# Topological Frobenius reciprocity for representations of nilpotent groups and motion groups

Robert J Archbold, Eberhard Kaniuth

Research output: Contribution to journalArticle

### Abstract

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 language English 745-769 25 Journal of Lie Theory 27 3 Published - 31 Dec 2017

### Keywords

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