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Transactions of the American Mathematical Society

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On Harish-Chandra's $ \mu $-function for $ p$-adic groups


Author: Allan J. Silberger
Journal: Trans. Amer. Math. Soc. 260 (1980), 113-121
MSC: Primary 22E50
DOI: https://doi.org/10.1090/S0002-9947-1980-0570781-7
MathSciNet review: 570781
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Abstract: The Harish-Chandra $ \mu $-function is, up to known constant factors, the Plancherel's measure associated to an induced series of representations. In this paper we show that, when the series is induced from special representations lifted to a parabolic subgroup, the $ \mu $-function is a quotient of translated $ \mu $-functions associated to series induced from supercuspidal representations. It is now known, in both the real and p-adic cases, that the $ \mu $-function is always an Euler factor.


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

  • [1] Harish-Chandra, Harmonic analysis on reductive 𝑝-adic groups, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 167–192. MR 0340486
  • [2] Harish-Chandra, Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. (2) 104 (1976), no. 1, 117–201. MR 0439994, https://doi.org/10.2307/1971058
  • [3] A. W. Knapp and E. M. Stein, Singular integrals and the principal series. III, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 4622–4624. MR 0367116
  • [4] Allan J. Silberger, Introduction to harmonic analysis on reductive 𝑝-adic groups, Mathematical Notes, vol. 23, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. Based on lectures by Harish-Chandra at the Institute for Advanced Study, 1971–1973. MR 544991
  • [5] -, Special representations of reductive p-adic groups are not integrable, Ann. of Math. (to appear).
  • [6] Nolan R. Wallach, On Harish-Chandra’s generalized 𝐶-functions, Amer. J. Math. 97 (1975), 386–403. MR 0399357, https://doi.org/10.2307/2373718

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0570781-7
Keywords: Plancherel's measure, tempered representation
Article copyright: © Copyright 1980 American Mathematical Society

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