Spherical distributions on Lie groups and 𝐶^{∞} vectors

Penney, R.

doi:10.1090/S0002-9947-1976-0457632-5

Spherical distributions on Lie groups and $C^{\infty }$ vectors
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Trans. Amer. Math. Soc. 223 (1976), 367-384 Request permission

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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The theory of group representations

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