Mathematische Zeitschrift

, Volume 285, Issue 3–4, pp 739–793 | Cite as

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

  • Eyal Kaplan
  • Shunsuke Yamana


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.


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

Mathematics Subject Classification

11F66 11F70 



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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Authors and Affiliations

  1. 1.Department of MathematicsOhio State UniversityColumbusUSA
  2. 2.Hakubi CenterKyotoJapan
  3. 3.Graduate School of MathematicsKyoto UniversityKitashirakawa, KyotoJapan
  4. 4.Department of Mathematics, Bar-Ilan UniversityRamat GanIsrael

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