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A GENERALISATION OF THE FROBENIUS RECIPROCITY THEOREM

Part of: Abstract harmonic analysis Locally compact groups and their algebras

Published online by Cambridge University Press:  18 February 2019

H. KUMUDINI DHARMADASA Open the ORCID record for H. KUMUDINI DHARMADASA [Opens in a new window]  and
WILLIAM MORAN Open the ORCID record for WILLIAM MORAN [Opens in a new window]
H. KUMUDINI DHARMADASA*
Affiliation:
Discipline of Mathematics, School of Natural Sciences, College of Science and Engineering, University of Tasmania, Hobart, Tasmania 7001, Australia email kumudini@utas.edu.au
WILLIAM MORAN
Affiliation:
Room 2.29 Building 193, Electrical and Electronic Engineering, The University of Melbourne, Victoria 3010, Australia email wmoran@unimelb.edu.au
*
✉kumudini@utas.edu.au
Rights & Permissions [Opens in a new window]

Abstract

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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.

Keywords

separable locally compact groupquasi-invariant measuremodular function𝜆- function

MSC classification

Primary: 43A65: Representations of groups, semigroups, etc.
Secondary: 22D30: Induced representations 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 2 , October 2019 , pp. 317 - 322
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

References

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