Skip to main content
SpringerLink
Log in

Linear periods for unitary representations

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Let F be a local non-Archimedean field of characteristic zero with a finite residue field. Based on Tadić’s classification of the unitary dual of \({\mathrm {GL}}_{2n}(F)\), we classify irreducible unitary representations of \({\mathrm {GL}}_{2n}(F)\) that have nonzero linear periods, in terms of Speh representations that have nonzero periods. We also give a necessary and sufficient condition for the existence of a nonzero linear period for a Speh representation.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Code Availability

Not applicable.

Materials Availability

Not applicable.

References

  1. Bernstein, I.N., Zelevinsky, A.V.: Induced representations of reductive \({\mathfrak{p} }\)-adic groups. I. Annales Scientifiques de l’École Normale Supérieure 10, 441–472 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  2. Blanc, P., Delorme, P.: Vecteurs distributions \(H\)-invariants de représentations induites, pour un espace symétrique réductif \(p\)-adique \(G/H\). Annales de l’Institut Fourier. 58, 213–261 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, F., Sun, B.: Uniqueness of twisted linear periods and twisted Shalika periods. Sci. China Math. 63, 1–22 (2020)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cogdell, J. W., Piatetski-Shapiro, I. I.: Derivatives and \(L\)-functions for \({\rm GL}(n)\). In: Cogdell, J., Kim, J.L., Zhu, C.B. (eds.) Representation theory, number theory, and invariant theory, pp. 115–173 Progress in Mathematics, vol. 323. Birkhäuser, Basel (2017)

  5. Fang, Y., Sun, B., Xue, H.: Godement–Jacquet \(L\)-functions and full theta lifts. Mathematische Zeitschrift. 289, 593–604 (2018)

    Article  MATH  MathSciNet  Google Scholar 

  6. Feigon, B., Lapid, E., Offen, O.: On representations distinguished by unitary groups. Publications Mathématiques de l’IHÉS 115(1), 185–323 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Friedberg, S., Jacquet, H.: Linear periods. J. Reine Angew. Math. 443, 91–140 (1993)

    MATH  MathSciNet  Google Scholar 

  8. Gan, W. T.: Periods and theta correspondence. In: Aizenbud, A., Gourevitch, D., Kazhdan, D., Lapid, E.M. (eds.) Representations of Reductive Groups, pp. 113–132 Proceedings of Symposia in Pure Mathematics, volume 101. American Mathematical Society (2019). https://doi.org/10.1090/pspum/101

  9. Gan, W.T., Gross, B.H., Prasad, D.: Branching laws for classical groups: the non-tempered case. Compos. Math. 156(11), 2298–2367 (2020). https://doi.org/10.1112/s0010437x20007496

    Article  MATH  MathSciNet  Google Scholar 

  10. Gurevich, M.: On a local conjecture of Jacquet, ladder representations and standard modules. Mathematische Zeitschrift 281(3–4), 1111–1127 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  11. Jacquet, H., Rallis, S.: Uniqueness of linear periods. Compos. Math. 102(1), 65–123 (1996)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Jiang, D., Soudry, D.: Generic representations and local langlands reciprocity law for \(p\)-adic \({\rm SO}_{2n+1}\). In: Hida, H., Ramakrishnan, D., Shahidi, F. (eds.) Contributions to automorphic forms, geometry, and number theory, pp. 457–519. The Johns Hopkins University Press, Baltimore and London (2004)

  14. Jiang, D., Nien, C., Qin, Y.: Local Shalika models and functoriality. Manusc. Math. 127(2), 187–217 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kable, A.C.: Asai \(L\)-functions and Jacquet’s conjecture. Am. J. Math. 126(4), 789–820 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kret, A., Lapid, E.: Jacquet modules of ladder representations. Comptes Rendus Math. 350(21–22), 937–940 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lapid, E., Mínguez, A.: On a determinantal formula of Tadić. Am. J. Math. 136(1), 111–142 (2014)

    Article  MATH  Google Scholar 

  18. Lapid, E., Mínguez, A.: On parabolic induction on inner forms of the general linear group over a non-archimedean local field. Selecta Math. 22(4), 2347–2400 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  19. Matringe, N.: Distinguished generic representations of \({\rm GL} (n)\) over \(p\)-adic fields. Int. Math. Res. Not. 2011(1), 74–95 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Matringe, N.: Linear and Shalika local periods for the mirabolic group, and some consequences. J. Number Theory 138, 1–19 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  21. Matringe, N.: Unitary representations of \({\rm GL}(n, K)\) distinguished by a Galois involution for a \(p\)-adic field \(K\). Pacific J. Math. 271(2), 445–460 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  22. Matringe, N.: On the local Bump–Friedberg \(L\)-function. J. Reine Angew. Math. 2015(709), 119–170 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  23. Mínguez, A.: Correspondance de Howe explicite: paires duales de type II. Annales Scientifiques de l’École Normale Supérieure. 41, 717–741 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mitra, A., Offen, O., Sayag, E.: Klyachko models for ladder representations. Docum. Math. 22, 611–657 (2017)

    MATH  MathSciNet  Google Scholar 

  25. Moeglin, C., Waldspurger, J.L.: Sur l’involution de Zelevinski. J. Reine Angew. Math. 1986(372), 136–177 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  26. Offen, O.: On parabolic induction associated with a \(p\)-adic symmetric space. J. Number Theory 170, 211–227 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  27. Offen, O., Sayag, E.: On unitary representations of \({\rm GL} _{2n}\) distinguished by the symplectic group. J. Number Theory 125(2), 344–355 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. Offen, O., Sayag, E.: Uniqueness and disjointness of Klyachko models. J. Funct. Anal. 254(11), 2846–2865 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Sécherre, V.: Représentations cuspidales de \({\rm GL}(r,D)\) distinguées par une involution intérieure. https://arxiv.org/pdf/2005.05615.pdf

  30. Tadić, M.: Classification of unitary representations in irreducible representations of general linear group (non-archimedean case). Annales Scientifiques de l’Ecole Normale Supérieure. 19, 335–382 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  31. Tadić, M.: Induced representations of \({\rm GL}(n, A)\) for p-adic division algebras \(A\). J. Reine Angew. Math. 405, 48–77 (1990)

    MATH  MathSciNet  Google Scholar 

  32. Tadić, M.: On characters of irreducible unitary representations of general linear groups. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 65(1), 341–363 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  33. Zelevinsky, A.V.: Induced representations of reductive p-adic groups. II. on irreducible representations of \({\rm GL}(n)\). Annales Scientifiques de l’École Normale Supérieure. 13(2), 165–210 (1980)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks Wee Teck Gan for his kindness of sharing his paper [8] with us. He would also like to thank Dipendra Prasad for helpful comments and suggestions to improve the paper. The author thanks the referee for many remarks and suggestions and for pointing out an error in an earlier draft of the paper. The author was supported by the National Natural Science Foundation of China (no.12001191).

Funding

This work was supported in part by the National Natural Science Foundation of China (no. 12001191).

Author information

Authors and Affiliations

  1. Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP), School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, China

    Chang Yang

Authors
  1. Chang Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Not applicable.

Corresponding author

Correspondence to Chang Yang.

Ethics declarations

Conflicts of interest

The author declares that there is no conflict of interests regarding the publication of 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 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

Yang, C. Linear periods for unitary representations. Math. Z. 302, 2253–2284 (2022). https://doi.org/10.1007/s00209-022-03136-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-022-03136-y

Keywords

Mathematics Subject Classification

Search

Navigation