Skip to main content
Log in

Archimedean distinguished representations and exceptional poles

  • Published:
manuscripta mathematica Aims and scope Submit manuscript


Let F be an archimedean local field and let E be \(F\times F\) (resp. a quadratic extension of F). We prove that an irreducible generic (resp. nearly tempered) representation of \(\textrm{GL}_n(E)\) is \(\textrm{GL}_n(F)\) distinguished if and only if its Rankin-Selberg (resp. Asai) L-function has an exceptional pole of level zero at 0. Further, we deduce a necessary condition for the ramification of such representations using the theory of weak test vectors developed by Humphries and Jo.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Anandavardhanan, U.K., Kable, A.C., Tandon, R.: Distinguished representations and poles of twisted tensor L-functions. Proc. Amer. Math. Soc. 132(10), 2875–2883 (2004)

    Article  MathSciNet  Google Scholar 

  2. Baruch, E.M.: A proof of Kirillov’s conjecture. Ann. Math. 158(1), 207–252 (2003)

    Article  MathSciNet  Google Scholar 

  3. Beuzart-Plessis, R.: Archimedean Theory and \(\epsilon \)-Factors for the Asai Rankin-Selberg Integrals, Relative Trace Formulas, Simons Symposia, Springer, Cham pp. 1–50 (2021)

  4. Chai, J.: Some results on Archimedean Rankin-Selberg integrals. Pacific J. Math. 273(2), 277–305 (2015)

    Article  MathSciNet  Google Scholar 

  5. Cogdell, J.W., Piatetski-Shapiro, I.I.: Remarks on Rankin- Selberg convolutions. Contributions to automorphic forms, geometry, and number theory. pp. 255-278, Johns Hopkins University Press, Baltimore, MD (2004)

  6. Flicker, Y.Z.: On zeroes of the twisted tensor L-function. Math. Ann. 297, 199–219 (1993)

    Article  MathSciNet  Google Scholar 

  7. Howe, R.: Transcending classical invariant theory. J. Amer. Math. Soc. 2(3), 535–552 (1989)

    Article  MathSciNet  Google Scholar 

  8. Humphries, P.: Archimedean newform theory for GL\(_n\). arXiv:2008.12406, (2020)

  9. Humphries, P., Jo, Y.: Test vectors for archimedean period integrals. arXiv:2112.06860 (2021)

  10. Jacquet, H.: Archimedean Rankin-Selberg integrals. Automorphic forms and L-functions II. Local aspects., 2022(783) , 49–94 (2022)

  11. Jacquet, H.: Distinction by the quasi-split unitary group. Israel J. Math. 178(1), 269–324 (2010)

    Article  MathSciNet  Google Scholar 

  12. Jacquet, H., Piatetski-Shapiro, I.I., Shalika, J.A.: Rankin-Selberg convolutions. Am. J. Math. 105(2), 367–464 (1983)

    Article  MathSciNet  Google Scholar 

  13. Kemarsky, A.: Distinguished representations of GL\(_n(\mathbb{C} )\). Israel J. Math. 207(1), 435–448 (2015)

    Article  MathSciNet  Google Scholar 

  14. Kemarsky, A.: A note on Kirillov model for representations of GL\(_n(\mathbb{C} )\). C. R. Math. Acad. Sci. Paris. 353(7), 579–582 (2015)

    Article  MathSciNet  Google Scholar 

  15. Langlands, R.P.: On the classification of irreducible representations of real algebraic groups. Representation theory and harmonic analysis on semisimple Lie groups. 101-170, Math. Surveys Monogr., 31, Amer. Math. Soc., Providence, RI, (1989)

  16. Matringe, N.: Distinguished representations and exceptional poles of the Asai-L-function. Manuscripta Math. 131, 415–426 (2010)

    Article  MathSciNet  Google Scholar 

  17. Tate, J.: Number theoretic background, in automorphic forms, representations, and L functions. Proc. Symposia Pure Math., AMS 33, 3–26 (1979)

  18. Vogan, D.: Gelf́and-Kirillov dimension for Harish-Chandra modules. Invent. Math. 48, 75–98 (1978)

  19. Vogan, D., Speh, B.: Reducibility of generalized principal series representations. Acta Math. 145(3–4), 227–299 (2021)

    MathSciNet  Google Scholar 

Download references


I would like to thank my advisor Professor Ravi Raghunathan for his constant encouragement, invaluable comments, and for carefully reading several preliminary versions of this paper. I also wish to thank Professor Dipendra Prasad and Professor U.K. Anandavardhanan for several useful remarks.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Akash Yadav.

Ethics declarations

Conflict of interest

The author declares that there are no conflict of interest.

Data availibility

No new data were created or analysed during this study. Data sharing is not applicable to this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yadav, A. Archimedean distinguished representations and exceptional poles. manuscripta math. (2024).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:

Mathematics Subject Classification