subgroups. Ramanujan J 59, 635–651 (2022). https://doi.org/10.1007/s11139-022-00611-2

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