### Abstract

Original language | English |
---|---|

Pages (from-to) | 745-769 |

Number of pages | 25 |

Journal | Journal of Lie Theory |

Volume | 27 |

Issue number | 3 |

Publication status | Published - 2017 |

### Fingerprint

### Keywords

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

### Cite this

*Journal of Lie Theory*,

*27*(3), 745-769.

**Topological Frobenius reciprocity for representations of nilpotent groups and motion groups.** / Archbold, Robert J; Kaniuth, Eberhard.

Research output: Contribution to journal › Article

*Journal of Lie Theory*, vol. 27, no. 3, pp. 745-769.

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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 -