The chain group $C(G)$ of a locally compact group $G$ has one generator $g_{ \rho}$ for each irreducible unitary $G$-representation $\rho$, a relation $g_{ \rho}=g_{ \rho'}g_{ \rho"}$ whenever $\rho$ is weakly contained in $\rho'\otimes \rho"$, and $g_{ \rho^*}=g_{ \rho}^{ -1}$ for the representation $\rho^*$ contragredient to $\rho$. $G$ satisfies chain-center duality if assigning to each $g_{ \rho}$ the central character of $\rho$ is an isomorphism of $C(G)$ onto the dual $\widehat{ Z(G)}$ of the center of $G$. We prove that $G$ satisfies chain-center duality if it is (a) a compact-by-abelian extension, (b) connected nilpotent, (c) countable discrete icc or (d) connected semisimple; this generalizes M. M\"{ u}ger's result compact groups satisfy chain-center duality.
Categories: math.GR math.FA math.OA math.RT