Journal of Lie Theory

no. 4
About the Relation between Multiplicity Free and Strong Multiplicity Free
Gerrit van Dijk1
1Mathematical Institute, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
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

Gerrit van Dijk  1

1 Mathematical Institute, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands 
Gerrit 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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     title = {About the {Relation} between {Multiplicity} {Free} and {Strong} {Multiplicity} {Free}},
     journal = {Journal of Lie Theory},
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