A GENERALISATION OF THE FROBENIUS RECIPROCITY THEOREM. Issue 2 (18th February 2019)

• Record Type: Journal Article
• Authors: DHARMADASA, H. KUMUDINI; MORAN, WILLIAM
• Is Part Of: Bulletin of the Australian Mathematical Society. Volume 100:Issue 2(2019)
• Page Start: 317
• Page End: 322
• Publication Date: 2019-02-18
• Subjects: primary 43A65, -- secondary 22D30, -- 43A15; separable locally compact group, -- quasi-invariant measure, -- modular function, -- 𝜆- function; Mathematics -- Societies, etc; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0004972719000042
• Languages: English
• ISSNs: 0004-9727

Abstract

Abstract : 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.