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Weil representations associated with finite quadratic modules

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Abstract

To any finite quadratic module, that is, a finite abelian group together with a non-degenerate quadratic form, it is possible to associate a representation of \(\mathrm{Mp}_{2}(\mathbb Z )\), the metaplectic cover of the modular group. This representation is usually referred to as a Weil representation and our main result is a general explicit formula for its matrix coefficients. This result completes earlier work by Scheithauer in the case when the representation factors through \(\mathrm{SL}_{2}(\mathbb Z )\). Furthermore, our formula is given in a such a way that it is easy to implement efficiently on a computer.

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Acknowledgments

I would like to thank Nils Scheithauer for clarifying details of [15], Nils-Peter Skoruppa for sharing thoughts about Weil representations in general, and the manuscript [19], in particular. I would also like to thank Stephan Ehlen for his extensive assistance with proof reading the manuscript and in simplifying Theorem 4.1.

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  1. Department of Mathematical Sciences, Durham University, Science Laboratories, South Rd., Durham, DH1 3LE, UK

    Fredrik Strömberg

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  1. Fredrik Strömberg
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Correspondence to Fredrik Strömberg.

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Strömberg, F. Weil representations associated with finite quadratic modules. Math. Z. 275, 509–527 (2013). https://doi.org/10.1007/s00209-013-1145-x

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