Twisted symmetric square L-functions for \(\mathrm {GL}_n\)  and invariant trilinear forms

Abstract

Following the works of Bump and Ginzburg and of Takeda, we develop a theory of twisted symmetric square L-functions for \(\mathrm {GL}_n\). We characterize their pole in terms of certain trilinear period integrals, determine all irreducible summands of the discrete spectrum of \(\mathrm {GL}_n\) having nonvanishing trilinear periods, and construct nonzero local invariant trilinear forms on a certain family of induced representations.

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Acknowledgments

Yamana would like to thank Michael Harris for inviting him as a postdoctoral fellow at the Institut de mathématiques de Jussieu, where this paper was written. The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 290766 (AAMOT). Yamana is partially supported by JSPS Grant-in-Aid for Young Scientists (B) 26800017. Kaplan was partially supported by the ISF Center of Excellence Grant # 1691/10. We are very grateful to the anonymous referee for a very careful reading and detailed comments, which helped improve the exposition of the earlier version.

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Correspondence to Shunsuke Yamana.

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Kaplan, E., Yamana, S. Twisted symmetric square L-functions for \(\mathrm {GL}_n\)  and invariant trilinear forms. Math. Z. 285, 739–793 (2017). https://doi.org/10.1007/s00209-016-1726-6

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Keywords

  • Symmetric square L-functions
  • Exceptional representations
  • Rankin–Selberg integral representation
  • Distinguished representations

Mathematics Subject Classification

  • 11F66
  • 11F70