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A generalisation of the Frobenius Reciprocity theorem

Citation

Dharmadasa, HK and Moran, W, A generalisation of the Frobenius Reciprocity theorem, Bulletin of the Australian Mathematical Society, 100, (2) pp. 317-322. ISSN 0004-9727 (2019) [Refereed Article]


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Copyright Statement

Copyright 2019 Australian Mathematical Publishing Association Inc.

DOI: doi:10.1017/S0004972719000042

Abstract

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.

Item Details

Item Type:Refereed Article
Keywords:separable locally compact groups, quasi-invariant measures, modular function, lambda function
Research Division:Mathematical Sciences
Research Group:Pure Mathematics
Research Field:Group Theory and Generalisations
Objective Division:Expanding Knowledge
Objective Group:Expanding Knowledge
Objective Field:Expanding Knowledge in the Mathematical Sciences
UTAS Author:Dharmadasa, HK (Dr Kumudini Dharmadasa)
ID Code:130894
Year Published:2019
Deposited By:Mathematics and Physics
Deposited On:2019-02-19
Last Modified:2020-05-20
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