Skip to main content
SpringerLink
Log in

Characters, genericity, and multiplicity one for U(3)

  • Published:
Journal d’Analyse Mathematique Aims and scope

Abstract

Let ψ:U→ℂ be a generic character of the unipotent radicalU of a Borel subgroup of a quasisplitp-adic groupG. The number (0 or 1) of ψ-Whittaker models on an admissible irreducible representation π ofG was expressed by Rodier in terms of the limit of values of the trace of π at certain measures concentrated near the origin. An analogous statement holds in the twisted case. This twisted analogue is used in [F, p. 47] to provide a local proof of the multiplicity one theorem for U(3). This assert that each discrete spectrum automorphic representation of the quasisplit unitary group U(3) associated with a quadratic extensionE/F of number fields occurs in the discrete spectrum with multiplicity one. It is pointed out in [F, p. 47] that a proof of the twisted analogue of Rodier's theorem does not appear in print. It is then given below. Detailing this proof is necessitated in particular by the fact that the attempt in [F, p. 48] at a global proof of the multiplicity one theorem for U(3), although widely quoted, is incomplete, as we point out here.

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

  • [BZ1] J. Bernstein and A. Zelevinskii,Representations of the group GL(n, F)where F is a nonarchimedean local field, Russian Math. Surveys31 (1976), 1–68.

    Article  MATH  MathSciNet  Google Scholar 

  • [BZ2] J. Bernstein and A. Zelevinsky,Induced representations of reductive p-adic groups, Ann. Sci. École Norm. Sup.10 (1977), 441–472.

    MATH  MathSciNet  Google Scholar 

  • [C] L. Clozel,Characters of non-connected, reductive p-adic groups, Canad. J. Math.39 (1987), 149–167.

    MATH  MathSciNet  Google Scholar 

  • [F] Y. Flicker,Packets and liftings for U(3), J. Analyse Math.50 (1988), 19–63.

    Article  MATH  MathSciNet  Google Scholar 

  • [F1] Y. Flicker,Base change trace identity for U(3), J. Analyse Math.52 (1989), 39–52.

    MATH  MathSciNet  Google Scholar 

  • [F2] Y. Flicker,Elementary proof of a fundamental lemma for a unitary group, Canad. J. Math.50 (1998), 74–98.

    MATH  MathSciNet  Google Scholar 

  • [F3] Y. Flicker,Lifting Automorphic Forms on PGSp(2)and SO(4)to PGL(4), Research monograph, 2001; see also:Automorphic forms on PGSp(2), Electron. Res. Ann. Amer. Math. Soc.10 (2004), 39–50;Automorphic forms on SO(4), Proc. Japan Acad.80 (2004), 100–104.

  • Y. Flicker,Automorphic Representations of Low Rak Groups, Research monograph, 2003.

  • [FK] Y. Flicker and D. Kazhdan,Metaplectic correspondence, Publ. Math. IHES64 (1987), 53–110.

    Google Scholar 

  • [GP] S. Gelbart and I. Piatetski-Shapiro,Automorphic forms and L-functions for the unitary group, inLie Groups Representations II, Lecture Notes in Math.1041, Springer, Berlin, 1984, pp. 141–184.

    Chapter  Google Scholar 

  • [H] G. Harder,Eisensteinkohomologie und die Konstruktion gemischter Motive, Lecture Notes in Math.1562, Springer, Berlin, 1993.

    MATH  Google Scholar 

  • [HC] Harish-Chandra,Admissible invariant distributions on reductive p-adic groups, Notes by S. DeBacker and P. Sally, AMS Univ. Lecture Series16 (1999); see also: Queen's Papers in Pure and Appl. Math.48 (1978), 281–346.

  • [MW] C. Moeglin and J. L. Waldspurger,Modèles de Whittaker dégénérés pour des groupes p-adiques, Math. Z.196 (1987), 427–452.

    Article  MATH  MathSciNet  Google Scholar 

  • [R] F. Rodier,Modèle de Whittaker et caractères de représentations, inNon-commutative Harmonic Analysis, Lecture Notes in Math.466, Springer, Berlin, 1975, pp. 151–171.

    Google Scholar 

  • [Ro] J. Rotman,An Introduction to Homological Algebra, Academic Press, New York, 1979.

    MATH  Google Scholar 

  • [S] G. Savin,Dual pair G J ×PGL2 where G J is the automorphism group of the Jordan algebra J. Ivent. Math.118 (1994), 141–160.

    Article  MATH  MathSciNet  Google Scholar 

  • [So] D. Soudry,A uniqueness theorem for representations of GSO(6)and the strong multiplicity one theorem for generic representations of GSp(4), Israel J. Math.58 (1987), 257–287.

    Article  MATH  MathSciNet  Google Scholar 

  • [W] J.-L. Waldspurger,Homogénéité de certaines distributions sur les groupes p-adiques, Publ. Math. IHES81 (1995), 25–72.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, 231 W. 18th Ave., 43210-1174, Columbus, OH, USA

    Yuval Z. Flicker

Authors
  1. Yuval Z. Flicker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuval Z. Flicker.

Additional information

Partially supported by a Lady Davis Visiting Professorship at the Hebrew University and the Max-Planck-Institut für Mathematik, Bonn.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Flicker, Y.Z. Characters, genericity, and multiplicity one for U(3). J. Anal. Math. 94, 1–15 (2004). https://doi.org/10.1007/BF02789039

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02789039

Keywords

Search

Navigation