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Transactions of the American Mathematical Society

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



Spherical distributions on Lie groups and $ C\sp{\infty }$ vectors

Author: R. Penney
Journal: Trans. Amer. Math. Soc. 223 (1976), 367-384
MSC: Primary 22E45; Secondary 43A90
MathSciNet review: 0457632
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Abstract: Given a Lie group G (not necessarily unimodular) and a subgroup K of G (not necessarily compact), it is shown how to associate with every finite-dimensional unitary irreducible representation $ \delta $ of K a class of distributions analogous to the class of spherical functions of height $ \delta $ familiar from the unimodular-maximal compact case. The two concepts agree as nearly as possible. A number of familiar theorems are generalized to our situation. As an application we obtain a generalization of the Frobenius reciprocity theorem and of Plancherel's theorem to arbitrary induced representations of Lie groups.

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

  • [1] R. Godement, A theory of spherical functions. I, Trans. Amer. Math. Soc. 73 (1952), 496-556. MR 14, 620. MR 0052444 (14:620c)
  • [2] Roe Goodman, Complex Fourier analysis on nilpotent Lie groups, Trans. Amer. Math. Soc. 160 (1971), 373-391. MR 0417334 (54:5390)
  • [3] G. W. Mackey, The theory of group representations, Notes by Dr. Fell and Dr. Lowdenslager, Department of Mathematics, University of Chicago, Chicago, Ill., 1955. MR 19, 117.
  • [4] R. Penney, Entire vectors and holomorphic extension of representations. II, Trans. Amer. Math. Soc. 191 (1974), 195-207. MR 0364556 (51:810)
  • [5] -, Abstract Plancherel theorems and a Frobenius reciprocity theorem, J. Functional Analysis 18 (1975), 177-190. MR 0444844 (56:3191)
  • [6] N. S. Poulsen, On $ {C^\infty }$-vectors and intertwining bilinear forms for representations of Lie groups, J. Functional Analysis 9 (1972), 87-120. MR 46 #9239. MR 0310137 (46:9239)

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Keywords: $ {C^\infty }$ vector, intertwining operator, spherical distribution
Article copyright: © Copyright 1976 American Mathematical Society

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