Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



The $ SL(2)$-type and base change

Authors: Omer Offen and Eitan Sayag
Journal: Represent. Theory 13 (2009), 228-235
MSC (2000): Primary 22E50; Secondary 11S37
Published electronically: June 23, 2009
MathSciNet review: 2515933
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The $ SL(2)$-type of any smooth, irreducible and unitarizable representation of $ GL_n$ over a $ p$-adic field was defined by Venkatesh. We provide a natural way to extend the definition to all smooth and irreducible representations. For unitarizable representations we show that the $ SL(2)$-type of a representation is preserved under the base change with respect to any finite extension. The Klyachko model of a smooth, irreducible and unitarizable representation $ \pi$ of $ GL_n$ depends only on the $ SL(2)$-type of $ \pi$. As a consequence we observe that the Klyachko model of $ \pi$ and of its base change are of the same type.

References [Enhancements On Off] (What's this?)

  • [Aub95] Anne-Marie Aubert.
    Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d'un groupe réductif $ p$-adique.
    Trans. Amer. Math. Soc., 347(6):2179-2189, 1995. MR 1285969 (95i:22025)
  • [Aub96] Anne-Marie Aubert.
    Erratum: ``Duality in the Grothendieck group of the category of finite-length smooth representations of a $ p$-adic reductive group'' [Trans. Amer. Math. Soc. 347 (1995), no. 6, 2179-2189.
    Trans. Amer. Math. Soc., 348(11):4687-4690, 1996. MR 1390967 (97c:22019)
  • [Ban06] Dubravka Ban.
    Symmetry of Arthur parameters under Aubert involution.
    J. Lie Theory, 16(2):251-270, 2006. MR 2197592 (2007d:22024)
  • [Hen85] Guy Henniart.
    Le point sur la conjecture de Langlands pour $ {\rm GL}(N)$ sur un corps local.
    In Séminaire de théorie des nombres, Paris 1983-84, volume 59 of Progr. Math., pp. 115-131. Birkhäuser Boston, Boston, MA, 1985. MR 902829 (88j:11083)
  • [Hen00] Guy Henniart.
    Une preuve simple des conjectures de Langlands pour $ {\rm GL}(n)$ sur un corps $ p$-adique.
    Invent. Math., 139(2):439-455, 2000. MR 1738446 (2001e:11052)
  • [HR90] Michael J. Heumos and Stephen Rallis.
    Symplectic-Whittaker models for $ {\rm Gl}\sb n$.
    Pacific J. Math., 146(2):247-279, 1990. MR 1078382 (91k:22036)
  • [HT01] Michael Harris and Richard Taylor.
    The geometry and cohomology of some simple Shimura varieties, volume 151 of Annals of Mathematics Studies.
    Princeton University Press, Princeton, NJ, 2001.
    With an appendix by Vladimir G. Berkovich. MR 1876802 (2002m:11050)
  • [OS07] Omer Offen and Eitan Sayag.
    On unitary representations of $ {\rm GL}\sb {2n}$ distinguished by the symplectic group.
    J. Number Theory, 125(2):344-355, 2007. MR 2332593 (2009a:22012)
  • [OS08a] Omer Offen and Eitan Sayag.
    Global mixed periods and local Klyachko models for the general linear group.
    Int. Math. Res. Not. IMRN, (1):Art. ID rnm 136, 25, 2008. MR 2417789 (2009e:22017)
  • [OS08b] Omer Offen and Eitan Sayag.
    Uniqueness and disjointness of Klyachko models.
    J. Funct. Anal., 254(11):2846-2865, 2008. MR 2414223 (2009e:22018)
  • [Pro98] Kerrigan Procter.
    Parabolic induction via Hecke algebras and the Zelevinsky duality conjecture.
    Proc. London Math. Soc. (3), 77(1):79-116, 1998. MR 1625483 (99h:22025)
  • [Tad86] Marko Tadić.
    Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case).
    Ann. Sci. École Norm. Sup. (4), 19(3):335-382, 1986. MR 870688 (88b:22021)
  • [Tad95] M. Tadić.
    On characters of irreducible unitary representations of general linear groups.
    Abh. Math. Sem. Univ. Hamburg, 65:341-363, 1995. MR 1359141 (96m:22039)
  • [Ven05] Akshay Venkatesh.
    The Burger-Sarnak method and operations on the unitary dual of $ {\rm GL}(n)$.
    Represent. Theory, 9:268-286 (electronic), 2005. MR 2133760 (2006e:22020)
  • [Zel80] A. V. Zelevinsky.
    Induced representations of reductive $ {\mathfrak{p}}$-adic groups. II. On irreducible representations of $ {\rm GL}(n)$.
    Ann. Sci. École Norm. Sup. (4), 13(2):165-210, 1980. MR 584084 (83g:22012)

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 22E50, 11S37

Retrieve articles in all journals with MSC (2000): 22E50, 11S37

Additional Information

Omer Offen
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000 Israel

Eitan Sayag
Affiliation: Department of Mathematics, Ben Gurion University, Be’er Sheva, 84105 Israel

Received by editor(s): August 25, 2008
Received by editor(s) in revised form: April 12, 2009
Published electronically: June 23, 2009
Additional Notes: In this research the first named author is supported by The Israel Science Foundation (grant No. 88/08)
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society