Journal of Lie Theory

no. 4
About the Relation between Multiplicity Free and Strong Multiplicity Free
Journal of Lie Theory, Volume 19 (2009) no. 4, pp. 661-670
Let $G$ be a unimodular Lie group with finitely many connected components and let $H$ be a closed unimodular subgroup of $G$. Let $\pi$ be an irreducible unitary representation of $G$ on $\cal H$ and $\tau$ one of $H$ on $V$. Denote by ${\rm Hom}_H\, ({\cal H}_\infty ,V)$ the vector space of continuous linear mappings ${\cal H}_\infty\to V$ that commute with the $H$-actions. Set ${\rm m}\, (\pi,\, \tau )={\rm dim}\, {\rm Hom}_H\, ({\cal H}_\infty ,V)$. The pair $(G,H)$ is called a multiplicity free pair if ${\rm m}\, (\pi,\,\tau )\leq 1$ for all $\pi$ and $\tau$. We show: if every $\pi$ has a distribution character, then $(G,H)$ is a multiplicity free pair if and only if $(G\times H,\, {\rm diag}\, (H\times H))$ is a generalized Gelfand pair.
DOI: 10.5802/jolt.573
Classification: 4301, 4302, 43A85, 22Dxx
Keywords: Gelfand pair, multiplicity free, strong multiplicity free 
@article{JOLT_2009_19_4_a1,
     author = {G. van Dijk},
     title = {About the {Relation} between {Multiplicity} {Free} and {Strong} {Multiplicity} {Free}},
     journal = {Journal of Lie Theory},
     pages = {661--670},
     year = {2009},
     volume = {19},
     number = {4},
     doi = {10.5802/jolt.573},
     zbl = {1190.43009},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.573/}
}
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G. van Dijk. About the Relation between Multiplicity Free and Strong Multiplicity Free. Journal of Lie Theory, Volume 19 (2009) no. 4, pp. 661-670. doi: 10.5802/jolt.573

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