Topological Frobenius Reciprocity for Representations of Nilpotent Groups and Motion Groups

R. J. Archbold; E. Kaniuth

doi:10.5802/jolt.968

R. J. Archbold; E. Kaniuth

Journal of Lie Theory, Volume 27 (2017) no. 3, pp. 745-769

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.

Zbl

DOI: 10.5802/jolt.968

Classification: 22D10, 22D30
Keywords: Locally compact group, nilpotent group, motion group, SIN-group, unitary representation, induced representation, weak containment, topological Frobenius reciprocity, tensor product

@article{JOLT_2017_27_3_a6,
     author = {R. J. Archbold and E. Kaniuth},
     title = {Topological {Frobenius} {Reciprocity} for {Representations} of {Nilpotent} {Groups} and {Motion} {Groups}},
     journal = {Journal of Lie Theory},
     pages = {745--769},
     year = {2017},
     volume = {27},
     number = {3},
     doi = {10.5802/jolt.968},
     zbl = {1388.22003},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.968/}
}

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R. J. Archbold; E. Kaniuth. Topological Frobenius Reciprocity for Representations of Nilpotent Groups and Motion Groups. Journal of Lie Theory, Volume 27 (2017) no. 3, pp. 745-769. doi: 10.5802/jolt.968

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