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The Knapp-Stein dimension theorem for $ p$-adic groups

Author: Allan J. Silberger
Journal: Proc. Amer. Math. Soc. 68 (1978), 243-246
MSC: Primary 22E50
Correction: Proc. Amer. Math. Soc. 76 (1979), 169-170.
MathSciNet review: 0492091
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Abstract: Knapp and Stein have proved for semisimple Lie groups that the dimension of the commuting algebra of an induced tempered representation equals the index of a certain reflection group in a larger group. A precise analogue of their result is stated and proved in this paper for p-adic groups.

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  • [1] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • [2] 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
  • [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 376964,
  • [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

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Keywords: Reductive p-adic groups, tempered unitary representations, commuting algebras
Article copyright: © Copyright 1978 American Mathematical Society