Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 19 (2009), No. 4, 661--670Copyright 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 dijk@math.leidenuniv.nl [Abstract-pdf] 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 [ Fulltext-pdf  (160  KB)] for subscribers only.