# 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 - 2017

### Fingerprint

Nilpotent Group
Locally Compact Group
Reciprocity
Frobenius
Pi
Motion
Induced Representations
Irreducible Representation
Assertion
Subgroup
If and only if
Closed
Theorem

### Keywords

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

### Cite this

In: Journal of Lie Theory, Vol. 27, No. 3, 2017, p. 745-769.

Research output: Contribution to journalArticle

@article{0c31c8ea5e474a3799034845d9815d81,
title = "Topological Frobenius reciprocity for representations of nilpotent groups and motion groups",
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.",
keywords = "locally compact group, nilpotent group, motion group, SIN-group, unitary representation, induced representation, weak containment, topological Frobenius reciprocity, tensor product",
author = "Archbold, {Robert J} and Eberhard Kaniuth",
year = "2017",
language = "English",
volume = "27",
pages = "745--769",
journal = "Journal of Lie Theory",
issn = "0949-5932",
publisher = "Heldermann Verlag",
number = "3",

}

TY - JOUR

T1 - Topological Frobenius reciprocity for representations of nilpotent groups and motion groups

AU - Archbold, Robert J

AU - Kaniuth, Eberhard

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - locally compact group

KW - nilpotent group

KW - motion group

KW - SIN-group

KW - unitary representation

KW - induced representation

KW - weak containment

KW - topological Frobenius reciprocity

KW - tensor product

M3 - Article

VL - 27

SP - 745

EP - 769

JO - Journal of Lie Theory

JF - Journal of Lie Theory

SN - 0949-5932

IS - 3

ER -