Skip to main content
SpringerLink
Log in

The newform K-type and p-adic spherical harmonics

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Let \(K: = {\rm{G}}{{\rm{L}}_n}({\cal O})\) denote the maximal compact subgroup of GLn(F), where F is a nonarchimedean local field with ring of integers \({\cal O}\). We study the decomposition of the space of locally constant functions on the unit sphere in Fn into irreducible K-modules; for F = ℚp, these are the p-adic analogues of spherical harmonics. As an application, we characterise the newform and conductor exponent of a generic irreducible admissible smooth representation of GLn(F) in terms of distinguished K-types. Finally, we compare our results to analogous results in the archimedean setting.

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

References

  1. A. O. L. Atkin and J. Lehner, Hecke operators on Γ0(m), Mathematische Annalen 185 (1970), 134–160.

    Article  MathSciNet  MATH  Google Scholar 

  2. K. Atkinson and W. Han, Spherical Harmonics and Approximations on the Unit Sphere: An Introduction, Lecture Notes in Mathematics, Vol. 2044, Springer, Heidelberg, 2012.

    Google Scholar 

  3. E. M. Baruch and S. Purkait, Hecke algebras, new vectors and new forms on Γ0(m), Mathematische Zeitschrift 287 (2017), 705–733.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. N. Bernstein, p-invariant distributions on GL(n) and the classification of unitary representations of GL(n) (non-archimedean case), in Lie Groups Representations, II, Lecture Notes in Mathematics, Vol. 1041, Springer, Berlin, 1984, pp. 50–102.

    Google Scholar 

  5. P. S. Campbell and M. Nevins, Branching rules for unramified principal series representations of GL(3) over a p-adic field, Journal of Algebra 321 (2009), 2422–2444.

    Article  MathSciNet  MATH  Google Scholar 

  6. P. S. Campbell and M. Nevins, Branching rules for ramified principal series representations of GL(3) over a p-adic field, Canadian Journal of Mathematics 62 (2010), 34–51.

    Article  MathSciNet  MATH  Google Scholar 

  7. W. Casselman, On some results of Atkin and Lehner, Mathematische Annalen 201 (1973), 301–314.

    Article  MathSciNet  MATH  Google Scholar 

  8. W. Casselman, The restriction of a representation of GL2(k) to \({\rm{G}}{{\rm{L}}_2}(\mathfrak{o})\), Mathematische Annalen 206 (1973), 311–318.

    Article  MathSciNet  Google Scholar 

  9. S.-C. Chang, On classification of special linear group over commutative rings, Communications in Algebra 26 (1998), 3993–4005.

    Article  MathSciNet  MATH  Google Scholar 

  10. K. Hansen, Restriction to \({\rm{G}}{{\rm{L}}_2}({\cal O})\) of supercuspidal representations of GL2(F), Pacific Journal of Mathematics 130 (1987), 327–349.

    Article  MathSciNet  Google Scholar 

  11. G. Hill, On the nilpotent representations of \({\rm{G}}{{\rm{L}}_n}({\cal O})\), Manuscripta Mathematica 82 (1994), 293–311.

    Article  MathSciNet  Google Scholar 

  12. P. Humphries, Archimedean newform theory for GLn, https://arxiv.org/abs/2008.12406.

  13. H. Jacquet, A correction to “Conducteur des représentations du groupe linéaire”, Pacific Journal of Mathematics 260 (2012), 515–525.

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Jacquet, I. I. Piatetski-Shapiro and J. Shalika, Conducteur des représentations du groupe linéaire, Mathematische Annalen 256 (1981), 199–214.

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Jacquet and J. Shalika, The Whittaker model of induced representations, Pacific Journal of Mathematics 109 (1983), 107–120.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Jana and P. D. Nelson, Analytic newvectors for GLn(ℝ), https://arxiv.org/abs/1911.01880.

  17. N. Matringe, Essential Whittaker functions for GL(n), Documenta Mathematica 18 (2013), 1191–1214.

    Article  MathSciNet  MATH  Google Scholar 

  18. U. Onn and P. Singla, On the unramified principal series of GL(3) over non-archimedean local fields, Journal of Algebra 397 (2014), 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  19. E. E. Petrov, Harmonic analysis on a \(\mathfrak{P}\)-adic sphere, Soviet Mathematics 26 (1982), 103–106.

    MATH  Google Scholar 

  20. A. A. Popa, Whittaker newforms for archimedean representations of GL(2), Journal of Number Theory 128 (2008), 1637–1645.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Reeder, Old forms on GLn, American Journal of Mathematics 113 (1991), 911–930.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Reeder, review of [Hil94], Mathematical Reviews MR1265002.

  23. W. Rudin, Function Theory in the Unit Ball of ℂn, Classics in Mathematics, Springer, Heidelberg, 2008.

    Google Scholar 

  24. R. Schmidt, Some remarks on local newforms for GL(2), Journal of the Ramanujan Mathematical Society 17 (2002), 115–147.

    MathSciNet  MATH  Google Scholar 

  25. A. J. Silberger, PGL2 over the p-adics: Its Representations, Spherical Functions, and Fourier Analysis, Lecture Notes in Mathematics, Vol. 166, Springer, Berlin, 1970.

    Google Scholar 

  26. N. Templier, Large values of modular forms, Cambridge Journal of Mathematics 2 (2014), 91–116.

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Venkatesh, Large sieve inequalities for GL(n)-forms in the conductor aspect, Advances in Mathematics 200 (2006), 336–356.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Thanks are owed to Subhajit Jana for helpful discussions regarding newforms and matrix coefficients.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904, USA

    Peter Humphries

Authors
  1. Peter Humphries
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Humphries.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Humphries, P. The newform K-type and p-adic spherical harmonics. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2581-x

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s11856-023-2581-x

Search

Navigation