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Complementary series for $ p$-adic groups. I


Author: Allan J. Silberger
Journal: Trans. Amer. Math. Soc. 259 (1980), 589-598
MSC: Primary 22E50
DOI: https://doi.org/10.1090/S0002-9947-1980-0567099-5
MathSciNet review: 567099
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Abstract: Let $ \Omega $ be a nonarchimedean local field, G the group of $ \Omega $-points of a connected reductive algebraic group defined over $ \Omega $. This paper establishes that to each zero of the Plancherel measure of G one can associate complementary series. Our result is the analogue for p-adic groups of a similar statement, announced separately by Knapp-Stein and Harish-Chandra, for real groups.


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  • [1] I. M. Gelfand and M. I. Graev, Representations of the group of second-order matrices with elements in a locally compact field and special functions on locally compact fields, Uspehi Mat. Nauk 18 (1963), no. 4(112), 29-99. MR 0155931 (27:5864)
  • [2a] Harish-Chandra, On the theory of the Eisenstein integral (Proc. Conf. Harmonic Analysis, Univ. of Maryland, College Park, Md., 1971), Lecture Notes in Math., vol. 266, Springer-Verlag, Berlin and New York, 1971. MR 0399355 (53:3200)
  • [2b] -, Harmonic analysis on reductive p-adic groups, Proc. Sympos. Pure Math., vol. 26, Amer. Math. Soc., Providence, R. I., 1974, pp. 167-192. MR 0340486 (49:5238)
  • [3] A. W. Knapp and E. M. Stein, Singular integrals and the principal series. IV, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 2459-2461. MR 0376964 (51:13139)
  • [4] R. G. Laha, Complementary series representations of $ {\text{GL}}(2,\,D)$, where D is a central division algebra over a non-archimedean local field, Notices Amer. Math. Soc. 25 (1978), p. A-262; Abstract #753-B44.
  • [5] G. I. Ol'sanskii, Intertwining operators and complementary series in the class of representations induced from parabolic subgroups of the general linear group over a locally compact division algebra, Mat. Sb. 93 (135), 1974 = Math. USSR Sb. 22 (1974), 217-355. MR 0499010 (58:16988)
  • [6] A. J. Silberger, Introduction to harmonic analysis on reductive p-adic groups (based on lectures by Harish-Chandra at the Institute for Advanced Study, Princeton, N. J., 1971-73), Math. Notes, no. 23, Princeton Univ. Press, Princeton, N. J., 1979. MR 544991 (81m:22025)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0567099-5
Keywords: Complementary series, unitary representations, induced representations
Article copyright: © Copyright 1980 American Mathematical Society

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