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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      δ  \delta       of  K  a class of distributions analogous to the class of spherical functions of height      δ  \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.

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