Skip to main content
SpringerLink
Log in

Theta lifts of generic representations for dual pairs \((\text {Sp}_{2n}, \text {O}(V))\)

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We determine the occurrence and explicitly describe the theta lifts on all levels of all the irreducible generic representations for the dual pair of groups \((\text {Sp}_{2n}, \text {O}(V))\) defined over a local nonarchimedean field \({\mathbb {F}}\) of characteristic 0. As a direct application of our results, we are able to produce a series of non-generic unitarizable representations of these groups.

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

Similar content being viewed by others

Notes

  1. This notation is motivated by the one used by Atobe and Gan, but does not have quite the same meaning as in the original paper [3].

References

  1. Arthur, J.: The Endoscopic Classification of Representations. American Mathematical Society Colloquium Publications, Providence (2013)

    Book  Google Scholar 

  2. Atobe, H.: On the uniqueness of generic representations in an L-packet. Int. Math. Res. Not. 2017(23), 7051–7068 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Atobe, H., Gan, W.T.: Local theta correspondence of tempered representations and Langlands parameters. Invent. Math. 210(2), 341–415 (2017)

    Article  MathSciNet  Google Scholar 

  4. Atobe, H.: The local theta correspondence and the local Gan-Gross-Prasad conjecture for the symplectic-metaplectic case. Math. Ann. 371(1), 225–295 (2018)

    Article  MathSciNet  Google Scholar 

  5. Gan, W.T., Ishcino, A.: Formal degrees and local theta correspondence. Invent. Math. 195, 509–672 (2014)

    Article  MathSciNet  Google Scholar 

  6. Gan, W.T., Takeda, S.: A proof of the Howe duality conjecture. J. Am. Math. Soc 29(2), 473–493 (2016)

    Article  MathSciNet  Google Scholar 

  7. Goldberg, D.: Reducibility of induced representations for Sp(2n) and SO(n). Am. J. Math. 116(5), 1101–1151 (1994)

    Article  MathSciNet  Google Scholar 

  8. Hanzer, M.: The generalized injectivity conjecture for classical p-adic groups. Int. Math. Res. Not. 2010(2), 195–237 (2009)

    Article  MathSciNet  Google Scholar 

  9. Heiermann, V., Muić, G.: On the standard modules conjecture. Math. Z. 255(4), 847–853 (2007)

    Article  MathSciNet  Google Scholar 

  10. Heiermann, V., Opdam, E.: On the tempered \(L\)-functions conjecture. Am. J. Math. 135, 777–799 (2013)

    Article  MathSciNet  Google Scholar 

  11. Howe, R.: \(\theta \)-series and invariant theory, in Automorphic forms, representations and L-functions, In: Proceedings of Symposium Pure Mathematics XXXIII, American Mathematical Society, pp. 275–285 (1979)

  12. Kaletha, T.: Genericity and contragredience in the local Langlands correspondence. Algebra Number Theory 7(10), 2447–2474 (2014)

    Article  MathSciNet  Google Scholar 

  13. Kudla, S.S.: Notes on the local theta correspondence, (Unpublished) (1996)

  14. Kudla, S.S.: On the local theta correspondence. Invent. Math. 83(2), 229–255 (1986)

    Article  MathSciNet  Google Scholar 

  15. Kudla, S.S., Rallis, S.: On first occurrence in the local theta correspondence. In: Cogdell, J., et al. (eds.) Automorphic Representations. Progress and Prospects, L-functions and Applications (2005)

  16. Lapid, E., Muić, G., Tadić, M.: On the generic unitary dual of quasisplit classical groups. Int. Math. Res. Not. 26, 335–1354 (2004)

    MathSciNet  MATH  Google Scholar 

  17. Li, J.-S.: Singular unitary representations of classical groups. Invent. Math. 97(2), 237–255 (1989)

    Article  MathSciNet  Google Scholar 

  18. Mœglin, C., Vignéras, M.-F., Waldspurger, J.-L.: Correspondances de Howe sur un corps p-adique

  19. Moeglin, C., Waldspurger, J.-L.: Le spectre résiduel de \(\mathit{GL}(n)\). Annales Scientifiques de l’École Normale Supérieure 22(4), 605–674 (1989)

    Article  MathSciNet  Google Scholar 

  20. Muić, G.: A proof of Casselman-Shahidi’s conjecture for quasi-split classical groups. Can. Math. Bull. 44(3), 298–312 (2001)

    Article  MathSciNet  Google Scholar 

  21. Muić, G.: Howe correspondence for discrete series representations; the case of (Sp(n); O(V)). Journal fur die Reine und Angewandte Mathematik 567, 99–150 (2004)

    MathSciNet  MATH  Google Scholar 

  22. Muić, G.: On the structure of theta lifts of discrete series for dual pairs (Sp(n), O(V)). Israel J. Math. 164(1), 87–124 (2008)

    Article  MathSciNet  Google Scholar 

  23. Muić, G.: Reducibility of standard representations. Pac. J. Math. 222(1), 133–168 (2005)

    Article  MathSciNet  Google Scholar 

  24. Muić, G.: Theta lifts of tempered representations for dual pairs (\(\text{ Sp }_{2n}\), \(O(V)\)). Can. J. Math 60(6), 1306–1335 (2008)

    Article  Google Scholar 

  25. Muić, G., Savin, G.: Symplectic-orthogonal theta lifts of generic discrete series. Duke Math. J. 101(2), 317–333 (2000)

    Article  MathSciNet  Google Scholar 

  26. Rodier, F.: Whittaker models for admissible representations of reductive p-adic split groups, in Harmonic Analysis on Homogeneous Spaces. In: Proceedings of Symposium Pure Mathematics XXVI, American Mathematical Society, pp. 425–430(1973)

  27. Sun, B., Zhu, C.-B.: Conservation relations for local theta correspondence. J. Am. Math. Soc. 28(4), 939–983 (2015)

    Article  MathSciNet  Google Scholar 

  28. Tadić, M.: Reducibility and discrete series in the case of classical p-adic groups; an approach based on examples, in Geometry and analysis of automorphic forms of several variables, World Scientific (2012), pp. 254–333

  29. Tadić, M.: Structure arising from induction and Jacquet modules of representations of classical p-adic groups. J. Algebra 177(1), 1–33 (1995)

    Article  MathSciNet  Google Scholar 

  30. Waldspurger, J.-L.: Demonstration d’une conjecture de duality de Howe dans le case p-adiques, \(p \ne 2\), in Festschrift in honor of I. Piatetski-Shapiro. In: Israel Mathematics Conference Proceedings vol. 2 pp. 267–324 (1990)

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

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000, Zagreb, Croatia

    Petar Bakić

Authors
  1. Petar Bakić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Petar Bakić.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bakić, P. Theta lifts of generic representations for dual pairs \((\text {Sp}_{2n}, \text {O}(V))\). manuscripta math. 165, 291–338 (2021). https://doi.org/10.1007/s00229-020-01221-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-020-01221-2

Mathematics Subject Classification

Search

Navigation