
Journal of Lie Theory 19 (2009), No. 4, 661670 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 dijk@math.leidenuniv.nl [Abstractpdf] 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 [ Fulltextpdf (160 KB)] for subscribers only. 