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Transactions of the American Mathematical Society

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



Spherical distributions on Lie groups and $C^{\infty }$ 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.

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Keywords: <!– MATH ${C^\infty }$ –> <IMG WIDTH="38" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${C^\infty }$"> vector, intertwining operator, spherical distribution
Article copyright: © Copyright 1976 American Mathematical Society