Skip to main content
SpringerLink
Log in

The Howe Duality Conjecture: Quaternionic Case

  • Chapter
  • First Online:
Representation Theory, Number Theory, and Invariant Theory

Part of the book series: Progress in Mathematics ((PM,volume 323))

Abstract

We complete the proof of the Howe duality conjecture in the theory of local theta correspondence by treating the remaining case of quaternionic dual pairs in arbitrary residual characteristic.

In celebration of Professor Roger Howe’s 70th birthday

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

Access this chapter

Log in via an institution

Institutional subscriptions

References

  1. J. Bernstein, Draft of: representations of p-adic groups, Lectures by Joseph Bernstein, Harvard University Fall 1992, written by Karl E. Rumelhart.

    Google Scholar 

  2. W. T. Gan and S. Takeda, A proof of the Howe duality conjecture, J. Amer. Math. Soc. 29 (2016), no. 2, 473–493.

    Article  MathSciNet  Google Scholar 

  3. R. Howe, θ-series and invariant theory, in Automorphic Forms, Representations and L-functions, Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 1, pp. 275–285.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. M. Hanzer, Inducirane reprezentacije hermitskih kvaternionskih grupa, Ph.D. thesis, University of Zagreb (2005).

    Google Scholar 

  6. S. S. Kudla, On the local theta-correspondence, Invent. Math. 83 (1986), 229–255.

    Article  MathSciNet  Google Scholar 

  7. S. S. Kudla and S. Rallis, On first occurrence in the local theta correspondence, Automorphic representations, L-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ. 11, de Gruyter, Berlin, 2005, pp. 273–308.

    Google Scholar 

  8. J.-S. Li, B. Sun and Y. Tian, The multiplicity one conjecture for local theta correspondences, Invent. Math. 184 (2011), 117–124.

    Article  MathSciNet  Google Scholar 

  9. Y. Lin, B. Sun and S. Tan, MVW-extensions of real quaternionic classical groups, Math. Z. 277 (2014), no. 1–2, 81–89.

    Article  MathSciNet  Google Scholar 

  10. A. Minguez, Correspondance de Howe explicite: paires duales de type II, Ann. Sci. Éc. Norm. Supér. 41 (2008), 717–741.

    Article  MathSciNet  Google Scholar 

  11. C. Mœglin, M.-F. Vignéras, and J.-L. Waldspurger, Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics 1291, Springer-Verlag, Berlin, 1987.

    Book  Google Scholar 

  12. S. Rallis, On the Howe duality conjecture, Compositio Math. 51 (1984), 333–399.

    MathSciNet  MATH  Google Scholar 

  13. V. Sécherre, Proof of the Tadić conjecture (U0) on the unitary dual of GLm(D), J. Reine Angew. Math. 626 (2009), 187–203.

    MathSciNet  MATH  Google Scholar 

  14. B. Sun, Dual pairs and contragredients of irreducible representations, Pacific. J. Math. 249 (2011), 485–494.

    Article  MathSciNet  Google Scholar 

  15. B. Sun and C.-B. Zhu, Conservation relations for local theta correspondence, to appear in Journal of AMS. arXiv:1204.2969.

    Google Scholar 

  16. M. Tadić, Structure arising from induction and Jacquet modules of representations of classical p-adic groups, Journal of Algebra vol. 177 (1995), Pg. 1–33.

    Article  MathSciNet  Google Scholar 

  17. J.-L. Waldspurger, Démonstration d’une conjecture de dualité de Howe dans le cas p-adique,p ≠ 2, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I, Israel Math. Conf. Proc. 2, Weizmann, Jerusalem, 1990, pp. 267–324.

    Google Scholar 

  18. A. Weil, Sur certaines d’opérateurs unitaires, Acta Math. 111 (1964), 143–211.

    Article  MathSciNet  Google Scholar 

  19. S. Yamana, Degenerate principal series representations for quaternionic unitary groups, Israel J. Math. 185 (2011), 77–124.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This paper is essentially completed during the conference in honour of Professor Roger Howe on the occasion of his 70th birthday. We thank the organizers of the conference (James Cogdell, Ju-Lee Kim, Jian-Shu Li, David Manderscheid, Gregory Margulis, Cheng-Bo Zhu and Gregg Zuckerman) for their kind invitation to speak at the conference and for providing local support. W.T. Gan is partially supported by an MOE Tier Two grant R-146-000-175-112. B. Sun is supported in part by the NSFC Grants 11222101, 11321101 and 11525105.

Author information

Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Wee Teck Gan

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Binyong Sun

Authors
  1. Wee Teck Gan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Binyong Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wee Teck Gan .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Ohio State University, Columbus, OH, USA

    Jim Cogdell

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Ju-Lee Kim

  3. Department of Mathematics, National University of Singapore, Singapore, Singapore

    Chen-Bo Zhu

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Gan, W.T., Sun, B. (2017). The Howe Duality Conjecture: Quaternionic Case. In: Cogdell, J., Kim, JL., Zhu, CB. (eds) Representation Theory, Number Theory, and Invariant Theory. Progress in Mathematics, vol 323. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-59728-7_6

Download citation

Publish with us

Policies and ethics

Search

Navigation