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Journal of Lie Theory 19 (2009), No. 4, 661--670
Copyright Heldermann Verlag 2009

About the Relation between Multiplicity Free and Strong Multiplicity Free

Gerrit van Dijk
Mathematical Institute, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands


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.

Keywords: Gelfand pair, multiplicity free, strong multiplicity free.

MSC: 4301, 4302, 43A85, 22Dxx

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