Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On supercuspidal representations of the metaplectic group

Author: James Meister
Journal: Trans. Amer. Math. Soc. 265 (1981), 575-598
MSC: Primary 22E50; Secondary 10C15
MathSciNet review: 610967
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Weil representations associated to anisotropic quadratic forms in one and three variables are used to study supercuspidal representations of the two-fold metaplectic covering group $ {\overline {{\text{GL}}} _2}(k)$, where $ k$ is a local nonarchimedean field of odd residual characteristic. The principal result is the explicit calculation of certain Whittaker functionals for any square-integrable irreducible admissible genuine representation of $ {\overline {{\text{GL}}} _2}(k)$. In particular, a recent conjecture of Gelbart and Piatetski-Shapiro is answered by obtaining a bijection between the set of quasicharacters of $ {k^ \ast }$ and the set of irreducible admissible genuine distinguished representations of $ {\overline {{\text{GL}}} _2}(k)$, i.e. those representations possessing only one Whittaker functional, or, equivalently, those having a unique Whittaker model. The distinguished representations are precisely the representations attached to the Weil representation associated to a one dimensional form.

The local piece of the generalized Shimura correspondence between automorphic forms of $ {\overline {{\text{GL}}} _2}({\mathbf{A}})$ and $ {\text{G}}{{\text{L}}_2}({\mathbf{A}})$ is also treated. Based upon a conjecture of the equivalences among the constituents of the Weil representations associated to two nonequivalent ternary forms, evidence for the explicit form of the local piece of this global correspondence, restricted to supercuspidal representations of $ {\overline {{\text{GL}}} _2}(k)$, is presented. In this form, the map is shown to be injective and its image is described.

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

  • [1] H. Boerner, Representations of groups, 2nd ed., North-Holland, Amsterdam, 1970.
  • [2] Y. Flicker, Automorphic forms on covering groups of $ {\text{G}}{{\text{L}}_2}$, Invent. Math. 57 (1980), 119-182. MR 567194 (81m:10057)
  • [3] S. Gelbart, Weil's representation and the spectrum of the metaplectic group, Lecture Notes in Math., vol. 530, Springer-Verlag, Berlin and New York, 1976. MR 0424695 (54:12654)
  • [4] S. Gelbart and I. I. Piatetski-Shapiro, Automorphic $ L$-functions of half-integral weight, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), 1620-1623. MR 0506039 (58:21945)
  • [5] -, Distinguished representations and modular forms of half-integral weight, Invent. Math. 59 (1980), 145-188. MR 577359 (82b:10035)
  • [6] -, On Shimura's correspondence for modular forms of half-integral weight, Proc. Colloq. on Automorphic Forms, Representation Theory and Arithmetic (Bombay, 1979).
  • [7] S. Gelbart, I. I. Piatetski-Shapiro and R. Howe, Uniqueness and existence of Whittaker models for the metaplectic group, Israel J. Math. 34 (1979), 21-37. MR 571393 (81j:22021)
  • [8] I. M. Gelfand and M. I. Graev, Representations of quaternion groups over locally compact and function fields, Functional Anal. Appl. 2 (1968). MR 0236314 (38:4611)
  • [9] H. Jacquet and R. P. Langlands, Automorphic forms on $ {\text{GL}}(2)$, Lecture Notes in Math., vol. 114, Springer-Verlag, Berlin and New York, 1970. MR 0401654 (53:5481)
  • [10] R. Lipsman, Group representations, Lecture Notes in Math., vol. 388, Springer-Verlag, Berlin and New York, 1974. MR 0372116 (51:8333)
  • [11] C. Moen, Ph. D. Thesis, University of Chicago, 1979.
  • [12] T. O'Meara, Quadratic forms, 2nd ed., Springer-Verlag, Berlin and New York, 1971.
  • [13] S. Rallis and G. Schiffman, Distributions invariantes pour la groupe orthogonal, Analyse Harmonique sur les Groupes de Lie, Lecture Notes in Math., vol. 497, Springer-Verlag, Berlin and New York, 1975. MR 0404140 (53:7944)
  • [14] -, Representations supercuspidales du groupe metaplectique, J. Math. Kyoto Univ. 17 (1977). MR 0498395 (58:16523)
  • [15] J. P. Serre, A course in arithmetic, Springer-Verlag, New York, 1973. MR 0344216 (49:8956)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E50, 10C15

Retrieve articles in all journals with MSC: 22E50, 10C15

Additional Information

Keywords: Metaplectic group, supercuspidal representations, Whittaker model, Shimura correspondence
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society