A generalisation of the Frobenius Reciprocity theorem

Let 𝐺 be a locally compact group and 𝐾 a closed subgroup of 𝐺. Let 𝛾, 𝜋 be representations of 𝐾 and 𝐺 respectively. Moore’s version of the Frobenius reciprocity theorem was established under the strong conditions that the underlying homogeneous space 𝐺/𝐾 possesses a right-invariant measure and the representation space 𝐻(𝛾) of the representation 𝛾 of 𝐾 is a Hilbert space. Here, the theorem is proved in a more general setting assuming only the existence of a quasi-invariant measure on 𝐺/𝐾 and that the representation spaces 𝔅(𝛾) and 𝔅(𝜋) are Banach spaces with 𝔅(𝜋) being reflexive. This result was originally established by Kleppner but the version of the proof given here is simpler and more transparent.