Topological Frobenius reciprocity for representations of nilpotent groups and motion groups

Robert J Archbold, Eberhard Kaniuth

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Let $G$ be a locally compact group and $H$ a closed subgroup of $G$, and let $\pi$ and $\tau$ be irreducible representations of $G$ and $H$, respectively. If $G$ is compact then, by the classical Frobenius reciprocity theorem, $\pi$ is contained in the induced representation ${\rm ind}_H^G \tau$ if and only if $\pi|_H$ contains $\tau$. Topological Frobenius properties, which a general locally compact group may or may not satisfy, are obtained by replacing containment by weak containment of representations. We investigate the `if' and the `only if' assertions for nilpotent locally compact groups and for motion groups.
Original languageEnglish
Pages (from-to)745-769
Number of pages25
JournalJournal of Lie Theory
Issue number3
Publication statusPublished - 31 Dec 2017



  • locally compact group
  • nilpotent group
  • motion group
  • SIN-group
  • unitary representation
  • induced representation
  • weak containment
  • topological Frobenius reciprocity
  • tensor product

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