Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-13T11:57:04.698Z Has data issue: false hasContentIssue false
  • English
  • Français

Genericity of supercuspidal representations of p-adic Sp4

Part of:Lie groups

Published online by Cambridge University Press:  01 January 2009

Corinne Blondel and
Shaun Stevens
Corinne Blondel
Affiliation:
CNRS, Théorie des Groupes, Case 7012, Institut de Mathématiques de Jussieu, Université Paris 7, F-75251 Paris cedex 05, France (email: blondel@math.jussieu.fr)
Shaun Stevens
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom (email: shaun.stevens@uea.ac.uk)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We describe the supercuspidal representations of Sp4(F), for F a non-archimedean local field of residual characteristic different from two, and determine which are generic.

Keywords

symplectic groupssupercuspidal representationsWhittaker model

MSC classification

Secondary:22E50: Representations of Lie and linear algebraic groups over local fields

Information

Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 1 , January 2009, pp. 213 - 246
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Blasco, L., Description du dual admissible de U(2,1)(F) par la théorie des types de C. Bushnell et P. Kutzko, Manuscripta Math. 107 (2002), 151–186.CrossRefGoogle Scholar
[2]Borel, A., Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (Springer, New York, 1991).CrossRefGoogle Scholar
[3]Bushnell, C. J., Hereditary orders, Gauss sums and supercuspidal representations of GLN, J. Reine Angew. Math. 375/376 (1987), 184–210.Google Scholar
[4]Bushnell, C. J. and Henniart, G., Local tame lifting for GL(N) I: simple characters, Publ. Math. Inst. Hautes Études Sci. 83 (1996), 105–233.CrossRefGoogle Scholar
[5]Bushnell, C. J. and Henniart, G., Supercuspidal representations of GL n: explicit Whittaker functions, J. Algebra 209 (1998), 270–287.CrossRefGoogle Scholar
[6]Bushnell, C. J. and Henniart, G., On the derived subgroups of certain unipotent subgroups of reductive groups over infinite fields, Transform. Groups 7 (2002), 211–230.CrossRefGoogle Scholar
[7]Bushnell, C. J. and Henniart, G., Generalized Whittaker models and the Bernstein center, Amer. J. Math. 125 (2003), 513–547.CrossRefGoogle Scholar
[8]Bushnell, C. J. and Kutzko, P. C., The admissible dual of GL(N) via compact open subgroups, Annals of Mathematics Studies, vol. 129 (Princeton University Press, Princeton, NJ, 1993).CrossRefGoogle Scholar
[9]Bushnell, C. J. and Kutzko, P. C., Semisimple types in GLn, Compositio Math. 119 (1999), 53–97.CrossRefGoogle Scholar
[10]DeBacker, S. and Reeder, M., Depth-zero supercuspidal L-packets and their stability, Ann. of Math. (2), to appear. Available online at http://annals.princeton.edu.Google Scholar
[11]Digne, F. and Michel, J., Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21 (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
[12]Gan, W. T. and Takeda, S., The local Langlands conjecture for GSp(4), Preprint (2007), arXiv:0706.0952.Google Scholar
[13]Gan, W. T. and Takeda, S., The local Langlands conjecture for Sp(4), Preprint (2008), arXiv:0805.2731.Google Scholar
[14]Howe, R. and Piatetski-Shapiro, I. I., A counterexample to the “generalized Ramanujan conjecture” for (quasi)-split groups, in Automorphic forms, representations and L-functions, Corvallis, OR, 1977, Proceedings of Symposia in Pure Mathematics, vol. 33, part 1 (American Mathematical Society, Providence, RI, 1979), 315–322.Google Scholar
[15]Lusztig, G., Representations of finite Chevalley groups, CBMS Regional Conference Series, vol. 39 (American Mathematical Society, Providence, RI, 1978).CrossRefGoogle Scholar
[16]O’Meara, O. T., Introduction to quadratic forms (Springer, Berlin, 1963).CrossRefGoogle Scholar
[17]Paskunas, V. and Stevens, S., On the realisation of maximal simple types and epsilon factors of pairs, Amer. J. Math. 130 (2008), 1211–1261.CrossRefGoogle Scholar
[18]Rodier, F., 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), 425–430.CrossRefGoogle Scholar
[19]Springer, T. A., Characters of special groups, in Seminar on algebraic groups and related finite groups, Lecture Notes in Mathematics, vol. 131 (Springer, Berlin, 1970).Google Scholar
[20]Srinivasan, B., The characters of the finite symplectic group Sp(4,q), Trans. American Math. Soc. 131 (1968), 488–525.Google Scholar
[21]Stevens, S., Intertwining and supercuspidal types for p-adic classical groups, Proc. London Math. Soc. 83 (2001), 120–140.CrossRefGoogle Scholar
[22]Stevens, S., Semisimple characters for p-adic classical groups, Duke Math. J. 127 (2005), 123–173.CrossRefGoogle Scholar
[23]Stevens, S., The supercuspidal representations of p-adic classical groups, Invent. Math. 172 (2008), 289–352.CrossRefGoogle Scholar
[24]Weil, A., Basic number theory (Springer, Berlin, 1974).CrossRefGoogle Scholar