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A GENERALISATION OF THE FROBENIUS RECIPROCITY THEOREM
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 2 / October 2019
- Published online by Cambridge University Press:
- 18 February 2019, pp. 317-322
- Print publication:
- October 2019
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- Article
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- You have access
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Let
$G$ be a locally compact group and 
$K$ a closed subgroup of 
$G$. Let 
$\unicode[STIX]{x1D6FE},$
$\unicode[STIX]{x1D70B}$ 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 
$H(\unicode[STIX]{x1D6FE})$ of the representation 
$\unicode[STIX]{x1D6FE}$ 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 
$\mathfrak{B}(\unicode[STIX]{x1D6FE})$ and 
$\mathfrak{B}(\unicode[STIX]{x1D70B})$ are Banach spaces with 
$\mathfrak{B}(\unicode[STIX]{x1D70B})$ being reflexive. This result was originally established by Kleppner but the version of the proof given here is simpler and more transparent.
