Skip to main content
SpringerLink
Log in

Klingen \({\mathfrak {p}}^2\) vectors for \(\mathrm{GSp}(4)\)

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We calculate the dimensions of the spaces of invariant vectors under the Klingen congruence subgroup of level \({\mathfrak {p}}^2\) for all irreducible, admissible representations of \(\mathrm{GSp}(4, F)\) with trivial central character for F a \({\mathfrak {p}}\)-adic field.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Atkin, A.O.L., Lehner, J.: Hecke operators on \(\varGamma _{0}(m)\). Math. Ann. 185, 134–160 (1970). https://doi.org/10.1007/BF01359701

    Article  MathSciNet  MATH  Google Scholar 

  2. Berndt, R., Schmidt, R.: Elements of the representation theory of the Jacobi group. In: Progress in Mathematics, vol. 163. Birkhäuser Verlag, Basel (1998). https://doi.org/10.1007/978-3-0348-0283-3

  3. Borel, A.: Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup. Invent. Math. 35, 233–259 (1976). https://doi.org/10.1007/BF01390139

    Article  MathSciNet  MATH  Google Scholar 

  4. Bump, D.: Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55. Cambridge University Press, Cambridge (1997). https://doi.org/10.1017/CBO9780511609572

    Book  MATH  Google Scholar 

  5. Bushnell, C.J., Henniart, G.: Supercuspidal representations of \({\rm GL}_n\): explicit Whittaker functions. J. Algebra 209(1), 270–287 (1998). https://doi.org/10.1006/jabr.1998.7542

    Article  MathSciNet  MATH  Google Scholar 

  6. Casselman, W.: On some results of Atkin and Lehner. Math. Ann. 201, 301–314 (1973). https://doi.org/10.1007/BF01428197

    Article  MathSciNet  MATH  Google Scholar 

  7. Casselman, W.: Introduction to admissible representations of p-adic groups. unpublished notes (1995)

  8. Jacquet, H., Piatetski-Shapiro, I.I., Shalika, J.: Conducteur des représentations du groupe linéaire. Math. Ann. 256(2), 199–214 (1981). https://doi.org/10.1007/BF01450798

    Article  MathSciNet  MATH  Google Scholar 

  9. Moy, A., Prasad, G.: Unrefined minimal \(K\)-types for \(p\)-adic groups. Invent. Math. 116(1–3), 393–408 (1994). https://doi.org/10.1007/BF01231566

    Article  MathSciNet  MATH  Google Scholar 

  10. Moy, A., Prasad, G.: Jacquet functors and unrefined minimal \(K\)-types. Comment. Math. Helv. 71(1), 98–121 (1996). https://doi.org/10.1007/BF02566411

    Article  MathSciNet  MATH  Google Scholar 

  11. Roberts, B., Schmidt, R.: Local newforms for GSp(4). In: Lecture Notes in Mathematics, vol. 1918. Springer, Berlin (2007)

  12. Roberts, B., Schmidt, R.: Some results on Bessel functionals for \({\rm GSp}(4)\). Doc. Math. 21, 467–553 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Roberts, B., Schmidt, R.: Stable Klingen vectors. Manuscript in preparation (2019)

  14. Rösner, M.A.: Parahoric restriction for gsp(4) and the inner cohomology of siegel modular threefolds. Ph.D. Thesis (2016)

  15. Sally Jr., P.J., Tadić, M.: Induced representations and classifications for \({\rm GSp}(2, F)\) and \({\rm Sp}(2, F)\). Mém. Soc. Math. France (N.S.) 52, 75–133 (1993)

    Article  MathSciNet  Google Scholar 

  16. Schmidt, R.: Some remarks on local newforms for \(\rm GL(2)\). J. Ramanujan Math. Soc. 17(2), 115–147 (2002)

    MathSciNet  MATH  Google Scholar 

  17. Schmidt, R.: Iwahori-spherical representations of \({\rm GSp}(4)\) and Siegel modular forms of degree 2 with square-free level. J. Math. Soc. Japan 57(1), 259–293 (2005). http://projecteuclid.org/euclid.jmsj/1160745825

  18. Schmidt, R.: On classical Saito-Kurokawa liftings. J. Reine Angew. Math. 604, 211–236 (2007). https://doi.org/10.1515/CRELLE.2007.024

    Article  MathSciNet  MATH  Google Scholar 

  19. Tunnell, J.B.: On the local Langlands conjecture for \(GL(2)\). Invent. Math. 46(2), 179–200 (1978). https://doi.org/10.1007/BF01393255

    Article  MathSciNet  MATH  Google Scholar 

  20. Yi, S.: Klingen vectors of level 2 for \({\rm GSp}(4)\). Ph.D. Thesis (2019). https://shareok.org/handle/11244/319553

Download references

Acknowledgements

I would like to thank my advisor Dr. Ralf Schmidt for introducing me to this interesting topic and guiding me patiently through it. I would also like to thank Dr. Brooks Roberts for his helpful comments and the referee for helping improve the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oklahoma, Norman, OK, 73019-3103, USA

    Shaoyun Yi

Authors
  1. Shaoyun Yi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shaoyun Yi.

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

Yi, S. Klingen \({\mathfrak {p}}^2\) vectors for \(\mathrm{GSp}(4)\). Ramanujan J 54, 511–554 (2021). https://doi.org/10.1007/s11139-020-00284-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-020-00284-9

Keywords

Mathematics Subject Classification

Search

Navigation