Skip to main content
SpringerLink
Log in

Values of Iwahori-fixed vectors in the non-split Bessel model on GSp(2n)

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

Abstract

We realize the non-split Bessel model of Novodvorsky and Piatetski-Shapiro in [22] as a generalized Gelfand-Graev representation of GSp(4), as suggested by Kawanaka in [17]. With uniqueness of the model already established in [22], we establish existence of a Bessel model for unramified principal series representations. We then connect the Iwahori-fixed vectors in the Bessel model to a linear character of the Hecke algebra of GSp(4) following the method outlined more generally in [3]. We use this connection to calculate the image of Iwahori-fixed vectors of unramified principal series in the model, and ultimately provide an explicit alternator expression for the spherical vector in the model. We show that the resulting alternator expression matches previous results of Bump, Friedberg, and Furusawa in [6]. We extend all results to GSp(2n), under the assumption that the aforementioned uniqueness property of the model holds for n > 2.

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

References

  1. A. Borel and J. Tits, Groupes réductifs, Institut des Hautes Etudes Scientifiques. Publications Mathématiques 27 (1965), 55–150.

    Article  Google Scholar 

  2. B. Brubaker, D. Bump and S. Friedberg, Unique functionals and representations of Hecke algebras, Pacific Journal of Mathematics 260 (2012), 381–394.

    Article  MathSciNet  Google Scholar 

  3. B. Brubaker, D. Bump and S. Friedberg, Matrix coefficients and Iwahori-Hecke algebra modules, Advances in Mathematics 299 (2016), 247–271.

    Article  MathSciNet  Google Scholar 

  4. B. Brubaker, D. Bump and A. Licata, Whittaker functions and Demazure operators, Journal of Number Theory 146 (2015), 41–68.

    Article  MathSciNet  Google Scholar 

  5. F. Bruhat, Sur les représentations induites des groupes de Lie, Bulletin de la Société Mathématique de France 84 (1956), 97–205.

    Article  MathSciNet  Google Scholar 

  6. D. Bump, S. Friedberg and M. Furusawa, Explicit formulas for the Waldspurger and Bessel models, Israel Journal of Mathematics 102 (1997), 125–177.

    Article  MathSciNet  Google Scholar 

  7. D. Bump, S. Friedberg and D. Ginzburg, Whittaker-orthogonal models, functoriality, and the Rankin-Selberg method, Inventiones Mathematicae 109 (1992), 55–96.

    Article  MathSciNet  Google Scholar 

  8. R. W. Carter, Finite Groups of Lie Type, Wiley Classics Library, John Wiley & Sons, Chichester, 1993.

    Google Scholar 

  9. W. Casselman, The unramified principal series of p-adic groups. I. The spherical function, Compositio Mathematica 40 (1980), 387–406.

    MathSciNet  MATH  Google Scholar 

  10. D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold, New York, 1993. MR 1251060 (94j:17001)

    Google Scholar 

  11. S. Friedberg and D. Goldberg, On local coefficients for non-generic representations of some classical groups, Compositio Mathematica 116 (1999), 133–166.

    Article  MathSciNet  Google Scholar 

  12. W. T. Gan, B. H. Gross and D. Prasad, Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups, Astérisque 346 (2012), 1–109.

    MathSciNet  MATH  Google Scholar 

  13. I. M. Gel'fand and M. I. Graev, Construction of irreducible representations of simple algebraic groups over a finite field, Doklady Akademii Nauk SSSR 147 (1962), 529–532.

    MathSciNet  MATH  Google Scholar 

  14. D. Ginzburg, Certain conjectures relating unipotent orbits to automorphic representations, Israel Journal of Mathematics 151 (2006), 323–355.

    Article  MathSciNet  Google Scholar 

  15. T. J. Haines, R. E. Kottwitz and A. Prasad, Iwahori-Hecke algebras, Journal of the Ramanujan Mathematical Society 25 (2010), 113–145.

    MathSciNet  MATH  Google Scholar 

  16. N. Kawanaka, Generalized Gel' fand-Graev representations and Ennola duality, in Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), Advanced Studies in Pure Mathematics, Vol. 6, North-Holland, Amsterdam, 1985, pp. 175–206.

    Article  Google Scholar 

  17. N. Kawanaka, Shintani lifting and Gel' fand-Graev representations, in The Arcata Conference on Representations of Finite Groups (Arcata, Calif, 1986), Proceedings of Symposia in Pure Mathematics, Vol. 47, American Mathematical Society, Providence, RI, 1987, pp. 147–163.

    Article  Google Scholar 

  18. D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Inventiones Mathematicae 87 (1987), 153–215.

    Article  MathSciNet  Google Scholar 

  19. G. Lusztig, Affine Hecke algebras and their graded version, Journal of the American Mathematical Society 2 (1989), 599–635.

    Article  MathSciNet  Google Scholar 

  20. C. Mœglin, Front d’onde des représentations des groupes classiques p-adiques, American Journal of Mathematical 118 (1996), 1313–1346.

    Article  Google Scholar 

  21. C. Mœglin and J.-L. Waldspurger, Modèles de Whittaker dégénérés pour des groupes p-adiques, Mathematische Zeitschrift 196 (1987), 427–452.

    Article  MathSciNet  Google Scholar 

  22. M. E. Novodvorskiĭ and I. I. Pjateckiĭ-Sapiro, Generalized Bessel models for the symplectic group of rank 2, Matematicheskii Sbornik 90 (132) (1973), 246–256, 326.

    Google Scholar 

  23. F. Rodier, Whittaker models for admissible representations of reductive p-adic split groups, in Harmonic Analysis on Homogeneous Spaces (Williams College, Williamstown, MA, 1972), Proceedings of Symposia in Pure Mathematics, Vol. 26, American Mathematical Society, Providence, RI, 1973, pp. 425–430.

    Chapter  Google Scholar 

  24. J. D. Rogawski, On modules over the Hecke algebra of a p-adic group, Inventiones Mathematicae 79 (1985), 443–465.

    Article  MathSciNet  Google Scholar 

  25. M. Rosenlicht, On quotient varieties and the affine embedding of certain homogeneous spaces, Transactions of the American Mathematical Society 101 (1961), 211–223.

    Article  MathSciNet  Google Scholar 

  26. A. J. Silberger, Introduction to Harmonic Analysis on Reductive p-adic Groups, Mathematical Notes, Vol. 23, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1979.

  27. D. Witte, Products of similar matrices, Proceedings of the American Mathematical Society 126 (1998), 1005–1015.

    Article  MathSciNet  Google Scholar 

  28. H. Yamashita, Finite multiplicity theorems for induced representations of semisimple Lie groups. II. Applications to generalized Gel'fand-Graev representations, Journal of Mathematics of Kyoto University 28 (1988), 383–444.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Wellesley College, 106 Central St., Wellesley, MA, 02482, USA

    Will Grodzicki

Authors
  1. Will Grodzicki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Will Grodzicki.

Additional information

Part of this work was done while the author was at St. Olaf College.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grodzicki, W. Values of Iwahori-fixed vectors in the non-split Bessel model on GSp(2n). Isr. J. Math. 237, 373–414 (2020). https://doi.org/10.1007/s11856-020-2009-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-020-2009-9

Search

Navigation